CMAES

The covariance matrix adaptation evolution strategy (CMA-ES) has been highly effective in black-box continuous optimization, as demonstrated by its success in both benchmark problems and various real-world applications. To address the need for an accessible and powerful tool in this domain, we developed cmaes, a simple and practical Python library for CMA-ES. cmaes is characterized by its simplicity, offering intuitive use and high code readability. This makes it suitable for quick use of CMA-ES, as well as for educational purposes and seamless integration into other libraries. Despite its simple design, cmaes maintains advanced functionality. It incorporates recent advancements in CMA-ES, such as learning rate adaptation for challenging scenarios, transfer learning, mixed-variable optimization, and multi-objective optimization capabilities. These advanced features are accessible through a user-friendly API, ensuring that cmaes can be easily adopted in practical applications. We present cmaes as a strong candidate for a practical Python CMA-ES library aimed at practitioners. The software is available under the MIT license at https://github.com/CyberAgentAILab/cmaes.

Temperature uniformity in Li-ion battery thermal management systems (BTMS) is crucial for achieving thermal safety and optimal performance in electric vehicles. This study proposes a method to enhance the temperature uniformity of a liquid-cooled BTMS by using a kriging metamodel and the Covariance Matrix Adaptation Evolution Strategy (CMA-ES). The heat generation rate of the battery was determined based on voltage and current data, followed by thermal fluid analysis to evaluate the cooling performance. Key design variables included the outer and inner dimensions of the cooling plate and the coolant flow rate, constrained by the BTMS mass. The kriging metamodel was used to reduce computational costs significantly, while CMA-ES provided a robust optimization method capable of efficiently handling complex, multimodal design spaces. The combined approach resulted in a 21.15% reduction in temperature difference without increasing the BTMS mass, demonstrating the effectiveness of this optimization strategy in improving the thermal management of Li-ion battery modules.

Evolutionary optimization algorithms often face defects and limitations that complicate the evolution processes or even prevent them from reaching the global optimum. A notable constraint pertains to the considerable quantity of function evaluations required to achieve the intended solution. This concern assumes heightened significance when addressing costly optimization problems. However, recent research has shown that integrating machine learning methods, specifically surrogate models, with evolutionary optimization can enhance various aspects of these algorithms. Among the evolutionary algorithms, the Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES) is particularly favored. This preference is due to its use of Gaussian distribution for calculating evolution and its ability to adapt optimization parameters, which reduces the need for user intervention in adjusting initial parameters. In this research endeavor, we propose the adoption of surrogate models within the CMA-ES framework called CMA-SAO to develop an initial surrogate model that facilitates the adaptation of optimization parameters through the acquisition of pertinent information derived from the associated surrogate model. Empirical validation reveals that CMA-SAO algorithm markedly diminishes the number of function evaluations in comparison to prevailing algorithms, thereby providing a significant enhancement in operational efficiency.

We introduce linear surrogate functions for modeling inequality constraints to solve constrained blackbox optimization problems with the Augmented Lagrangian CMA-ES. Each surrogate is constructed from a binary classifier that predicts the sign of the constraint value. The classifier, and consequently the resulting algorithm, is invariant under sign preserving transformations of the constraint values and can handle binary, flat, and deceptive constraints. Somewhat surprisingly, we find that adopting a sign-based classification model of the constraints allows to solve classes of constrained problems which can not be solved with the original Augmented Lagrangian method using the true constraint value.

Several practical applications of evolutionary computation possess objective functions that receive the design variables and externally given parameters. Such problems are termed contextual optimization problems. These problems require finding the optimal solutions corresponding to the given context vectors. Existing contextual optimization methods train a policy model to predict the optimal solution from context vectors. However, the performance of such models is limited by their representation ability. By contrast, warm starting methods have been used to initialize evolutionary algorithms on a given problem using the optimization results on similar problems. Because warm starting methods do not consider the context vectors, their performances can be improved on contextual optimization problems. Herein, we propose a covariance matrix adaptation evolution strategy with contextual warm starting (CMA-ES-CWS) to efficiently optimize the contextual optimization problem with a given context vector. The CMA-ES-CWS utilizes the optimization results of past context vectors to train the multivariate Gaussian process regression. Subsequently, the CMA-ES-CWS performs warm starting for a given context vector by initializing the search distribution using posterior distribution of the Gaussian process regression. The results of the numerical simulation suggest that CMA-ES-CWS outperforms the existing contextual optimization and warm starting methods.

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The covariance matrix adaptive evolution strategy (CMA-ES) has been widely used in the field of 2D/3D registration in recent years. This optimization method exhibits exceptional robustness and usability for complex surgical scenarios. However, due to the inherent ill-posed nature of the 2D/3D registration task and the presence of numerous local minima in the landscape of similarity measures, evolution strategies often require a larger population size in each generation to ensure the stability of registration and the globality and effectiveness of search, which makes the entire process computationally expensive. In this paper, we build a 2D/3D registration framework based on a learning rate adaptation CMA-ES manner. The framework employs a fixed and small population size, leading to minimized runtime and optimal utilization of computing resources. We conduct experimental comparisons between the proposed framework and other intensity-based baselines using a substantial volume of synthetic data. The results suggest that our method demonstrates superiority in both registration accuracy and running time compared with other optimization-based baselines. Code is available at github.com/m1nhengChen/CMAES-reg.

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Recent research in Cooperative Coevolution~(CC) have achieved promising progress in solving large-scale global optimization problems. However, existing CC paradigms have a primary limitation in that they require deep expertise for selecting or designing effective variable decomposition strategies. Inspired by advancements in Meta-Black-Box Optimization, this paper introduces LCC, a pioneering learning-based cooperative coevolution framework that dynamically schedules decomposition strategies during optimization processes. The decomposition strategy selector is parameterized through a neural network, which processes a meticulously crafted set of optimization status features to determine the optimal strategy for each optimization step. The network is trained via the Proximal Policy Optimization method in a reinforcement learning manner across a collection of representative problems, aiming to maximize the expected optimization performance. Extensive experimental results demonstrate that LCC not only offers certain advantages over state-of-the-art baselines in terms of optimization effectiveness and resource consumption, but it also exhibits promising transferability towards unseen problems.

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The covariance matrix adaptation evolution strategy (CMA-ES) is one of the most successful methods for solving continuous black-box optimization problems. A practically useful aspect of CMA-ES is that it can be used without hyperparameter tuning. However, the hyperparameter settings still have a considerable impact on performance, especially for difficult tasks, such as solving multimodal or noisy problems. This study comprehensively explores the impact of learning rate on CMA-ES performance and demonstrates the necessity of a small learning rate by considering ordinary differential equations. Thereafter, it discusses the setting of an ideal learning rate. Based on these discussions, we develop a novel learning rate adaptation mechanism for CMA-ES that maintains a constant signal-to-noise ratio. Additionally, we investigate the behavior of CMA-ES with the proposed learning rate adaptation mechanism through numerical experiments and compare the results with those obtained for CMA-ES with a fixed learning rate and with population size adaptation. The results show that CMA-ES with the proposed learning rate adaptation works well for multimodal and/or noisy problems without extremely expensive learning rate tuning.

A robust and efficient optimization-based 2D/3D registration framework is crucial for the navigation system of orthopedic surgical robots. It can provide precise position information of surgical instruments and implants during surgery. While artificial intelligence technology has advanced rapidly in recent years, traditional optimization-based registration methods remain indispensable in the field of 2D/3D registration.he exceptional precision of this method enables it to be considered as a post-processing step of the learning-based methods, thereby offering a reliable assurance for registration. In this paper, we present a coarse-to-fine registration framework based on the CMA-ES algorithm. We conducted intensive testing of our method using data from different parts of the spine. The results shows the effectiveness of the proposed framework on real orthopedic spine surgery clinical data. This work can be viewed as an additional extension that complements the optimization-based methods employed in our previous studies.

In several real-world applications in medical and control engineering, there are unsafe solutions whose evaluations involve inherent risk. This optimization setting is known as safe optimization and formulated as a specialized type of constrained optimization problem with constraints for safety functions. Safe optimization requires performing efficient optimization without evaluating unsafe solutions. A few studies have proposed the optimization methods for safe optimization based on Bayesian optimization and the evolutionary algorithm. However, Bayesian optimization-based methods often struggle to achieve superior solutions, and the evolutionary algorithm-based method fails to effectively reduce unsafe evaluations. This study focuses on CMA-ES as an efficient evolutionary algorithm and proposes an optimization method termed safe CMA-ES. The safe CMA-ES is designed to achieve both safety and efficiency in safe optimization. The safe CMA-ES estimates the Lipschitz constants of safety functions transformed with the distribution parameters using the maximum norm of the gradient in Gaussian process regression. Subsequently, the safe CMA-ES projects the samples to the nearest point in the safe region constructed with the estimated Lipschitz constants. The numerical simulation using the benchmark functions shows that the safe CMA-ES successfully performs optimization, suppressing the unsafe evaluations, while the existing methods struggle to significantly reduce the unsafe evaluations.

In this paper, we propose a method for efficiently solving the mixed-integer black-box optimization problem by utilizing the probability distribution models of integer variables in the CMA-ES algorithm. Firstly, some elite points among the generated ones during the evolution process of the CMA-ES algorithm are collected after some consecutive iterations to successively build the probability distribution model of integer variables. Then, the model is partially used to generate the values for the integer variables of candidate solutions in some next iterations. The numerical experiments on the MI-BBO benchmark problems will show that the probability distribution models of integer variables can guide the CMA-ES better to the optimal solution of the problem, as well as demonstrate the efficiency of the proposed method.

The Covariance Matrix Adaptation Evolution Strategy (CMA-ES) is one of the most successful examples of a derandomized evolution strategy. However, it still relies on randomly sampling offspring, which can be done via a uniform distribution and subsequently transforming into the required Gaussian. Previous work has shown that replacing this uniform sampling with a low-discrepancy sampler, such as Halton or Sobol sequences, can improve performance over a wide set of problems. We show that iterating through small, fixed sets of low-discrepancy points can still perform better than the default uniform distribution. Moreover, using only 128 points throughout the search is sufficient to closely approximate the empirical performance of using the complete pseudorandom sequence up to dimensionality 40 on the BBOB benchmark. For lower dimensionalities (below 10), we find that using as little as 32 unique low discrepancy points performs similar or better than uniform sampling. In 2D, for which we have highly optimized low discrepancy samples available, we demonstrate that using these points yields the highest empirical performance and requires only 16 samples to improve over uniform sampling. Overall, we establish a clear relation between the $L_2$ discrepancy of the used point set and the empirical performance of the CMA-ES.

Multimodal optimization presents a significant challenge in optimization problems due to the existence of multiple attraction basins. Balancing exploration and exploitation is essential for the efficiency of algorithms designed to solve these problems. In this paper, we propose the KbP-LaF-CMAES algorithm to address multimodal optimization problems based on the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) framework. The Leaders and Followers (LaF) and Knowledge-based Perturbation (KbP) strategies are the primary components of the KbP-LaF-CMAES algorithm. The LaF strategy is utilized to extensively explore the potential local spaces, where two cooperative populations evolve in synergy. The KbP strategy is employed to enhance exploration capabilities. Improved variants of CMA-ES are used to exploit specific domains containing local optima, thereby potentially identifying the global optimum. Simulation results on the test suite demonstrate that KbP-LaF-CMAES significantly outperforms other meta-heuristic algorithms.

This study modifies the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) algorithm for multi-modal optimization problems. The enhancements focus on addressing the challenges of multiple global minima, improving the algorithm's ability to maintain diversity and explore complex fitness landscapes. We incorporate niching strategies and dynamic adaptation mechanisms to refine the algorithm's performance in identifying and optimizing multiple global optima. The algorithm generates a population of candidate solutions by sampling from a multivariate normal distribution centered around the current mean vector, with the spread determined by the step size and covariance matrix. Each solution's fitness is evaluated as a weighted sum of its contributions to all global minima, maintaining population diversity and preventing premature convergence. We implemented the algorithm on 8 tunable composite functions for the GECCO 2024 Competition on Benchmarking Niching Methods for Multi-Modal Optimization (MMO), adhering to the competition's benchmarking framework. The results are presenting in many ways such as Peak Ratio, F1 score on various dimensions. They demonstrate the algorithm's robustness and effectiveness in handling both global optimization and MMO- specific challenges, providing a comprehensive solution for complex multi-modal optimization problems.

Discrete and mixed-variable optimization problems have appeared in several real-world applications. Most of the research on mixed-variable optimization considers a mixture of integer and continuous variables, and several integer handlings have been developed to inherit the optimization performance of the continuous optimization methods to mixed-integer optimization. In some applications, acceptable solutions are given by selecting possible points in the disjoint subspaces. This paper focuses on the optimization on sets of points and proposes an optimization method by extending the covariance matrix adaptation evolution strategy (CMA-ES), termed the CMA-ES on sets of points (CMA-ES-SoP). The CMA-ES-SoP incorporates margin correction that maintains the generation probability of neighboring points to prevent premature convergence to a specific non-optimal point, which is an effective integer-handling technique for CMA-ES. In addition, because margin correction with a fixed margin value tends to increase the marginal probabilities for a portion of neighboring points more than necessary, the CMA-ES-SoP updates the target margin value adaptively to make the average of the marginal probabilities close to a predefined target probability. Numerical simulations demonstrated that the CMA-ES-SoP successfully optimized the optimization problems on sets of points, whereas the naive CMA-ES failed to optimize them due to premature convergence.

The covariance matrix adaptation evolution strategy (CMA-ES) is a powerful optimization method for continuous black-box optimization problems. Several noise-handling methods have been proposed to bring out the optimization performance of the CMA-ES on noisy objective functions. The adaptations of the population size and the learning rate are two major approaches that perform well under additive Gaussian noise. The reevaluation technique is another technique that evaluates each solution multiple times. In this paper, we discuss the difference between those methods from the perspective of stochastic relaxation that considers the maximization of the expected utility function. We derive that the set of maximizers of the noise-independent utility, which is used in the reevaluation technique, certainly contains the optimal solution, while the noise-dependent utility, which is used in the population size and leaning rate adaptations, does not satisfy it under multiplicative noise. Based on the discussion, we develop the reevaluation adaptation CMA-ES (RA-CMA-ES), which computes two update directions using half of the evaluations and adapts the number of reevaluations based on the estimated correlation of those two update directions. The numerical simulation shows that the RA-CMA-ES outperforms the comparative method under multiplicative noise, maintaining competitive performance under additive noise.

Integrating non-elitist evolution strategies into MOEA/D is challenging because the former usually requires many samples for updates, which is costly for MOEAID. In contrast, we suggest using (1+ 1)-ES for three reasons: fewer samples needed for updates, lower computational overhead, and better flexibility for subproblem collaboration. To verify this, we introduce (1+1)-MOEA/D-CMA, where each subproblem is solved by a different (1+1)-ES solver, and the solvers collaborate through a novel solution injection scheme. Comprehensive experiments show that the proposed algorithm performs better than several widely used algorithms. More importantly, owing to the lightweight nature of (1+1)-CMA-ES, the algorithm is shown to run faster and scale better to large population sizes, than other MOEA/D variants based on (µ/µw, λ)-CMA-ES.

Many real-world optimization tasks suffer from noise. So far, the research on noise-tolerant optimization algorithms is still restricted to low-dimensional problems with less than 100 decision variables. In reality, many problems are high dimensional. Cooperative coevolutionary (CC) algorithms based on a divide-and-conquer strategy are promising in solving complex high-dimensional problems. However, noisy fitness evaluations pose a challenge in problem decomposition for CC. The state-of-the-art grouping methods, such as differential grouping (DG) and recursive DG, are unable to work properly in noisy environments. Because it is impossible to distinguish whether the change of one variable’s difference value is caused by noise or the perturbation of its interacting variables. As a result, every pair of variables will be identified as nonseparable in these methods. In this article, we study how to group decision variables with the covariance matrix adaptation evolution strategy (CMA-ES) in noisy environments and subsequently propose a landscape-aware grouping (LAG) method. Instead of detecting pairwise interacting variables, we directly identify a nonseparable subcomponent. To this end, we propose to use two convergence features: 1) variable convergence time and 2) accumulative path, to describe variables’ fitness landscapes; then, variables are clustered according to these two features. Numerical experiments show that LAG can more effectively identify interactive decision variables in the presence of multiplicative noise than the DG and some of its variants. Up to 500 dimensions, the performance of CC CMA-ES with landscape-aware grouping (CC-CMAES-LAG) is competitive compared with existing CC algorithms and uncertainty-handling CMA-ES (UH-CMA-ES).

The covariance matrix adaptation evolution strategy (CMA-ES) is one of the most successful methods for solving black-box continuous optimization problems. One practically useful aspect of the CMA-ES is that it can be used without hyperparameter tuning. However, the hyperparameter settings still have a considerable impact, especially for difficult tasks such as solving multimodal or noisy problems. In this study, we investigate whether the CMA-ES with default population size can solve multimodal and noisy problems. To perform this investigation, we develop a novel learning rate adaptation mechanism for the CMA-ES, such that the learning rate is adapted so as to maintain a constant signal-to-noise ratio. We investigate the behavior of the CMA-ES with the proposed learning rate adaptation mechanism through numerical experiments, and compare the results with those obtained for the CMA-ES with a fixed learning rate. The results demonstrate that, when the proposed learning rate adaptation is used, the CMA-ES with default population size works well on multimodal and/or noisy problems, without the need for extremely expensive learning rate tuning.

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Physics-Informed Neural Networks (PINNs) have recently received increasing attention, however, optimizing the loss function of PINNs is notoriously difficult, where the landscape of the loss function is often highly non-convex and rugged. Local optimization methods based on gradient information can converge quickly but are prone to being trapped in local minima for training PINNs. Evolutionary algorithms (EAs) are well known for the global search ability, which can help escape from local minima. It has been reported in the literature that EAs show some advantages over gradient-based methods in training PINNs. Inspired by the Memetic Algorithm, we combine global-search based EAs and local-search based batch gradient descent in order to make the best of both word. In addition, since the PINN loss function is composed of multiple terms, balancing these terms is also a challenging issue. Therefore, we also attempt to combine EAs with multiple-gradient descent algorithm for multi-objective optimization. Our experiments provide strong evidence for the superiority of the above algorithms.

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This study targets the mixed-integer black-box optimization (MI-BBO) problem where continuous and integer variables should be optimized simultaneously. The CMA-ES, our focus in this study, is a population-based stochastic search method that samples solution candidates from a multivariate Gaussian distribution (MGD), which shows excellent performance in continuous BBO. The parameters of MGD, mean and (co)variance, are updated based on the evaluation value of candidate solutions in the CMA-ES. If the CMA-ES is applied to the MI-BBO with straightforward discretization, however, the variance corresponding to the integer variables becomes much smaller than the granularity of the discretization before reaching the optimal solution, which leads to the stagnation of the optimization. In particular, when binary variables are included in the problem, this stagnation more likely occurs because the granularity of the discretization becomes wider, and the existing modification to the CMA-ES does not address this stagnation. To overcome these limitations, we propose a simple modification of the CMA-ES based on lower-bounding the marginal probabilities associated with the generation of integer variables in the MGD. The numerical experiments on the MI-BBO benchmark problems demonstrate the efficiency and robustness of the proposed method.

Many state-of-the-art automated machine learning (AutoML) systems use greedy ensemble selection (GES) by Caruana et al. (2004) to ensemble models found during model selection post hoc. Thereby, boosting predictive performance and likely following Auto-Sklearn 1's insight that alternatives, like stacking or gradient-free numerical optimization, overfit. Overfitting in Auto-Sklearn 1 is much more likely than in other AutoML systems because it uses only low-quality validation data for post hoc ensembling. Therefore, we were motivated to analyze whether Auto-Sklearn 1's insight holds true for systems with higher-quality validation data. Consequently, we compared the performance of covariance matrix adaptation evolution strategy (CMA-ES), state-of-the-art gradient-free numerical optimization, to GES on the 71 classification datasets from the AutoML benchmark for AutoGluon. We found that Auto-Sklearn's insight depends on the chosen metric. For the metric ROC AUC, CMA-ES overfits drastically and is outperformed by GES -- statistically significantly for multi-class classification. For the metric balanced accuracy, CMA-ES does not overfit and outperforms GES significantly. Motivated by the successful application of CMA-ES for balanced accuracy, we explored methods to stop CMA-ES from overfitting for ROC AUC. We propose a method to normalize the weights produced by CMA-ES, inspired by GES, that avoids overfitting for CMA-ES and makes CMA-ES perform better than or similar to GES for ROC AUC.

Box-constraints limit the domain of decision variables and are common in real-world optimization problems, for example, due to physical, natural or spatial limitations. Consequently, solutions violating a box-constraint may not be evaluable. This assumption is often ignored in the literature, e.g., existing benchmark suites, such as COCO/BBOB, allow the optimizer to evaluate infeasible solutions. This paper presents an initial study on the strict-box-constrained benchmarking suite (SBOX-COST), which is a variant of the well-known BBOB benchmark suite that enforces box-constraints by returning an invalid evaluation value for infeasible solutions. Specifically, we want to understand the performance difference between BBOB and SBOX-COST as a function of two initialization methods and six constraint-handling strategies all tested with modular CMA-ES. We find that, contrary to what may be expected, handling box-constraints by saturation is not always better than not handling them at all. However, across all BBOB functions, saturation is better than not handling, and the difference increases with the number of dimensions. Strictly enforcing box-constraints also has a clear negative effect on the performance of classical CMA-ES (with uniform random initialization and no constraint handling), especially as problem dimensionality increases.

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The covariance matrix adaptation evolution strategy (CMA-ES) is an efficient continuous black-box optimization method. The CMA-ES possesses many attractive features, including invariance properties and a well-tuned default hyperparameter setting. Moreover, several components to specialize the CMA-ES have been proposed, such as noise handling and constraint handling. To utilize these advantages in mixed-integer optimization problems, the CMA-ES with margin has been proposed. The CMA-ES with margin prevents the premature convergence of discrete variables by the margin correction, in which the distribution parameters are modified to leave the generation probability for changing the discrete variable. The margin correction has been applied to (μ/μw,Λ)-CMA-ES, while this paper introduces the margin correction into (1+1)-CMA-ES, an elitist version of CMA-ES. The (1+1)-CMA-ES is often advantageous for unimodal functions and can be computationally less expensive. To tackle the performance deterioration on mixed-integer optimization, we use the discretized elitist solution as the mean of the sampling distribution and modify the margin correction not to move the elitist solution. The numerical simulation using benchmark functions on mixed-integer, integer, and binary domains shows that (1+1)-CMA-ES with margin outperforms the CMA-ES with margin and is better than or comparable with several specialized methods to a particular search domain.

We compare the performances of one implementation of CMA-ES (pycma version 3.3.0) for optimizing functions with both continuous and integer variables. The implementation incorporates a lower bound on the variance along the integer coordinates to keep the optimization from stalling. This benchmark will serve as a baseline for further works on pycma. Results show substantial improvement since the last benchmarked version of pycma. Also this implementation is competitive with other mixed integer algorithms.

Covariance Matrix Adaptation Evolution Strategy (CMA-ES) is a highly effective optimization technique. A primary challenge when applying CMA-ES in high dimensionality is sampling from a multivariate normal distribution with an arbitrary covariance matrix, which involves its decomposition. The cubic complexity of this process is the main obstacle to applying CMA-ES in high-dimensional spaces. We introduce a version of CMA-ES that uses no covariance matrix at all. In the proposed matrix-free CMA-ES, an archive stores the vectors of differences between individuals and the midpoint, normalized by the step size. New individuals are generated as the weighted combinations of the vectors from the archive. The probability distribution of individuals generated by the proposed method is identical to that of the standard CMA-ES. Experimental results show that reducing the archive size to store only a fixed number of the most recent populations is sufficient, without compromising optimization efficiency. The matrix-free and matrix-based CMA-ES achieve comparable results on the quadratic function when the step-size adaptation is turned off. When coupled with the step-size adaptation method, the matrix-free CMA-ES yields comparable results to those of plain CMA-ES, according to the results obtained for the CEC'2017 benchmark suite.

Bilinear Matrix Inequalities (BMIs) are fundamental to control system design but are notoriously difficult to solve due to their nonconvexity. This study addresses BMI-based control optimization problems by adapting and integrating advanced evolutionary strategies. Specifically, a memetic Covariance Matrix Adaptation Evolution Strategy (memetic CMA-ES) is proposed, which incorporates a local refinement phase via a (1+1)-CMA-ES within the global search process. While these algorithmic components are established in evolutionary computing, their tailored integration and specific tuning for control design tasks represent a novel application in this context. Experimental evaluations on $H_{\infty}$ controller synthesis and spectral abscissa optimization demonstrate that the proposed method achieves superior performance compared to existing BMI solvers in terms of both solution quality and robustness. This work bridges the gap between evolutionary computation and control theory, providing a practical and effective approach to tackling challenging BMI-constrained problems.

In recent years, with the practical application of next-generation power semiconductors such as SiC and GaN, suppressing current and voltage overshoot during high-speed switching and reducing switching losses have become key issues. One solution to this problem is active gate control. In this paper, we propose an automatic gate pattern optimization method based on the Covariance Matrix Adaptation Evolution Strategy (CMA-ES). Compared to a fixed gate pattern, this method reduces switching losses by 25.7% and overshoot by 29.2%. CMA-ES does not require prior adjustment of hyperparameters and can be applied without readjustment, even when changing devices or expanding to high-dimensional design spaces.

We propose using a covariance matrix adaptation evolution strategy (CMA-ES) to optimize launch power in the $C+L+S$ band long-haul system, significantly reducing computation time while maintaining performance, and enabling faster response to channel variations.

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Abstract Despite the recent development in evolutionary multi- and many-objective optimization, the problems with large-scale decision variables still remain challenging. In this work, we propose a scalable small subpopulations based covariance matrix adaptation evolution strategy, namely S3-CMA-ES, for solving many-objective optimization problems with large-scale decision variables. The proposed S3-CMA-ES attempts to approximate the set of Pareto-optimal solutions using a series of small subpopulations instead of a whole population, where each subpopulation converges to only one solution. In the proposed S3-CMA-ES, a diversity improvement strategy is designed to generate and select new solutions. The performance of S3-CMA-ES is compared with five representative algorithms on 36 test instances with 5–15 objectives and 500–1500 decision variables. The empirical results demonstrate the superiority of the proposed S3-CMA-ES.

Despite the state-of-the-art performance of the covariance matrix adaptation evolution strategy (CMA-ES), high-dimensional black-box optimization problems are challenging tasks. Such problems often involve a property called low effective dimensionality (LED), in which the objective function is formulated with redundant dimensions relative to the intrinsic objective function and a rotation transformation of the search space. The CMA-ES suffers from LED for two reasons: the default hyperparameter setting is determined by the total number of dimensions, and the norm calculations in step-size adaptations are performed including elements on the redundant dimensions. In this paper, we incorporate countermeasures for LED into the CMA-ES and propose CMA-ES-LED. We tackle with the rotation transformation using the eigenvectors of the covariance matrix. We estimate the effectiveness of each dimension in the rotated search space using the element-wise signal-to-noise ratios of the mean vector update and the rank-$\mu$ update, both of which updates can be explained as the natural gradient ascent. Then, we adapt the hyperparameter using the estimated number of effective dimensions. In addition, we refine the cumulative step-size adaptation and the two-point step-size adaptation to measure the norms only on the effective dimensions. The experimental results show the CMA-ES-LED outperforms the CMA-ES on benchmark functions with LED.

Intelligent reflecting surface (IRS) is applied in 5G and 6G to improve communication performance by reconfiguring signal propagation environments while maintaining low computational complexity and high energy efficiency. This paper proposes a novel methodology of joint active and passive beamforming optimization for wireless communications with the aid of multiple IRSs. In the considered system, the IRSs are deployed to assist the communication link from a multi-antenna access point (AP) to single-antenna users. We maximize the received signal to noise ratio (SNR) and sum rate of the multi-IRS-assisted communication system for single user and multiple users respectively, by solving a joint optimization problem of the active beamforming at the AP and the reflection beamforming at the IRSs. Specifically, we develop a mechanism of joint beamforming optimization for multi-IRS-assisted communications based on covariance matrix adaptation evolution strategy (CMA-ES), namely the BO-CMA algorithm, to jointly consider both the AP active beamforming and IRS reflection beamforming in the dynamic signal propagation environment. Furthermore, simulation results are provided to show that the BO-CMA algorithm is capable of significantly outperforming the benchmark schemes. Consequently, the solution based on the BO-CMA algorithm can dynamically rebuild the signal propagation environment, and improve the received SNR and the sum rate respectively for multi-IRS-assisted one user and multiple users communications.

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This article studies the problem of trajectory correction optimization for aerodynamic load shedding in the rocket's ascending phase considering the wind. Based on the statistical horizontal wind field, the weakest wind field is proposed to plan the nominal trajectory and the strongest wind field is proposed to test the performance of the designed trajectory. Considering the weakest wind field, two time-varying correction coefficients are calculated by one deep neural network unlike the traditional methods, and used to plan the rocket's flight attitudes when optimizing the rocket's trajectory. There are a large number of parameters to be optimized in this problem, so the traditional trajectory optimization techniques may suffer from poor convergence issues. By introducing the chaotic function into the traditional evolution strategy with covariance matrix adaptation, a novel variant C-CMA-ES is proposed to solve the trajectory optimization problem, and its efficiency is demonstrated by some popular test functions. Besides, the cost function to be minimized is shaped based on the rocket's maximal normal aerodynamic load and final state deviations. Finally, compared with the other three strategies, the efficiency of the proposed trajectory correction strategy is demonstrated by multiple simulation scenarios considering the strongest wind field and the Monte Carlo method considering the random wind fields.

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In black-box optimization, the user inevitably encounters noise in an objective function. Many noise treatment techniques have been developed to properly evaluate the effectiveness of solutions in the optimization process. A noise treatment called sign averaging was proposed recently, and it is proved that the adverse effects of noise can be reduced and the ranking of the candidate solutions on the median of the objective function can be estimated, even when the mean of the objective function is not well-defined. Although a theoretical guarantee exists, empirical studies on sign averaging are yet to be conducted. In this study, we implemented sign averaging in a covariance matrix adaptation evolution strategy, named the SA-CMA-ES, with an adaptive mechanism controlling the strength of the noise treatment. We experimentally demonstrated that 1) the SA-CMA-ES successfully continues to lower the median of the objective function given more budgets and is more sample efficient than the SA-CMA-ES without the adaptive mechanism for sign averaging; 2) the SA-CMA-ES is competitive with the UH-CMA-ES with Monte-Carlo median estimation and that with conventional averaging for optimizing the mean and median, and it is better than that with conventional averaging when the variance of the objective function is not well-defined.

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Accurate measurement of temperature and velocity fields in combustion and flow diagnostic plays an important role in analyzing the heat transfer mechanism and ensuring the safe operation of equipment. In this work, considering the refraction effect of sound waves, temperature and velocity fields are reconstructed simultaneously based on the nonlinear acoustic tomography (NAT) by using the covariance matrix adaptation evolution strategy (CMA-ES) algorithm. In the forward problem, the fast two-point ray-tracing algorithm is introduced to track curved sound waves more efficiently. Due to the sparse measurement signals caused by the limited space and curved acoustic paths, Tikhonov regularization is introduced as the constraint of smoothing prior information to reduce the ill-posedness of the problem. The influence of the velocity regularization parameter and temperature regularization parameter on simultaneously visualizing the temperature and corresponding velocity fields are investigated, respectively. Then the selection strategies of these two regularization parameters are given, which are proved to have high selection efficiency. On this basis, inhomogeneous temperature and velocity fields are reconstructed simultaneously by using the CMA-ES algorithm, simulated annealing, and genetic algorithms. By solving the inverse problem of the NAT, which is highly nonlinear and ill-posed, the feasibility and effectiveness of the CMA-ES algorithm in the simultaneous reconstruction of velocity and temperature fields are confirmed. The proposed method for visualizing the temperature and velocity fields will supply key feedback for the combustion process control.

Existing research on auto-berthing of ships has mainly focused on the design and implementation of controllers for automatic berthing. For the real automatic docking processes, not only do external environmental perturbations need to be taken into account but also motion paths, docking strategies and ship mechanical constraints, which are important influential factors to measure autonomous docking methods. Through a literature review of ship path planning and motion control for automatic berthing, it is found that many studies ignore the interference of the actual navigational environment, especially for ships sailing at slow speed when berthing, or do not consider the physical constraints of the steering gear and the main engine. In this paper, we propose a hybrid approach for autonomous berthing control systems based on a Linear Quadratic Regulator (LQR) and Covariance Matrix Adaptation Evolution Strategy (CMA-ES), which systematically addresses the problems involved in the berthing process, such as path planning, optimal control, adaptive berthing strategies, dynamic environmental perturbations and physically enforced structural constraints. The berthing control system based on the LQR and modified LQR-CMA-ES have been validated by simulation work. The simulation results show that the proposed method is able to achieve the automatic docking of the ship well and the system is robust and well adapted to environmental disturbances at slow speed when docking.

Most of the real-world black-box optimization problems are associated with multiple non-linear as well as non-convex constraints, making them difficult to solve. In this work, we introduce a variant of the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) with linear timing complexity to adopt the constraints of Constrained Optimization Problems (COPs). CMA-ES is already well-known as a powerful algorithm for solving continuous, non-convex, and black-box optimization problems by fitting a second-order model to the underlying objective function (similar in spirit, to the Hessian approximation used by Quasi-Newton methods in mathematical programming). The proposed algorithm utilizes an e-constraint-based ranking and a repair method to handle the violation of the constraints. The experimental results on a group of real-world optimization problems show that the performance of the proposed algorithm is better than several other state-of-the-art algorithms in terms of constraint handling and robustness.

No abstract available

The security defenses that are not comparable to sophisticated adversary tools, let the cloud as an open environment for attacks and intrusions. In this paper, an intelligent protection framework for intrusion detection in a cloud computing environment based on a covariance matrix self-adaptation evolution strategy (CMSA-ES) and multi-criteria decision-making (MCDM) is proposed. The proposed framework constructs an optimal intrusion detector by using CMSA-ES algorithm which adjusts the best parameter set for the attack detector. Moreover, the proposed framework uses a MEREC-VIKOR, a hybrid standardized evaluation technique. MEREC-VIKOR generates the own performance metrics (S, R, and Q) of the proposed framework which is a combination of multi-conflicting criteria. The proposed framework is evaluated for attack detection by using CICIDS 2017 dataset. The experiments show that the proposed framework can detect cloud attacks accurately with low S (utility), R (regret), and Q (integration between S and R). The proposed framework is analyzed with respect to several evolutionary algorithms such as GA, IGASAA, and CMA-ES. The performance analysis demonstrates that the proposed framework that depends on CMSA-ES converges faster than the other evolutionary algorithms such as GA, IGASAA, and CMA-ES. The outcomes also demonstrate that the proposed model is comparable to the state-of-the-art techniques.

Uncertainties caused by material variation can significantly impair the characteristics of devices. Therefore, it is important to design devices whose performance is not significantly damaged even when material variations occur. Robust optimization seeks for the optimal solutions that are robust to fluctuations due to uncertainties caused by material variation, geometrical variation due to assembly tolerances, and changes in physical properties over time in real-world problems. However, naive robust optimization requires iterative calculations to compute the expected values, which need a huge computational burden. This paper introduces a novel robust optimization method for magnetic devices using the covariance matrix adaptation evolution strategy (CMA-ES). In this method, called RCMA-ES (robust CMA-ES), the expected value of the objective function is evaluated using the local average of neighboring individuals without increasing the computation cost. For validation, RCM-ES and robust genetic algorithm (RGA), one of the robust optimization methods without increasing the computational load, was applied to the topology optimization of a magnetic shield and actuator, considering the uncertainty in the BH characteristics. RCM-ES was demonstrated to be particularly more effective for topology optimization with a large number of dimensions compared to RGA and provides robust optimal shapes that are insensitive to variations in BH characteristics.

No abstract available

Abstract Evolution-based neural architecture search methods have shown promising results, but they require high computational resources because these methods involve training each candidate architecture from scratch and then evaluating its fitness, which results in long search time. Covariance Matrix Adaptation Evolution Strategy (CMA-ES) has shown promising results in tuning hyperparameters of neural networks but has not been used for neural architecture search. In this work, we propose a framework called CMANAS which applies the faster convergence property of CMA-ES to the deep neural architecture search problem. Instead of training each individual architecture seperately, we used the accuracy of a trained one shot model (OSM) on the validation data as a prediction of the fitness of the architecture, resulting in reduced search time. We also used an architecture-fitness table (AF table) for keeping a record of the already evaluated architecture, thus further reducing the search time. The architectures are modeled using a normal distribution, which is updated using CMA-ES based on the fitness of the sampled population. Experimentally, CMANAS achieves better results than previous evolution-based methods while reducing the search time significantly. The effectiveness of CMANAS is shown on two different search spaces using four datasets: CIFAR-10, CIFAR-100, ImageNet, and ImageNet16-120. All the results show that CMANAS is a viable alternative to previous evolution-based methods and extends the application of CMA-ES to the deep neural architecture search field.

The covariance matrix self-adaptation evolution strategy with repelling subpopulations (RS-CMSA-ES) is one of the most successful multimodal optimization (MMO) methods currently available. However, some of its components may become inefficient in certain situations. This study introduces the second variant of this method, called RS-CMSA-ESII. It improves the adaptation schemes for the normalized taboo distances of the archived solutions and the covariance matrix of the subpopulation, the termination criteria for the subpopulations, and the way in which the infeasible solutions are treated. It also improves the time complexity of RS-CMSA-ES by updating the initialization procedure of a subpopulation and developing a more accurate metric for determining critical taboo regions. The effects of these modifications are illustrated by designing controlled numerical simulations. RS-CMSA-ESII is then compared with the most successful and recent niching methods for MMO on a widely adopted test suite. The results obtained reveal the superiority of RS-CMSA-ESII over these methods, including the winners of the competition on niching methods for MMO in previous years. Besides, this study extends RS-CMSA-ESII to dynamic MMO and compares it with a few recently proposed methods on the modified moving peak benchmark functions.

Among population-based metaheuristics, both Differential Evolution (DE) and Covariance Matrix Adaptation Evolution Strategy (CMA-ES) perform outstanding for real parameter single objective optimization. Compared with DE, CMA-ES stagnates much earlier in many occasions. In this paper, we propose CMA-ES with individuals redistribution based on DE, IR-CMA-ES, to address stagnation in CMA-ES. We execute experiments based on two benchmark test suites to compare our algorithm with nine peers. Experimental results show that our IR-CMA-ES is competitive in the field of real parameter single objective optimization.

The paper represents a competition entry for the competition on bound constrained single objective numerical optimization at The Genetic and Evolutionary Computation Conference (GECCO) 2022[2] by a hybrid algorithm named Zeroth-Order Covariance Matrix Adaptation Evolution Strategy (ZO-CMAES).

During the last two decades, the notion of multiobjective optimization (MOO) has been successfully adopted to solve the nonconvex constrained optimization problems (COPs) in their most general forms. However, such works mainly utilized the Pareto dominance-based MOO framework while the other successful MOO frameworks, such as the reference vector (RV) and the decomposition-based ones, have not drawn sufficient attention from the COP researchers. In this article, we utilize the concepts of the RV-based MOO to design a ranking strategy for the solutions of a COP. We first transform the COP into a biobjective optimization problem (BOP) and then solve it by using the covariance matrix adaptation evolution strategy (CMA-ES), which is arguably one of the most competitive evolutionary algorithms of current interest. We propose an RV-based ranking strategy to calculate the mean and update the covariance matrix in CMA-ES. Besides, the RV is explicitly tuned during the optimization process based on the characteristics of COPs in a RV-based MOO framework. We also propose a repair mechanism for the infeasible solutions and a restart strategy to facilitate the population to escape from the infeasible region. We test the proposal extensively on two well-known benchmark suites comprised of 36 and 112 test problems (at different scales) from the IEEE CEC (Congress on Evolutionary Computation) 2010 and 2017 competitions along with a real-world problem related to power flow. Our experimental results suggest that the proposed algorithm can meet or beat several other state-of-the-art constrained optimizers in terms of the performance on a wide variety of problems.

No abstract available

Fuzzy cognitive maps (FCMs) are an efficient soft computing tool commonly used for modeling and analyzing complex systems, and have significant advantages in knowledge representation and fast reasoning. At present, many researchers have devoted themselves to solving the learning problem of FCMs. However, most of the existing learning algorithms are prone to local convergence and are not suitable for large-scale FCMs learning problems. Moreover, the learning problem of FCMs is often a complex non-convex optimization problem, which is easy to cause the final learning result of the algorithm to be only a local optimal solution. To address these limitations, this paper proposes a novel algorithm for learning FCMs by combining multimodal optimization and modified covariance matrix adaptation evolution strategy (CMA-ES), termed as MCMA-ES. First, the learning problem of FCMs is modeled as a multimodal optimization problem (MMOP), and then a multimodal optimization algorithm is proposed by combining the detect multimodal clustering (DMC) and CMA-ES, and the cubic chaos factor is combined to enhance the capability of local search. Finally the proposed algorithm is combined with the decomposition strategy to learn FCMs. Experiments are carried out on 5 benchmark datasets, and MCMA-ES is also applied to 15 groups of large-scale gene regulatory system reconstruction problems. The experimental results show that MCMA-ES has high learning generalization and accuracy.

No abstract available

In this paper, we discuss a method for generating new individuals such that their mean vector and the covariance matrix are defined by formulas analogous to the covariance matrix adaptation evolution strategy (CMA-ES). In contrast to CMA-ES, which generates new individuals using multivariate Gaussian distribution with an explicitly defined covariance matrix, the introduced method uses combinations of difference vectors between archived individuals and univariate Gaussian random vectors along directions of past shifts of the population midpoints. We use this method to formulate the differential evolution strategy (DES)—an algorithm that is a crossover between differential evolution (DE) and CMA-ES. The numerical results presented in this paper indicate that DES is competitive against CMA-ES in performing both local and global optimization.

Source mask optimization (SMO) is one of the indispensable resolution enhancement techniques to guarantee the image fidelity and process robustness for the 2Xnm technology node and beyond. The optimization capacity and convergence efficiency of SMO are important, especially for full-chip SMO. An SMO method using the covariance matrix adaptation evolution strategy (CMA-ES), together with a new source representation method, is proposed in this paper. Based on the forward vector imaging formulation, the encoding and decoding methods of the source and the mask, and the constructed merit function, the source and the mask are optimized using the CMA-ES algorithm. The solution search space and the search step size are adaptively updated during the optimization procedure. Considering the sparsity of the optimal source, the source is represented by a set of ideal point sources with unit intensity and adjustable positions. The advantageous spatial frequency components of the source for imaging performance improvement are identified through the aggregation of the point sources. Simulations and comparisons verify the superior optimization capacity and convergence efficiency of the proposed method.

The emergence of the Industrial Internet of Things (IIoT) has accelerated the adoption of Industrial Wireless Sensor Networks (IWSNs) for numerous applications. Effective communication in such applications requires reduced end-to-end transmission time, balanced energy consumption and increased communication reliability. Graph routing, the main routing method in IWSNs, has a significant impact on achieving effective communication in terms of satisfying these requirements. Graph routing algorithms involve applying the first-path available approach and using path redundancy to transmit data packets from a source sensor node to the gateway. However, this approach can affect end-to-end transmission time by creating conflicts among transmissions involving a common sensor node and promoting imbalanced energy consumption due to centralised management. The characteristics and requirements of these networks encounter further complications due to the need to find the best path on the basis of the requirements of IWSNs to overcome these challenges rather than using the available first-path. Such a requirement affects the network performance and prolongs the network lifetime. To address this problem, we adopt a Covariance-Matrix Adaptation Evolution Strategy (CMA-ES) to create and select the graph paths. Firstly, this article proposes three best single-objective graph routing paths according to the IWSN requirements that this research focused on. The sensor nodes select best paths based on three objective functions of CMA-ES: the best Path based on Distance (PODis), the best Path based on residual Energy (POEng) and the best Path based on End-to-End transmission time (POE2E). Secondly, to enhance energy consumption balance and achieve a balance among IWSN requirements, we adapt the CMA-ES to select the best path with multiple-objectives, otherwise known as the Best Path of Graph Routing with a CMA-ES (BPGR-ES). A simulation using MATALB with different configurations and parameters is applied to evaluate the enhanced graph routing algorithms. Furthermore, the performance of PODis, POEng, POE2E and BPGR-ES is compared with existing state-of-the-art graph routing algorithms. The simulation results reveal that the BPGR-ES algorithm achieved 87.53% more balanced energy consumption among sensor nodes in the network compared to other algorithms, and the delivery of data packets of BPGR-ES reached 99.86%, indicating more reliable communication.

Abstract This paper proposes a novel covariance matrix adaptation evolution strategy (CMA-ES) variant, named AEALSCE, for single-objective numerical optimization problems in the continuous domain. To avoid premature convergence and strengthen the exploration capacity of the basic CMA-ES, AEALSCE is obtained by integrating the CMA-ES with two strategies that can adjust the evolutionary directions and enrich the population diversity. The first strategy is named the anisotropic eigenvalue adaptation (AEA) technique, which adapts the search scope towards the optimal evolutionary directions. It scales the eigenvalues of the covariance matrix anisotropically based on local fitness landscape detection. The other strategy is named the local search (LS) strategy, which is executed under the eigen coordinate system and can be subdivided into two parts. In the first part, the new candidates of superior solutions are sampled around the best solution to perform local exploration. In the other part, the new candidates of inferior solutions are generated using a modified mean point along the fitness descent direction. The proposed AEALSCE algorithm is compared with other top competitors, including the CEC 2014 champion, L-SHADE, and the promising NBIPOP-aCMA-ES, by benchmarking the CEC 2014 testbed. Moreover, AEALSCE is applied in solving three constrained engineering design problems and parameter estimation of photovoltaic (PV) models. According to the statistical results of the experiments, our proposed AEALSCE is competitive with other algorithms in convergence efficiency and accuracy. AEALSCE benefits from a good balance of exploration and exploitation, and it exhibits a potential to address real-world optimization problems.

No abstract available

Over the past decades, more and more methods gain a giant development due to the development of technology. Evolutionary Algorithms are widely used as a heuristic method. However, the budget of computation increases exponentially when the dimensions increase. In this paper, we will use the dimensionality reduction method Principal component analysis (PCA) to reduce the dimension during the iteration of Covariance Matrix Adaptation Evolution Strategy (CMA-ES), which is a good Evolutionary Algorithm that is presented as the numeric type and useful for different kinds of problems. We assess the performance of our new methods in terms of convergence rate on multi-modal problems from the Black-Box Optimization Benchmarking (BBOB) problem set and we also use the framework COmparing Continuous Optimizers (COCO) to see how the new method going and compare it to the other algorithms.

Gaussian processes modeling technique has been shown as a valuable surrogate model for the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) in continuous single-objective black-box optimization tasks, where the optimized function is expensive. In this paper, we investigate how different Gaussian process settings influence the error between the predicted and genuine population ordering in connection with features representing the fitness landscape. Apart from using features for landscape analysis known from the literature, we propose a new set of features based on CMA-ES state variables. We perform the landscape analysis of a large set of data generated using runs of a surrogate-assisted version of the CMA-ES on the noiseless part of the Comparing Continuous Optimisers benchmark function testbed.

The covariance matrix adaptation evolution strategy (CMA-ES) is an efficient derivative-free optimization algorithm. It optimizes a black-box objective function over a well-defined parameter space in which feature functions are often defined manually. Therefore, the performance of those techniques strongly depends on the quality of the chosen features or the underlying parametric function space. Hence, enabling CMA-ES to optimize on a more complex and general function class has long been desired. In this paper, we consider modeling the input spaces in black-box optimization non-parametrically in reproducing kernel Hilbert spaces (RKHS). This modeling leads to a functional optimisation problem whose domain is a RKHS function space that enables optimisation in a very rich function class. We propose CMA-ES-RKHS, a generalized CMA-ES framework that is able to carry out black-box functional optimisation in RKHS. A search distribution on non-parametric function spaces, represented as a Gaussian process, is adapted by updating both its mean function and covariance operator. Adaptive and sparse representation of the mean function and the covariance operator can be retained for efficient computation in the updates and evaluations of CMA-ES-RKHS by resorting to sparsification. We will also show how to apply our new black-box framework to search for an optimum policy in reinforcement learning in which policies are represented as functions in a RKHS. CMA-ES-RKHS is evaluated on two functional optimization problems and two bench-marking reinforcement learning domains.

Covariance matrix adaptive evolution strategy (CMAES) is an excellent local search strategy that does not rely on gradient information. It is widely used in real-valued optimization because it fast converges to the local optimum. In addition, manta ray foraging optimization (MRFO) is an evolutionary algorithm inspired by the manta rays’ three particular foraging strategies which can carry out a good solution for global optimization problems. In this paper, we use CMAES to strengthen the local search ability of MRFO and propose CMAES-MRFO to generate better solutions. The CEC’2017 benchmark functions are employed to test the hybrid algorithm. Experimental results show that the proposed CMAES-MRFO significantly outperforms its original algorithms and some other newly-coming ones.

Local path planning algorithm is one of the key technologies for unmanned ground vehicle(UGV). In order to reduce the computational complexity of the local path planning algorithm and ensure the real-time performance of the algorithm, a non-uniform column grid lines modeling method is introduced, and on this basis, a modeling method for planning paths is proposed. Aiming at the path planning problem in a multi -obstacle environment, the evaluation function of the path is constructed from four aspects, and the covariance matrix adaptive evolution strategy(CMA-ES) is used to solve the nonlinear optimization problem. In order to further reduce the amount of calculation, the CMA-ES algorithm is improved by the dynamic adjustment strategy of population size. Experiment results show that the path planning algorithm can effectively realize the local path planning in complex environment.

Quality-Diversity (QD) optimization is an emerging field that focuses on finding a set of behaviorally diverse and high-quality solutions. While the quality is typically defined w.r.t. a single objective function, recent work on Multi-Objective Quality-Diversity (MOQD) extends QD optimization to simultaneously optimize multiple objective functions. This opens up multi-objective applications for QD, such as generating a diverse set of game maps that maximize difficulty, realism, or other properties. Existing MOQD algorithms use non-adaptive methods such as mutation and crossover to search for non-dominated solutions and construct an archive of Pareto Sets (PS). However, recent work in QD has demonstrated enhanced performance through the use of covariance-based evolution strategies for adaptive solution search. We propose bringing this insight into the MOQD problem, and introduce MO-CMA-MAE, a new MOQD algorithm that leverages Covariance Matrix AdaptationEvolution Strategies (CMA-ES) to optimize the hypervolume associated with every PS within the archive. We test MO-CMA-MAE on three MOQD domains, and for generating maps of a co-operative video game, showing significant improvements in performance.

In this article, an effective method, called an adaptive covariance strategy based on reference points (RPCMA-ES) is proposed for multi-objective optimization. In the proposed algorithm, search space is divided into independent sub-regions by calculating the angle between the objective vector and the reference vector. The reference vectors can be used not only to decompose the original multi-objective optimization problem into a number of single-objective subproblems, but also to elucidate user preferences to target a preferred subset of the whole Pareto front (PF). In this respect, any single objective optimizers can be easily used in this algorithm framework. Inspired by the multi-objective estimation of distribution algorithms, covariance matrix adaptation evolution strategy (CMA-ES) is involved in RPCMA-ES. A state-of-the-art optimizer for single-objective continuous functions is the CMA-ES, which has proven to be able to strike a good balance between the exploration and the exploitation of search space. Furthermore, in order to avoid falling into local optimality and make the new mean closer to the optimal solution, chaos operator is added based on CMA-ES. By comparing it with four state-of-the-art multi-objective optimization algorithms, the simulation results show that the proposed algorithm is competitive and effective in terms of convergence and distribution.

To improve the quality of industrial internet network security assessment, this paper proposes an improved industrial internet network security situation assessment model. First,we analyze the factors that affect the network security of the industrial internet and determine the evaluation indicators; Second, the ER iterative algorithm and mutation method are used to fuse the evaluation indicators; Then, according to the fusion results, the industrial Internet network security situation assessment model is established; Finally, the selection covariance matrix adaptive evolution strategy (S-CMA-ES) is used to optimize the model parameters. This method establishes one-way selection of better strategies among different subclass groups, achieving better evaluation results. This method effectively solves the problem of reduced modeling accuracy caused by insufficient data. This paper conducts evaluation experiments on an industrial data set, and verifies the feasibility and effectiveness of the industrial Internet network security situation assessment model and S-CMA-ES optimization algorithm proposed in this paper.

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The “Policy Improvement with Path Integrals” (PI2) [25] and “Covariance Matrix Adaptation — Evolutionary Strategy” [8] are considered to be state-of-the-art in direct reinforcement learning and stochastic optimization respectively. We have recently shown that incorporating covariance matrix adaptation into PI2- which yields the PICMA2 algorithm — enables adaptive exploration by continually and autonomously reconsidering the exploration/exploitation trade-off. In this article, we provide an overview of our recent work on covariance matrix adaptation for direct reinforcement learning [22–24], highlight its relevance to developmental robotics, and conduct further experiments to analyze the results. We investigate two complementary phenomena from developmental robotics. First, we demonstrate PICMA2’s ability to adapt to slowly or abruptly changing tasks due to its continual and adaptive exploration. This is an important component of life-long skill learning in dynamic environments. Second, we show on a reaching task how PICMA2 subsequently releases degrees of freedom from proximal to more distal limbs as learning progresses. A similar effect is observed in human development, where it is known as ‘proximodistal maturation’.

No abstract available

We introduce an acceleration for covariance matrix adaptation evolution strategies (CMA-ES) by means of adaptive diagonal decoding (dd-CMA). This diagonal acceleration endows the default CMA-ES with the advantages of separable CMA-ES without inheriting its drawbacks. Technically, we introduce a diagonal matrix D that expresses coordinate-wise variances of the sampling distribution in DCD form. The diagonal matrix can learn a rescaling of the problem in the coordinates within a linear number of function evaluations. Diagonal decoding can also exploit separability of the problem, but, crucially, does not compromise the performance on nonseparable problems. The latter is accomplished by modulating the learning rate for the diagonal matrix based on the condition number of the underlying correlation matrix. dd-CMA-ES not only combines the advantages of default and separable CMA-ES, but may achieve overadditive speedup: it improves the performance, and even the scaling, of the better of default and separable CMA-ES on classes of nonseparable test functions that reflect, arguably, a landscape feature commonly observed in practice. The article makes two further secondary contributions: we introduce two different approaches to guarantee positive definiteness of the covariance matrix with active CMA, which is valuable in particular with large population size; we revise the default parameter setting in CMA-ES, proposing accelerated settings in particular for large dimension. All our contributions can be viewed as independent improvements of CMA-ES, yet they are also complementary and can be seamlessly combined. In numerical experiments with dd-CMA-ES up to dimension 5120, we observe remarkable improvements over the original covariance matrix adaptation on functions with coordinate-wise ill-conditioning. The improvement is observed also for large population sizes up to about dimension squared.

Abstract This paper addresses the analysis of covariance matrix self-adaptive Evolution Strategies (CMSA-ES) on a subclass of quadratic functions subject to additive Gaussian noise: the noisy ellipsoid model. To this end, it is demonstrated that the dynamical systems approach from the context of isotropic mutations can be transferred to ES that also control the covariance matrix. Theoretical findings such as the component-wise quadratic progress rate or the self-adaptation response function can thus be reused for the CMSA-ES analysis. By deriving the steady state quantities approached on the noisy ellipsoid model for constant population size, a detailed description of the asymptotic CMSA-ES behavior is obtained. By providing self-adaptive ES with a population control mechanism, despite noise disturbances, the algorithm is able to realize continuing progress towards the optimum. Regarding the population control CMSA-ES (pcCMSA-ES), the analytical findings allow to specify its asymptotic long-term behavior and to identify influencing parameters. The finally obtained convergence rate matches the theoretical lower bound of all comparison-based direct search algorithms.

Water distribution system design is a challenging optimisation problem with a high number of search dimensions and constraints. In this way, Evolutionary Algorithms (EAs) have been widely applied to optimise WDS to minimise cost subject whilst meeting pressure constraints. This paper proposes a new hybrid evolutionary framework that consists of three distinct phases. The first phase applied CMA-ES, a robust adaptive meta-heuristic for continuous optimisation. This is followed by an upward-greedy search phase to remove pressure violations. Finally, a downward greedy search phase is used to reduce oversized pipes. To assess the effectiveness of the hybrid method, it was applied to five well-known WDSs case studies. The results reveal that the new framework outperforms CMA-ES by itself and other previously applied heuristics on most benchmarks in terms of both optimisation speed and network cost.

This paper presents a novel mechanism to adapt surrogate-assisted population-based algorithms. This mechanism is applied to ACM-ES, a recently proposed surrogate-assisted variant of CMA-ES. The resulting algorithm, s*ACM-ES, adjusts online the lifelength of the current surrogate model (the number of CMA-ES generations before learning a new surrogate) and the surrogate hyper-parameters. Both heuristics significantly improve the quality of the surrogate model, yielding a significant speed-up of s*ACM-ES compared to the ACM-ES and CMA-ES baselines. The empirical validation of s*ACM-ES on the BBOB-2012 noiseless testbed demonstrates the efficiency and the scalability w.r.t the problem dimension and the population size of the proposed approach, that reaches new best results on some of the benchmark problems.

The covariance matrix adaptive evolution strategy (CMA-ES) is a random search evolution strategy with superior performance and high accuracy. However, when faced with multimodal complex functions, it also has the shortcomings of converging too fast and easily falling into local optimization. Matrix operations in high dimensions also greatly reduce the performance of the algorithm. This paper proposes an improved CMA-ES algorithm based on the cloud model and Cholesky factor update. The cloud model has a good ability to deal with uncertain problems, and the step size is controlled by cloud reasoning, which can better avoid falling into problems such as local optimization and premature convergence. At the same time, the Cholesky factor greatly reduces the computational cost of the algorithm by effectively updating the covariance, especially in high dimensions. Through multiple function tests, multiple experimental verifications and compared with CMA-ES and its Cholesky variant algorithm, the algorithm has the advantages of higher efficiency and more accurate convergence.

No abstract available

No abstract available

Gait phase (GP) estimation is a critical component in control of exoskeletons and prostheses, enabling seamless user interaction in various controllers. In recent years, methods based on machine learning and sensor fusion have offered advances in GP estimation, but their high computational costs and reliance on training and numerous sensors present practical challenges. Estimation methods using phase variables, such as phase-portrait-based methods, can circumvent these drawbacks. However, their lower accuracy has limited their application. To address this limitation, we introduce a novel human-in-the-loop (HIL) optimization approach for improving the accuracy of GP estimation in phase-portrait-based methods. The approach is based on geometric manipulation of the phase portraits with linear transformations, which are adapted online by employing Covariance Matrix Adaptation Evolution Strategy (CMA-ES). The performance of this adaptive method (termed AM) is compared against using a fixed transformation (FM) at different walking speeds on level and inclined treadmill. The results demonstrate the superior performance of AM in all tested conditions in terms of accuracy and linearity, with an average RMS error of $1.97 \pm 0.20\%$ . Convergence times for one round of optimization on a low-end single-board computer were less than 11 s on average. This study confirms the potential of leveraging HIL optimization for enhancing the performance of phase-portrait-based methods to reach accuracy levels comparable to more complex state-of-the-art methods.

In response to the challenges posed by multifactorial nonlinear relationships and uncertainties, and to address the limitations of the existing Belief Rule Base (BRB) in nonlinear fitting, uncertainty representation, and parameter optimization, this paper presents an improved reliable modeling method using a nonlinear belief rule base (R-NBRB). First, the linear inference mechanism is replaced by a smooth nonlinear S-function. This replacement better adapts to nonlinear dynamics in complex industrial systems. Second, attribute reliability is quantified through a reliability assessment method. Data, reliability, and expert knowledge are integrated using the Evidential Reasoning (ER) algorithm. Uncertainty is expressed in the form of belief degrees. Finally, the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) algorithm is applied to optimize the inference parameters. Decision bias caused by insufficient expert knowledge is thereby reduced. Experiments were conducted on a task involving the detection of a petroleum pipeline leak. The mean squared error (MSE) of the R-NBRB model is only 0.2569. This represents a 28.24% reduction compared with the BRB model. The proposed method’s effectiveness and adaptability in complex industrial situations are confirmed.

In the control of robot manipulators, input torque constraints and system nonlinearities present significant challenges for precise trajectory tracking. However, fuzzy adaptive logic control (FALC) often fails to generate the optimal membership functions or function intervals. This paper proposes a modified greater cane rat algorithm (MGCRA) to optimize a fuzzy adaptive logic controller (FALC) for minimizing input torques during trajectory tracking tasks. The main innovation lies in integrating the improved MGCRA with FALC, which enhances the controller’s adaptability and performance. For benchmarking, several state-of-the-art swarm intelligence algorithms—including particle swarm optimization (PSO), artificial bee colony (ABC), ant colony optimization (ACO), gray wolf optimization (GWO), covariance matrix adaptation evolution strategy (CMA-ES), adaptive guided differential evolution (AGDE), the basic greater cane rat algorithm (GCRA), and a trial-and-error method—are compared under identical conditions. Experimental results show that the MGCRA-tuned FALC achieves lower input torques and improved trajectory tracking accuracy compared to other methods. The findings demonstrate the effectiveness and potential of the proposed MGCRA-FALC framework for advanced robotic manipulator control.

In adversarial attacks, most existing methods adopt global attack methods, which attack by changing all image pixels, but this is not realistic. On the contrary, sparse attacks indicate that only perturbing local regions of the input image can deceive DNN models into making incorrect predictions. However, this method requires a large number of queries to generate adversarial examples, and the key issues it faces are locating the perturbation area and optimizing the magnitude of the perturbation. Currently, generating high-quality adversarial examples and improving query efficiency in restricted environments is a challenge for black-box attacks. In this paper, we propose a sparse black-box attack method based on the Active Subspace Evolution Strategy AS-ES , which locates the active subspace of the input image through the multi-arm bandit method, and uses the Covariance Matrix Adaptive Evolution Strategy algorithm for perturbation search in the low-dimensional subspace. We model this problem as a bi-level optimization problem, optimizing both the perturbation position and magnitude to generate high-quality adversarial examples while achieving efficient attacks. We conducted extensive experiments on multiple datasets and verified that the AS-ES method generates adversarial examples with higher quality and query efficiency than existing state-of-the-art attack methods.

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This article addresses the inspection task of confined power busbars in structurally constrained environments by proposing an Industrial Internet of Things (IIoT)-driven intelligent motion strategy generation framework that integrates discrete reasoning with adaptive control. The framework is centered on a hierarchical navigation guidance structure and incorporates three key components: fuzzy regulation-based adaptive motion sampling, dynamic programming-based discrete sequence optimization, and a covariance matrix adaptation evolution strategy (CMA-ES)-enhanced local controller self-adjustment mechanism. Together, these components form an intelligent control system suited for highly dynamic environments with frequent perception disturbances. To validate the effectiveness of the proposed approach, both simulation and physical verification platforms replicating typical industrial structures were developed. A series of systematic evaluations was conducted across multiple representative obstacle scenarios, covering key performance dimensions such as static path coherence, dynamic obstacle response, and control parameter adaptability. Experimental results demonstrate that the proposed control strategy outperforms conventional methods in terms of behavior generation efficiency, obstacle avoidance robustness, and deployment feasibility, indicating strong potential for practical application in industrial settings.

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In deep multi-task learning, weights of task-specific networks are shared between tasks to improve performance on each single one. Since the question, which weights to share between layers, is difficult to answer, human-designed architectures often share everything but a last task-specific layer. In many cases, this simplistic approach severely limits performance. Instead, we propose an algorithm to learn the assignment between a shared set of weights and task-specific layers. To optimize the non-differentiable assignment and at the same time train the differentiable weights, learning takes place via a combination of natural evolution strategy and stochastic gradient descent. The end result are task-specific networks that share weights but allow independent inference. They achieve lower test errors than baselines and methods from literature on three multi-task learning datasets.

Because stochastic gradient descent (SGD) has shown promise optimizing neural networks with millions of parameters and few if any alternatives are known to exist, it has moved to the heart of leading approaches to reinforcement learning (RL). For that reason, the recent result from OpenAI showing that a particular kind of evolution strategy (ES) can rival the performance of SGD-based deep RL methods with large neural networks provoked surprise. This result is difficult to interpret in part because of the lingering ambiguity on how ES actually relates to SGD. The aim of this paper is to significantly reduce this ambiguity through a series of MNIST-based experiments designed to uncover their relationship. As a simple supervised problem without domain noise (unlike in most RL), MNIST makes it possible (1) to measure the correlation between gradients computed by ES and SGD and (2) then to develop an SGD-based proxy that accurately predicts the performance of different ES population sizes. These innovations give a new level of insight into the real capabilities of ES, and lead also to some unconventional means for applying ES to supervised problems that shed further light on its differences from SGD. Incorporating these lessons, the paper concludes by demonstrating that ES can achieve 99% accuracy on MNIST, a number higher than any previously published result for any evolutionary method. While not by any means suggesting that ES should substitute for SGD in supervised learning, the suite of experiments herein enables more informed decisions on the application of ES within RL and other paradigms.

This tutorial introduces the CMA Evolution Strategy (ES), where CMA stands for Covariance Matrix Adaptation. The CMA-ES is a stochastic, or randomized, method for real-parameter (continuous domain) optimization of non-linear, non-convex functions. We try to motivate and derive the algorithm from intuitive concepts and from requirements of non-linear, non-convex search in continuous domain.

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Evolution strategies have been demonstrated to have the strong ability to roughly train deep neural networks and well accomplish reinforcement learning tasks. However, existing evolution strategies designed specially for deep reinforcement learning only involve the plain variants which can not realize the adaptation of mutation strength or other advanced techniques. The research of applying advanced and effective evolution strategies to reinforcement learning in an efficient way is still a gap. To this end, this paper proposes a restart-based rank-1 evolution strategy for reinforcement learning. When training the neural network, it adapts the mutation strength and updates the principal search direction in a way similar to the momentum method, which is an ameliorated version of stochastic gradient ascent. Besides, two mechanisms, i.e., the adaptation of the number of elitists and the restart procedure, are integrated to deal with the issue of local optima. Experimental results on classic control problems and Atari games show that the proposed algorithm is superior to or competitive with state-of-the-art algorithms for reinforcement learning, demonstrating the effectiveness of the proposed algorithm.

We consider solving a class of unconstrained optimization problems in which only stochastic estimates of the objective functions are available. Existing stochastic optimization methods are mainly extended from gradient-based methods, faced with the challenges of noisy function evaluations, hardness in choosing step-sizes, and probably ill-conditioned landscapes. This paper presents a stochastic evolution strategy (SES) framework and several adaptation schemes to avoid these challenges. The SES framework combines the ideas of population sampling and minibatch sampling in exploiting the zeroth-order gradient information, efficiently reducing the noise in both data selection and gradient approximation. In addition, it admits approximating the gradients using a non-isotropic Gaussian distribution to better capture the curvature information of the landscapes. Based on this framework, we implement a step-size adaptation rule and two covariance matrix adaptation rules, where the former can automatically tune the step-sizes and the latter are intended to cope with ill-conditioning. For SES with certain fixed step-sizes, we establish a nearly optimal convergence rate over smooth landscapes. We also show that using the adaptive step-sizes allows convergence at a slightly slower rate but without the need to know the smoothness constant. Several numerical experiments on machine learning problems verify the above theoretical results and suggest that the adaptive SES methods show much promise.

This work concerns the evolutionary approaches to distributed stochastic black-box optimization, in which each worker can individually solve an approximation of the problem with nature-inspired algorithms. We propose a distributed evolution strategy (DES) algorithm grounded on a proper modification to evolution strategies, a family of classic evolutionary algorithms, as well as a careful combination with existing distributed frameworks. On smooth and nonconvex landscapes, DES has a convergence rate competitive to existing zeroth-order methods, and can exploit the sparsity, if applicable, to match the rate of first-order methods. The DES method uses a Gaussian probability model to guide the search and avoids the numerical issue resulted from finite-difference techniques in existing zeroth-order methods. The DES method is also fully adaptive to the problem landscape, as its convergence is guaranteed with any parameter setting. We further propose two alternative sampling schemes which significantly improve the sampling efficiency while leading to similar performance. Simulation studies on several machine learning problems suggest that the proposed methods show much promise in reducing the convergence time and improving the robustness to parameter settings.

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This paper deals with the forecasting of tropical cyclone (TC) landed intensity change problem in which multi-level and multi-attribute decision are considered. A novel index model of tropical cyclone intensity change based on projection pursuit (PP) and evolution strategy (ES) is proposed to forecast the TC intensity change. We propose to use projection pursuit to project the high-dimensional TC intensity observation samples with 18 different attributes into one-dimensional projection index value. According to the projection index value distribution of learning samples, including TC intensifying and weakening, we can determine the cut off index value which distinguishes two different index value of intensifying and weakening samples. The final best projection unit vector can reflect degree of each attribute influence on TC intensity change. In order to solve the high-dimensional globally optimal problem in PP, evolution strategy with stochastic ranking is used to optimize the projection vector. The role of stochastic ranking is to balance the dominance of objective and penalty functions. Based on the index model, experimental results indicate that the accuracy of 693 TC intensity change samples reaches 89.2% when we take the index value 1.40 as the cut off value between TC intensifying and TC weakening, and the seven core attributes can also reflect the main meteorological characters of TC intensity change accurately.

In this article, we address the problem of the Canonical Polyadic decomposition (a.k.a. CP, Candecomp or Parafac decomposition) of N-th order tensors that can be very large. In our case, this decomposition is performed under nonnegativity constraints. While this problem is often tackled thanks to deterministic approaches, we focus here, on a stochastic approach based on a memetic algorithm. It relies on the evolution of a population and a local search stage. The main drawback of such algorithms can be their relative slowness. It is the reason why we suggest and implement a parallel strategy to increase the acceptance rate of the original algorithm and thus to accelerate its convergence speed. Numerical simulations are performed in order to illustrate the effectiveness of our approach on simulated 3D fluorescence spectroscopy tensors.

Black-box optimization problems often require simultaneously optimizing different types of variables, such as continuous, integer, and categorical variables. Unlike integer variables, categorical variables do not necessarily have a meaningful order, and the discretization approach of continuous variables does not work well. Although several Bayesian optimization methods can deal with mixed-category black-box optimization (MC-BBO), they suffer from a lack of scalability to high-dimensional problems and internal computational cost. This paper proposes CatCMA, a stochastic optimization method for MC-BBO problems, which employs the joint probability distribution of multivariate Gaussian and categorical distributions as the search distribution. CatCMA updates the parameters of the joint probability distribution in the natural gradient direction. CatCMA also incorporates the acceleration techniques used in the covariance matrix adaptation evolution strategy (CMA-ES) and the stochastic natural gradient method, such as step-size adaptation and learning rate adaptation. In addition, we restrict the ranges of the categorical distribution parameters by margin to prevent premature convergence and analytically derive a promising margin setting. Numerical experiments show that the performance of CatCMA is superior and more robust to problem dimensions compared to state-of-the-art Bayesian optimization algorithms.

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Large pre-trained language models (PLMs) have garnered significant attention for their versatility and potential for solving a wide spectrum of natural language processing (NLP) tasks. However, the cost of running these PLMs may be prohibitive. Furthermore, PLMs may not be open-sourced due to commercial considerations and potential risks of misuse, such as GPT-3. The parameters and gradients of PLMs are unavailable in this scenario. To solve the issue, black-box tuning has been proposed, which utilizes derivative-free optimization (DFO), instead of gradient descent, for training task-specific continuous prompts. However, these gradient-free methods still exhibit a significant gap compared to gradient-based methods. In this paper, we introduce gradient descent into black-box tuning scenario through knowledge distillation. Furthermore, we propose a novel method GDFO, which integrates gradient descent and derivative-free optimization to optimize task-specific continuous prompts in a harmonized manner. Experimental results show that GDFO can achieve significant performance gains over previous state-of-the-art methods.

This paper describes a simple, but effective sampling method for optimizing and learning a discrete approximation (or surrogate) of a multi-dimensional function along a one-dimensional line segment of interest. The method does not rely on derivative information and the function to be learned can be a computationally-expensive “black box” function that must be queried via simulation or other means. It is assumed that the underlying function is noise-free and smooth, although the algorithm can still be effective when the underlying function is piecewise smooth. The method constructs a smooth surrogate on a set of equally-spaced grid points by evaluating the true function at a sparse set of judiciously chosen grid points. At each iteration, the surrogate’s non-tabu local minima and maxima are identified as candidates for sampling. Tabu search constructs are also used to promote diversification. If no non-tabu extrema are identified, a simple exploration step is taken by sampling the midpoint of the largest unexplored interval. The algorithm continues until a user-defined function evaluation limit is reached. Numerous examples are shown to illustrate the algorithm’s efficacy and superiority relative to state-of-the-art methods, including Bayesian optimization and NOMAD, on primarily nonconvex test functions.

In this work, we utilize a Trust Region based Derivative Free Optimization (DFO-TR) method to directly maximize the Area Under Receiver Operating Characteristic Curve (AUC), which is a nonsmooth, noisy function. We show that AUC is a smooth function, in expectation, if the distributions of the positive and negative data points obey a jointly normal distribution. The practical performance of this algorithm is compared to three prominent Bayesian optimization methods and random search. The presented numerical results show that DFO-TR surpasses Bayesian optimization and random search on various black-box optimization problem, such as maximizing AUC and hyperparameter tuning.

We address the well-known NP-hard problem of packing rectangular items into a strip, a problem of significant importance in electronics (e.g., packing components on printed circuit boards and macro-cell placement in Very-Large- Scale Integration design) and telecommunications (e.g., allocating data packets over transmission channels). Traditional heuristics and metaheuristics struggle with generalization, efficiency, and adaptability, as they rely on predefined rules or require extensive computational effort for each new problem instance. In this paper, we propose a neural-driven constructive heuristic that leverages a lightware neural network trained via black-box optimization to dynamically evaluate item placement decisions. Instead of relying on static heuristic rules, our approach adapts to the characteristics of each problem instance, enabling more efficient and effective packing strategies. To train the neural network, we employ the Covariance Matrix Adaptation Evolution Strategy (CMA-ES), a state-ofthe- art derivative-free optimization method. Our method learns decision policies by optimizing fill factor improvements over a large dataset of problem instances. Unlike conventional heuristics, our approach dynamically adapts placement decisions based on a broad set of features describing the current partial solution and remaining items. Through extensive computational experiments, we compare our method against well-known strip packing heuristics, including MaxRects and Skyline-based algorithms. The results demonstrate that our approach consistently outperforms the best traditional heuristics, achieving up to 6.74 percentage points of improvement in packing efficiency. Furthermore, our method improves 87.87% of tested instances. Our study highlights the potential of machine learning-driven heuristics in combinatorial optimization and opens avenues for further research into adaptive decision-making strategies in packing and scheduling problems

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Automated algorithm performance prediction in numerical blackbox optimization often relies on problem characterizations, such as exploratory landscape analysis features. These features are typically used as inputs to machine learning models and are represented in a tabular format. However, such approaches often overlook algorithm configurations, a key factor influencing performance. The relationships between algorithm operators, parameters, problem characteristics, and performance outcomes form a complex structure best represented as a graph. This work explores the use of heterogeneous graph data structures and graph neural networks to predict the performance of optimization algorithms by capturing the complex dependencies between problems, algorithm configurations, and performance outcomes. We focus on two modular frameworks, modCMA-ES and modDE, which decompose two widely used derivative-free optimization algorithms: the covariance matrix adaptation evolution strategy (CMA-ES) and differential evolution (DE). We evaluate 324 modCMA-ES and 576 modDE variants on 24 BBOB problems across six runtime budgets and two problem dimensions. Achieving up to 36.6% improvement in MSE over traditional tabular-based methods, this work highlights the potential of geometric learning in black-box optimization.

We consider the problem of optimizing the sum of a smooth, nonconvex function for which derivatives are unavailable and a convex, nonsmooth function with an easy-to-evaluate proximal operator. Of particular focus is the case where the smooth part has a nonlinear least-squares (LS) structure. We adapt two existing approaches for derivative-free optimization (DFO) of nonsmooth compositions of smooth functions to this setting. Our main contribution is adapting our algorithm to handle inexactly computed stationarity measures, where the inexactness is adaptively adjusted as required by the algorithm (where previous approaches assumed access to exact stationary measures, which is not realistic in this setting). Numerically, we provide two extensions of the state-of-the-art DFO-LS solver for nonlinear least-squares problems and demonstrate their strong practical performance.

In this paper, we introduce a novel framework for black-box prompt tuning with a subspace learning and selection strategy, leveraging derivative-free optimization algorithms. This approach is crucial for scenarios where user interaction with language models is restricted to API usage, without direct access to their internal structures or gradients, a situation typical in Language-Model-as-a-Service (LMaaS). Our framework focuses on exploring the low-dimensional subspace of continuous prompts. Previous work on black-box prompt tuning necessitates a substantial number of API calls due to the random choice of the subspace. To tackle this problem, we propose to use a simple zeroth-order optimization algorithm to tackle nonconvex optimization challenges with nonsmooth nonconvex regularizers: the Zeroth-Order Mini-Batch Stochastic Proximal Gradient method (ZO-MB-SPG). A key innovation is the incorporation of nonsmooth nonconvex regularizers, including the indicator function of the l0 constraint, which enhances our ability to select optimal subspaces for prompt optimization. The experimental results show that our proposed black-box prompt tuning method on a few labeled samples can attain similar performance to the methods applicable to LMaaS with much fewer API calls.

The growing ubiquity of machine learning (ML) has led it to enter various areas of computer science, including black-box optimization (BBO). Recent research is particularly concerned with Bayesian optimization (BO). BO-based algorithms are popular in the ML community, as they are used for hyperparameter optimization and more generally for algorithm configuration. However, their efficiency decreases as the dimensionality of the problem and the budget of evaluations increase. Meanwhile, derivative-free optimization methods have evolved independently in the optimization community. Therefore, we urge to understand whether cross-fertilization is possible between the two communities, ML and BBO, i.e., whether algorithms that are heavily used in ML also work well in BBO and vice versa. Comparative experiments often involve rather small benchmarks and show visible problems in the experimental setup, such as poor initialization of baselines, overfitting due to problem-specific setting of hyperparameters, and low statistical significance. With this article, we update and extend a comparative study presented by Hutter et al. in 2013. We compare BBO tools for ML with more classical heuristics, first on the well-known Black-Box Optimization Benchmarking test suite from the COCO environment and then on Direct Policy Search for OpenAI Gym, a reinforcement learning benchmark. Our results confirm that BO-based optimizers perform well on both benchmarks when budgets are limited, albeit with a higher computational cost, while they are often outperformed by algorithms from other families when the evaluation budget becomes larger. We also show that some algorithms from the BBO community perform surprisingly well on ML tasks.

This paper presents the results and insights from the black-box optimization (BBO) challenge at NeurIPS 2020 which ran from July-October, 2020. The challenge emphasized the importance of evaluating derivative-free optimizers for tuning the hyperparameters of machine learning models. This was the first black-box optimization challenge with a machine learning emphasis. It was based on tuning (validation set) performance of standard machine learning models on real datasets. This competition has widespread impact as black-box optimization (e.g., Bayesian optimization) is relevant for hyperparameter tuning in almost every machine learning project as well as many applications outside of machine learning. The final leaderboard was determined using the optimization performance on held-out (hidden) objective functions, where the optimizers ran without human intervention. Baselines were set using the default settings of several open-source black-box optimization packages as well as random search.

In this paper, we introduce the Ensemble Kalman-Stein Gradient Descent (EnKSGD) class of algorithms. The EnKSGD class of algorithms builds on the ensemble Kalman filter (EnKF) line of work, applying techniques from sequential data assimilation to unconstrained optimization and parameter estimation problems. The essential idea is to exploit the EnKF as a black box (i.e. derivative-free, zeroth order) optimization tool if iterated to convergence. In this paper, we return to the foundations of the EnKF as a sequential data assimilation technique, including its continuous-time and mean-field limits, with the goal of developing faster optimization algorithms suited to noisy black box optimization and inverse problems. The resulting EnKSGD class of algorithms can be designed to both maintain the desirable property of affine-invariance, and employ the well-known backtracking line search. Furthermore, EnKSGD algorithms are designed to not necessitate the subspace restriction property and variance collapse property of previous iterated EnKF approaches to optimization, as both these properties can be undesirable in an optimization context. EnKSGD also generalizes beyond the $L^{2}$ loss, and is thus applicable to a wider class of problems than the standard EnKF. Numerical experiments with both linear and nonlinear least squares problems, as well as maximum likelihood estimation, demonstrate the faster convergence of EnKSGD relative to alternative EnKF approaches to optimization.

This paper proposes a surrogate based, global-search derivative free algorithm which is specifically designed for computationally expensive black-box (simulation-based) optimization problems with constraints. The algorithm, called SCR (Surrogate-CMAES-RQLIF), uses kriging to generate surrogate models of the black-box objective function and black-box constraints. These surrogate models are optimized using the global-search algorithm CMA-ES with the quadratic penalty approach for the constraints. The quality of the kriging surrogates is checked, and the surrogates are updated at each iteration by adding the point found by CMA-ES and additional training points. Once the region of the global optimum has been approximately defined, local search is performed using the hybrid direct-search/model based algorithm RQLIF. After each iteration the points sampled by RQLIF and some additional points found within the optimal region are used to update the surrogate model. Tests on 25 unconstrained and 21 constrained literature test problems show that SCR outperforms benchmark optimization algorithms. The outstanding performance of SCR is also confirmed on two real-world black-box problems arising in process engineering with computationally expensive simulations: the techno-economic optimization of a CO2 Purification Unit (CPU) and a Vacuum Pressure Swing Adsorption Unit (VPSA).

The Low Order-Value Optimization (LOVO) problem involves minimizing the minimum among a finite number of function values within a feasible set. LOVO has several practical applications such as robust parameter estimation, protein alignment, portfolio optimization, among others. In this work, we are interested in the constrained nonlinear optimization LOVO problem of minimizing the minimum between a finite number of function values subject to a nonempty closed convex set where each function is a black-box and continuously differentiable, but the derivatives are not available. We develop the first derivative-free trust-region algorithm for constrained LOVO problems with convergence to weakly critical points. Under suitable conditions, we establish the global convergence of the algorithm and also its worst-case iteration complexity analysis. An initial open-source implementation using only linear interpolation models is developed. Extensive numerical experiments and comparison with existing alternatives show the properties and the efficiency of the proposed approach when solving LOVO problems.

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This paper presents a modified trust-region approach for computing approximations to the complete Pareto front of multiobjective derivative-free optimization problems. It is assumed that the derivatives of the objective function components are not available, impossible or very expensive to estimate, such as in simulation optimization, bandit optimization, and adversarial black-box machine learning. The algorithm alternates between two main steps, namely, the extreme point step and the scalarization step, until predefined stopping criteria are met. The goal of the extreme point step is to expand the approximation to the complete Pareto front, by moving towards the extreme points of it, corresponding to the individual minimization of each objective function component. The scalarization step attempts to minimize the size of gaps in the Pareto front approximation, by solving a suitable scalarization problem. The scalarization step includes a pivotal additional step, referred to as the middle point step. This step plays a significant role in determining initial points for solving the scalarization problem. To overcome the absence of derivatives, a new technique based on polynomial interpolation and minimum Frobenius norm approaches is proposed to build models that approximate different objective function components. The convergence analysis is well established, even with the extra complexity introduced by the challenge of lacking derivative information. Numerical results are presented, indicating that this algorithm is efficiently and robustly competitive against state-of-the-art multiobjective derivative-free optimization algorithms that also aim to approximate complete Pareto fronts.

In this work we present TRFD, a derivative-free trust-region method based on finite differences for minimizing composite functions of the form $f(x)=h(F(x))$, where $F$ is a black-box function assumed to have a Lipschitz continuous Jacobian, and $h$ is a known convex Lipschitz function, possibly nonsmooth. The method approximates the Jacobian of $F$ via forward finite differences. We establish an upper bound for the number of evaluations of $F$ that TRFD requires to find an $\epsilon$-approximate stationary point. For L1 and Minimax problems, we show that our complexity bound reduces to $\mathcal{O}(n\epsilon^{-2})$ for specific instances of TRFD, where $n$ is the number of variables of the problem. Assuming that $h$ is monotone and that the components of $F$ are convex, we also establish a worst-case complexity bound, which reduces to $\mathcal{O}(n\epsilon^{-1})$ for Minimax problems. Numerical results are provided to illustrate the relative efficiency of TRFD in comparison with existing derivative-free solvers for composite nonsmooth optimization.

In the field of derivative-free optimization, both of its main branches, the deterministic and nature-inspired techniques, experienced in recent years substantial advancement. In this paper, we provide an extensive computational comparison of selected methods from each of these branches. The chosen representatives were either standard and well-utilized methods, or the best-performing methods from recent numerical comparisons. The computational comparison was performed on five di ff erent benchmark sets and the results were analyzed in terms of performance, time complexity, and convergence properties of the selected methods. The results showed that, when dealing with situations where the objective function evaluations are relatively cheap, the nature-inspired methods have a significantly better performance than their deterministic counterparts. However, in situations when the function evaluations are costly or otherwise prohibited, the deterministic methods might provide more consistent and overall better results.

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Direct Multisearch (DMS) and MultiGLODS are two derivative-free solvers for approximating the entire set of Pareto-optimal solutions of a multiobjective (blackbox) problem. They both follow the search/poll step approach of direct search methods, employ Pareto dominance to avoid aggregating objectives, and have theoretical limit guarantees. Although the original publications already compare the two algorithms empirically with a variety of multiobjective solvers, an analysis on their scaling behavior with dimension was missing. Here, we run the publicly available implementations on the bbob-biobj test suite of the COCO platform and by investigating their performances in more detail, observe (i) a small defect in the default initialization of DMS, (ii) for both algorithms a decrease in relative performance to other algorithms of the original studies (even matching the performance of random search for MultiGLODS in higher dimension), and (iii) consequently, an under-performance to previously untested stochastic solvers from the evolutionary computation field, especially when the dimension is higher.

Spare parts management affects significantly costs and service level for supply chains. This paper deals with an inventory management problem for multi-item repairable systems via a systemic perspective based on a new efficient integer black-box optimization model. With respect to the traditionally used marginal allocation that considers items individually, the proposed black-box optimization model is a holistic approach in the fact that it exploits relationships among items. The authors propose a derivative-free algorithm specifically tied to the application which exploits a new selection strategy for choosing entire subsets of items with the aim to get the best expected improvement in the objective function. The approach has been tested on a real case study for optimizing stocks in an airline's inventory network. The case study provides evidence about the good behavior of the exploratory geometry of the proposed approach in finding quickly a feasible and optimal solution for inventory control.

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Optimization methods play a crucial role in various fields and applications. In some optimization problems, the derivative information of the objective function is unavailable. Such black-box optimization problems need to be solved by derivative-free optimization methods. At the same time, optimization problems with ellipsoidal constraints are important and have widespread applications in various fields as well. Following the development of the late professor M. J. D. Powell’s efficient derivative-free trust-region optimization methods, this paper considers solving derivative-free optimization problems on the ellipsoid. Our new optimization solver EC-NEWUOA for problems on the ellipsoid in ℜ n is designed based on Powell’s derivative-free software NEWUOA for unconstrained optimization problems. The proposed techniques for our new method mainly include using the Courant penalty function, the augmented Lagrangian method, and the projection technique. Details about the method and theoretical analysis are included in this paper. We also compare our new method with other algorithms by solving test problems and then show the numerical advantages of our new method.

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Derivative-free optimization algorithms play an important role in scientific and engineering design optimization problems, especially when derivative information is not accessible. In this paper, we study the framework of sequential classification-based derivative-free optimization algorithms. By introducing learning theoretic concept hypothesis-target shattering rate, we revisit the computational complexity upper bound of SRACOS Inspired by the revisited upper bound, we propose an algorithm named RACE-CARS, which adds a random region-shrinking step compared with SRACOS. We further establish theorems showing the acceleration by region shrinking. Experiments on the synthetic functions as well as black-box tuning for language-model-as-a-service demonstrate empirically the efficiency of RACE-CARS. An ablation experiment on the introduced hyper-parameters is also conducted, revealing the mechanism of RACE-CARS and putting forward an empirical hyperparameter tuning guidance.

Abstract This paper presents an algorithm for constrained black-box and grey-box optimization. It is based on surrogate models developed using input-output data in a trust-region framework. Unlike many current methods, the proposed approach does not require feasible initial point and can handle hard constraints via a novel optimization-based constrained sampling scheme. A two-phase strategy is employed, where the first phase involves finding feasible point through minimizing a smooth constraint violation function (feasibility phase). The second phase improves the objective in the feasible region using the solution of the feasibility phase as starting point (optimization phase). The method is applied to solve 92 test problems and the performance is compared with established derivative-free solvers. The two-phase algorithm outperforms these solvers in terms of number of problems solved and number of samples used. We also apply the algorithm to solve a chemical process design problem involving highly-coupled, nonlinear algebraic and partial differential equations.

Recently, neural networks trained as optimizers under the "learning to learn" or meta-learning framework have been shown to be effective for a broad range of optimization tasks including derivative-free black-box function optimization. Recurrent neural networks (RNNs) trained to optimize a diverse set of synthetic non-convex differentiable functions via gradient descent have been effective at optimizing derivative-free black-box functions. In this work, we propose RNN-Opt: an approach for learning RNN-based optimizers for optimizing real-parameter single-objective continuous functions under limited budget constraints. Existing approaches utilize an observed improvement based meta-learning loss function for training such models. We propose training RNN-Opt by using synthetic non-convex functions with known (approximate) optimal values by directly using discounted regret as our meta-learning loss function. We hypothesize that a regret-based loss function mimics typical testing scenarios, and would therefore lead to better optimizers compared to optimizers trained only to propose queries that improve over previous queries. Further, RNN-Opt incorporates simple yet effective enhancements during training and inference procedures to deal with the following practical challenges: i) Unknown range of possible values for the black-box function to be optimized, and ii) Practical and domain-knowledge based constraints on the input parameters. We demonstrate the efficacy of RNN-Opt in comparison to existing methods on several synthetic as well as standard benchmark black-box functions along with an anonymized industrial constrained optimization problem.

We introduce a novel distributed derivative-free optimization framework that is resilient to stragglers. The proposed method employs coded search directions at which the objective function is evaluated, and a decoding step to find the next iterate. Our framework can be seen as an extension of evolution strategies and structured exploration methods where structured search directions were utilized. As an application, we consider black-box adversarial attacks on deep convolutional neural networks. Our numerical experiments demonstrate a significant improvement in the computation times.

The optimization of complex real-world systems often presents a challenge when explicit derivatives of the objective function are unavailable. In this paper, we address a real-world black-box optimization problem arising from the design and operation of Concentrated Solar Power (CSP) systems. CSP systems present a unique challenge due to their nonlinear, multi-modal nature, which complicates optimization using traditional gradient-based methods. Derivative-free optimization (DFO) techniques are well-suited to tackle such black-box problems, but even these methods can become impractical when the number of function evaluations required is too large since the evaluation can be expensive. To overcome this limitation, we utilize an adaptive sampling DFO approach that requires a smaller number of function evaluations by intelligently selecting informative points based on surrogate models. The surrogate models are then optimized using derivative-based optimization algorithms to find new sampling points that may be (near-) optimal. We apply this methodology to optimize a CSP problem from the SOLAR benchmark tool, specifically focusing on minimizing the cost of thermal storage, and the total investment cost. The results demonstrate that our adaptive sampling method, ADASNOBFIT, outperforms the well-known SNOBFIT algorithm in terms of solution quality.

Model merging refers to the process of integrating multiple distinct models into a unified model that preserves and combines the strengths and capabilities of the individual models. Most existing approaches rely on task vectors to combine models, typically under the assumption that model parameters are accessible. However, for extremely large language models (LLMs) such as GPT-4, which are often provided solely as black-box services through API interfaces (Language-Model-as-a-Service), model weights are not available to end users. This presents a significant challenge, which we refer to as black-box model merging (BMM) with massive LLMs. To address this challenge, we propose a derivative-free optimization framework based on the evolutionary algorithm (Evo-Merging) that enables effective model merging using only inference-time API queries. Our method consists of two key components: (1) sparsity-based denoising, designed to identify and filter out irrelevant or redundant information across models, and (2) sign-aware scaling, which dynamically computes optimal combination weights for the relevant models based on their performance. We also provide a formal justification, along with a theoretical analysis, for our asymmetric sparsification. Extensive experimental evaluations demonstrate that our approach achieves state-of-the-art results on a range of tasks, significantly outperforming existing strong baselines.

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We introduce the quantum adaptive distribution search (QuADS), a quantum continuous optimization algorithm that integrates Grover adaptive search (GAS) with the covariance matrix adaptation evolution strategy (CMA-ES), a classical technique for continuous optimization. QuADS utilizes the quantum-based search capabilities of GAS and enhances them with the principles of CMA-ES for more efficient optimization. It employs a multivariate normal distribution for the initial state of the quantum search and repeatedly updates it throughout the optimization process. Our numerical simulations show that QuADS outperforms both GAS and CMA-ES. This is achieved through adaptive refinement of the initial state distribution rather than consistently using a uniform state, resulting in fewer oracle calls. This study presents an important step toward exploiting the potential of quantum computing for continuous optimization. Published by the American Physical Society 2024

In this work, we propose a new variant of natural evolution strategies (NES) for high-dimensional black-box opti-mization problems. The proposed method, CR-FM-NES, extends a recently proposed state-of-the-art NES, Fast Moving Natural Evolution Strategy (FM-NES), in order to be applicable in high-dimensional problems. CR-FM-NES builds on an idea using a restricted representation of a covariance matrix instead of using a full covariance matrix, while inheriting an efficiency of FM-NES. The restricted representation of the covariance matrix enables CR-FM-NES to update parameters of a multivariate normal distribution in linear time and space complexity, which can be applied to high-dimensional problems. Our experimental results reveal that CR-FM-NES does not lose the efficiency of FM-NES, and on the contrary, CR-FM-NES has achieved significant speedup compared to FM-NES on some benchmark problems. Furthermore, our numerical experiments using 200, 600, and 1000-dimensional benchmark problems demonstrate that CR-FM-NES is effective over scalable baseline methods, VD-CMA and Sep-CMA.

This paper presents a natural evolution strategy for implicitly constrained black-box function optimization. The black-box function optimization is challenging because explicit representations of objective functions are not given, and only evaluation values of solutions can be used. In implicitly constrained black-box function problems, constraints are not explicitly given, and only the feasibility of a solution is obtained when the objective function is evaluated. In other words, the amount of constraint violation cannot be obtained, which makes the optimization difficult. Natural Evolution Strategies (NES) is one of the promising frameworks for black-box function optimization. DX-NES is an improved version of xNES which is a promising NES using a multivariate normal distribution as the probability distribution. DX-NES has been reported to show good performance on unconstrained black-box function optimization problems. However, DX-NES has a serious problem in that its performance degrades when applied to implicitly constrained problems. In order to address the problem, we propose DX-NES taking account of Implicit Constraint (DX-NES-IC). In experiments using benchmark problems and a lens system design problem, DX-NES-IC showed better performance than DX-NES, xNES, CMA-ES, and those with the resampling technique in terms of the number of successful trials and that of evaluations, where the resampling technique is a constraint handling method which can be used for implicitly constrained problems.

Evolution strategies (ESs) are zero-order stochastic black-box optimization heuristics invariant to monotonic transformations of the objective function. They evolve a multivariate normal distribution, from which candidate solutions are generated. Among different variants, CMA-ES is nowadays recognized as one of the state-of-the-art zero-order optimizers for difficult problems. Albeit ample empirical evidence that ESs with a step-size control mechanism converge linearly, theoretical guarantees of linear convergence of ESs have been established only on limited classes of functions. In particular, theoretical results on convex functions are missing, where zero-order and also first order optimization methods are often analyzed. In this paper, we establish almost sure linear convergence and a bound on the expected hitting time of an ES, namely the (1 + 1)-ES with (generalized) one-fifth success rule and an abstract covariance matrix adaptation with bounded condition number, on a broad class of functions. The analysis holds for monotonic transformations of positively homogeneous functions and of quadratically bounded functions, the latter of which particularly includes monotonic transformation of strongly convex functions with Lipschitz continuous gradient. As far as the authors know, this is the first work that proves linear convergence of ES on such a broad class of functions.

Evolution strategies (ESs) are powerful probabilistic search and optimization algorithms gleaned from biological evolution theory. They have been successfully applied to a wide range of real world applications. The modern ESs are mainly designed for solving continuous parameter optimization problems. Their ability to adapt the parameters of the multivariate normal distribution used for mutation during the optimization run makes them well suited for this domain. In this article we describe and study mixed integer evolution strategies (MIES), which are natural extensions of ES for mixed integer optimization problems. MIES can deal with parameter vectors consisting not only of continuous variables but also with nominal discrete and integer variables. Following the design principles of the canonical evolution strategies, they use specialized mutation operators tailored for the aforementioned mixed parameter classes. For each type of variable, the choice of mutation operators is governed by a natural metric for this variable type, maximal entropy, and symmetry considerations. All distributions used for mutation can be controlled in their shape by means of scaling parameters, allowing self-adaptation to be implemented. After introducing and motivating the conceptual design of the MIES, we study the optimality of the self-adaptation of step sizes and mutation rates on a generalized (weighted) sphere model. Moreover, we prove global convergence of the MIES on a very general class of problems. The remainder of the article is devoted to performance studies on artificial landscapes (barrier functions and mixed integer NK landscapes), and a case study in the optimization of medical image analysis systems. In addition, we show that with proper constraint handling techniques, MIES can also be applied to classical mixed integer nonlinear programming problems.

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The covariance matrix adaptation evolution strategy (CMA-ES) is a stochastic search algorithm using a multivariate normal distribution for continuous black-box optimization. In addition to strong empirical results, part of the CMA-ES can be described by a stochastic natural gradient method and can be derived from information geometric optimization (IGO) framework. However, there are some components of the CMA-ES, such as the rank-one update, for which the theoretical understanding is limited. While the rank-one update makes the covariance matrix to increase the likelihood of generating a solution in the direction of the evolution path, this idea has been difficult to formulate and interpret as a natural gradient method unlike the rank-$\mu$ update. In this work, we provide a new interpretation of the rank-one update in the CMA-ES from the perspective of the natural gradient with prior distribution. First, we propose maximum a posteriori IGO (MAP-IGO), which is the IGO framework extended to incorporate a prior distribution. Then, we derive the rank-one update from the MAP-IGO by setting the prior distribution based on the idea that the promising mean vector should exist in the direction of the evolution path. Moreover, the newly derived rank-one update is extensible, where an additional term appears in the update for the mean vector. We empirically investigate the properties of the additional term using various benchmark functions.

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This paper explores the theoretical basis of the covariance matrix adaptation evolution strategy (CMA-ES) from the information geometry viewpoint.To establish a theoretical foundation for the CMA-ES, we focus on a geometric structure of a Riemannian manifold of probability distributions equipped with the Fisher metric. We define a function on the manifold which is the expectation of fitness over the sampling distribution, and regard the goal of update of the parameters of sampling distribution in the CMA-ES as maximization of the expected fitness. We investigate the steepest ascent learning for the expected fitness maximization, where the steepest ascent direction is given by the natural gradient, which is the product of the inverse of the Fisher information matrix and the conventional gradient of the function.Our first result is that we can obtain under some types of parameterization of multivariate normal distribution the natural gradient of the expected fitness without the need for inversion of the Fisher information matrix. We find that the update of the distribution parameters in the CMA-ES is the same as natural gradient learning for expected fitness maximization. Our second result is that we derive the range of learning rates such that a step in the direction of the exact natural gradient improves the parameters in the expected fitness. We see from the close relation between the CMA-ES and natural gradient learning that the default setting of learning rates in the CMA-ES seems suitable in terms of monotone improvement in expected fitness. Then, we discuss the relation to the expectation-maximization framework and provide an information geometric interpretation of the CMA-ES.

Modern machine learning uses more and more advanced optimization techniques to find optimal hyper parameters. Whenever the objective function is non-convex, non continuous and with potentially multiple local minima, standard gradient descent optimization methods fail. A last resource and very different method is to assume that the optimum(s), not necessarily unique, is/are distributed according to a distribution and iteratively to adapt the distribution according to tested points. These strategies originated in the early 1960s, named Evolution Strategy (ES) have culminated with the CMA-ES (Covariance Matrix Adaptation) ES. It relies on a multi variate normal distribution and is supposed to be state of the art for general optimization program. However, it is far from being optimal for discrete variables. In this paper, we extend the method to multivariate binomial correlated distributions. For such a distribution, we show that it shares similar features to the multi variate normal: independence and correlation is equivalent and correlation is efficiently modeled by interaction between different variables. We discuss this distribution in the framework of the exponential family. We prove that the model can estimate not only pairwise interactions among the two variables but also is capable of modeling higher order interactions. This allows creating a version of CMA ES that can accomodate efficiently discrete variables. We provide the corresponding algorithm and conclude.

In the context of unconstraint numerical optimization, this paper investigates the global linear convergence of a simple probabilistic derivative-free optimization algorithm (DFO). The algorithm samples a candidate solution from a standard multivariate normal distribution scaled by a step-size and centered in the current solution. This solution is accepted if it has a better objective function value than the current one. Crucial to the algorithm is the adaptation of the step-size that is done in order to maintain a certain probability of success. The algorithm, already proposed in the 60's, is a generalization of the well-known Rechenberg's $(1+1)$ Evolution Strategy (ES) with one-fifth success rule which was also proposed by Devroye under the name compound random search or by Schumer and Steiglitz under the name step-size adaptive random search. In addition to be derivative-free, the algorithm is function-value-free: it exploits the objective function only through comparisons. It belongs to the class of comparison-based step-size adaptive randomized search (CB-SARS). For the convergence analysis, we follow the methodology developed in a companion paper for investigating linear convergence of CB-SARS: by exploiting invariance properties of the algorithm, we turn the study of global linear convergence on scaling-invariant functions into the study of the stability of an underlying normalized Markov chain (MC). We hence prove global linear convergence by studying the stability (irreducibility, recurrence, positivity, geometric ergodicity) of the normalized MC associated to the $(1+1)$-ES. More precisely, we prove that starting from any initial solution and any step-size, linear convergence with probability one and in expectation occurs. Our proof holds on unimodal functions that are the composite of strictly increasing functions by positively homogeneous functions with degree $\alpha$ (assumed also to be continuously differentiable). This function class includes composite of norm functions but also non-quasi convex functions. Because of the composition by a strictly increasing function, it includes non continuous functions. We find that a sufficient condition for global linear convergence is the step-size increase on linear functions, a condition typically satisfied for standard parameter choices. While introduced more than 40 years ago, we provide here the first proof of global linear convergence for the $(1+1)$-ES with generalized one-fifth success rule and the first proof of linear convergence for a CB-SARS on such a class of functions that includes non-quasi convex and non-continuous functions. Our proof also holds on functions where linear convergence of some CB-SARS was previously proven, namely convex-quadratic functions (including the well-know sphere function).

Many real-world problems encounter varying environments while searching for the optimal solutions. Problems of these kinds are known as dynamic optimization problems. Evolutionary algorithms (EAs) have shown their high capability of solving dynamic optimization problems. The large number of fitness evaluations, however, limits the utility of EAs in dynamic optimization with expensive fitness evaluation. This study proposes a fitness inheritance method based on $k$-nearest neighbors (kNN) to reduce the number of fitness evaluations. This fitness inheritance method is applied to the covariance matrix adaptation evolution strategy (CMAES), a state-of-the-art evolutionary algorithm for complex numerical optimization problems, to save fitness evaluations and improve search efficiency. This study further presents a convergence checking operator to maintain a reasonable accuracy of fitness inheritance by reinitializing the population and multivariate normal distribution model timely. Experimental results verify the effectiveness and efficiency of the proposed method on six benchmark problems of dynamic optimization.

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There are several studies on fuzzy univariate hypothesis tests corresponding to a normal distribution. A fuzzy statistical test was proposed in this study for mean and variance–covariance matrix of a multivariate normal with fuzzy random variables. For this purpose, a notion of fuzzy multivariate normal random variable with fuzzy mean and non-fuzzy variance–covariance matrix was first developed. Then, the concepts of the fuzzy type-I error, fuzzy type-II error, fuzzy power, non-fuzzy significance level and fuzzy p-value were extended. A degree-based criterion was also suggested to compare the fuzzy p-values as well as a specific significance level to decide whether accepting or rejecting the underlying hypotheses. The effectiveness of the proposed fuzzy hypothesis test was also examined through some numerical examples.

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We introduce a quantile-based multivariate log-normal distribution, providing a new multivariate skewed distribution with positive support. The parameters of this distribution are interpretable in terms of quantiles of marginal distributions and associations between pairs of variables, a desirable feature for statistical modeling purposes. We derive statistical properties of the quantile-based multivariate log-normal distribution involving the transformations, closed-form expressions for the mixed moments, expected value, covariance matrix, mode, Shannon entropy, and Kullback–Leibler divergence. We also present results on marginalization, conditioning, and independence. Additionally, we discuss parameter estimation and verify its performance through simulation studies. We evaluate the model fitting based on Mahalanobis-type distances. An application to children data is presented.

Abstract In this paper, we study the fractional moments of a truncated centered multivariate normal distribution, with a focus on their computation. We develop computational methods, including ones based on the holonomic gradient method, the second-order Laplace approximation, and the Monte Carlo method. These methods enable us to compute higher order fractional moments without evaluating multiple integrals. Via numerical experiments, we investigate their performances. Some applications, including robust graphical modeling based on the alternative multivariate t-distribution, are also presented.

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The path planning for mobile robots has attracted extensive attention, and evolutionary algorithms have been applied to this problem increasingly. In this paper, we propose a novel gradient eigendecomposition invariance biogeography-based optimization (GEI-BBO) for mobile robot path planning, which has the merits of high rotation invariance and excellent search performance. In GEI-BBO, we design an eigendecomposition mechanism for migration operation, which can reduce the dependency of biogeography-based optimization (BBO) on the coordinate system, improve the rotation invariance and share the information between eigensolutions more effectively. Meanwhile, to find the local optimal solution better, gradient descent is added, and the system search strategy can reduce the occurrence of local trapping phenomenon. In addition, combining the GEI-BBO with cubic spline interpolation will solve the problem of mobile robot path planning through a defined coding method and fitness function. A series of experiments are implemented on benchmark functions, whose results indicated that the optimization performance of GEI-BBO is superior to other algorithms. And the successful application of GEI-BBO for path planning in different environments confirms its effectiveness and practicability.

Metaheuristics are required in invariance when transforming the solution space or objective function (transformation invariance) for the robustness of the optimization algorithm because they are used in various environments. In this study, we first construct an analysis framework based on transformation invariance using a proof. Second, we point out that particle swarm optimization (PSO) lacks invariance under rotation of the solution space (rotational invariance) using the analysis framework. Third, we develop PSO with rotational invariance using correlativity (CRIPSO), which has a coordinate transformation function based on a covariance matrix. Fourth, we confirm that CRIPSO shows rotational invariance using the analysis framework. Finally, the performance of CRIPSO is verified through numerical experiments for typical separable benchmark functions with and without rotation of the solution space by comparing PSO values.

This paper evaluates the robustness and structural invariance of hybrid population-based metaheuristics under various objective space transformations. A lightweight plug-and-play hybridization operator is applied to nineteen state-of-the-art algorithms-including differential evolution (DE), particle swarm optimization (PSO), and recent bio-inspired methods-without modifying their internal logic. Benchmarking on the CEC-2017 suite across four dimensions (10, 30, 50, 100) is performed under five transformation types: baseline, translation, scaling, rotation, and constant shift. Statistical comparisons based on Wilcoxon and Friedman tests, Bayesian dominance analysis, and convergence trajectory profiling consistently show that differential-based hybrids (e.g., hIMODE, hSHADE, hDMSSA) maintain high accuracy, stability, and invariance under all tested deformations. In contrast, classical algorithms-especially PSO- and HHO-based variants-exhibit significant performance degradation under non-separable or distorted landscapes. The findings confirm the superiority of adaptive, structurally resilient hybrids for real-world optimization tasks subject to domain-specific transformations.

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Compared with the registration methods based on local optimizations, the heuristic registration methods are less sensitive to the initial position, and a reasonable bound range is essential to ensure the registration validity. In practice, compared with a rotation bound range, which is periodic, the setting of the translation range is more difficult and manual interventions required, especially when the initial position is complex. Moreover, it has yet to be discussed in past research. Therefore, a normal-based registration method based on the flower pollination algorithm is proposed in this paper, in which only rotation parameters ( $r_{x},r_{y},r_{z}$ ) are considered. In our method, the point correspondences are guided by their normal due to their invariance to position translation. Considering the normal degeneration caused by noise, outliers, and partial overlapping, the Pauta criterion is employed to remove distorted correspondences and acquire reliable translation. Moreover, the population of optimal pollens is guaranteed by the use of the searching radius adjustment and periodic boundary. A number of experiments demonstrate that the proposed method exhibits competitive or better performance in terms of initial position, noise, outliers and partial overlapping. Furthermore, a real quality inspection is also implemented to confirm the availability and superiority of the proposed method in the manufacturing process.

In the past three decades, a large number of metaheuristics have been proposed and shown high performance in solving complex optimization problems. While most variation operators in existing metaheuristics are empirically designed, this paper aims to design new operators automatically, which are expected to be search space independent and thus exhibit robust performance on different problems. For this purpose, this work first investigates the influence of translation invariance, scale invariance, and rotation invariance on the search behavior and performance of some representative operators. Then, we deduce the generic form of translation, scale, and rotation invariant operators. Afterwards, a principled approach is proposed for the automated design of operators, which searches for high-performance operators based on the deduced generic form. The experimental results demonstrate that the operators generated by the proposed approach outperform state-of-the-art ones on a variety of problems with complex landscapes and up to 1000 decision variables.

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Multivariate time-series prediction is a challenging research topic in the field of time-series analysis and modeling, and is continually under research. The echo state network (ESN), a type of efficient recurrent neural network, has been widely used in time-series prediction, but when using ESN, two crucial problems have to be confronted: 1) how to select the optimal subset of input features and 2) how to set the suitable parameters of the model. To solve this problem, the modified biogeography-based optimization ESN (MBBO-ESN) system is proposed for system modeling and multivariate time-series prediction, which can simultaneously achieve feature subset selection and model parameter optimization. The proposed MBBO algorithm is an improved evolutionary algorithm based on biogeography-based optimization (BBO), which utilizes an $S$ -type population migration rate model, a covariance matrix migration strategy, and a Lévy distribution mutation strategy to enhance the rotation invariance and exploration ability. Furthermore, the MBBO algorithm cannot only optimize the key parameters of the ESN model but also uses a hybrid-metric feature selection method to remove the redundancies and distinguish the importance of the input features. Compared with the traditional methods, the proposed MBBO-ESN system can discover the relationship between the input features and the model parameters automatically and make the prediction more accurate. The experimental results on the benchmark and real-world datasets demonstrate that MBBO outperforms the other traditional evolutionary algorithms, and the MBBO-ESN system is more competitive in multivariate time-series prediction than other classic machine-learning models.

Multi-objective (MO) optimization problems require simultaneously optimizing two or more objective functions. An MO algorithm needs to find solutions that reach different optimal balances of the objective functions, i.e., optimal Pareto front, therefore, high dimensionality of the solution space can hurt MO optimization much severer than single-objective optimization, which was little addressed in previous studies. This paper proposes a general, theoretically-grounded yet simple approach ReMO, which can scale current derivative-free MO algorithms to the high-dimensional non-convex MO functions with low effective dimensions, using random embedding. We prove the conditions under which an MO function has a low effective dimension, and for such functions, we prove that ReMO possesses the desirable properties of optimal Pareto front preservation, time complexity reduction, and rotation perturbation invariance. Experimental results indicate that ReMO is effective for optimizing the high-dimensional MO functions with low effective dimensions, and is even effective for the high-dimensional MO functions where all dimensions are effective but most only have a small and bounded effect on the function value.

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Backtracking search algorithm (BSA) has been applied to solve the various optimization problems in recent years. However, BSA is difficult to solve non-separable problems due to its single search mechanism. In this paper, backtracking search algorithm based on knowledge of different populations, named DKBSA, is proposed to solve continuous optimization problems. In DKBSA, sub-population partitioning method is used to enhance the local search ability and alleviate the loss rate of the diversity of population. Afterwards, a mutation strategy with knowledge guidance and rotation invariance, which is based on the current sub-population information and historical information, is designed to improve the convergence speed of the DKBSA. Furthermore, a control parameter of adaptive search factor is embedded in the mutation strategy to balance the exploitation and exploration of the proposed algorithm. Finally, a probabilistic model-based strategy is proposed to generate dominant individuals to further improve the search ability of the proposed algorithm. The experimental results of the state-of-the-art algorithms in the CEC2017 benchmark test suit reveal that the DKBSA is effective for solving non-separable problems.

Due to invariance to significant intensity differences, similarity metrics have been widely used as criteria for an area-based method for registering optical remote sensing image. However, for images with large scale and rotation difference, the robustness of similarity metrics can greatly determine the registration accuracy. In addition, area-based methods usually require appropriately selected initial values for registration parameters. This paper presents a registration approach using spatial consistency (SC) and average regional information divergence (ARID), called spatial-consistency and average regional information divergence minimization via quantum-behaved particle swarm optimization (SC-ARID-QPSO) for optical remote sensing images registration. Its key idea minimizes ARID with SC to select an ARID-minimized spatial consistent feature point set. Then, the selected consistent feature set is tuned randomly to generate a set of M registration parameters, which provide initial particle warms to implement QPSO to obtain final optimal registration parameters. The proposed ARID is used as a criterion for the selection of consistent feature set, the generation of initial parameter sets, and fitness functions used by QPSO. The iterative process of QPSO is terminated based on a custom-designed automatic stopping rule. To evaluate the performance of SC-ARID-QPSO, both simulated and real images are used for experiments for validation. In addition, two data sets are particularly designed to conduct a comparative study and analysis with existing state-of-the-art methods. The experimental results demonstrate that SC-ARID-QPSO produces better registration accuracy and robustness than compared methods.

We propose a novel optimization algorithm which named Nonlinear Map-model Optimization (abbr. NMO) method. The NMO is classified as swarm intelligence (abbr. SI) optimizer and consists of some search individuals whose dynamics is driven by a simple nonlinear map. The search point distribution is controlled by the simple nonlinear map. Based on the theoretical analysis results about the dynamics of the particle swarm optimization, we set so that the searching point distribution of the NMO becomes an optimal distribution. Also, the simple nonlinear map generates a chaotic search point time series while keeping the search range. Such a time series can efficiently search within the search range. As a result, NMO can search along the valley of the evaluation function. Namely, NMO is considered to have a rotation invariance and a scaling invariance. In general, the computation amount of SI optimizer is proportional to the number of search elements included in the SI optimizer. However, the NMO requires only a few particles comparing with other swarm intelligence optimizers. Therefore, the computation amount is the smaller than the other methods. As the result, the search performance of the NMO exhibits better than Standard PSO 2011.

In solving large scale optimization problems, CMA-ES has the disadvantages of high complexity and premature stagnation. To solve this problem, this paper proposes an improved CMA-ES, called GI-ES, for large-scale optimization problems. GI-ES uses all the historical information of the previous generation of individuals to evaluate the parameters of the distribution of the next generation. These estimates can be considered as approximate gradient information, which complete covariance information is not required. Thus GI-ES is friendly to large scale optimization problems. Comparative experiments have been done on state-of-the-art algorithms. The results proved the effectiveness and efficiency of GI-ES for large scale optimization problems.

In this paper we benchmark five variants of CMA-ES for optimization in large dimension on the novel large scale testbed of COCO under default or modified parameter settings. In particular, we compare the performance of the separable CMA-ES, of VD-CMA-ES and VkD-CMA-ES, of two implementations of the Limited Memory CMA-ES and of the Rank m Evolution Strategy, RmES. For VkD-CMA-ES we perform experiments with different complexity models of the search distribution and for RmES we study the impact of the number of evolution paths employed by the algorithm. The quasi-Newton L-BFGS-B algorithm is also benchmarked and we investigate the effect of choosing the maximum number of variable metric corrections for the Hessian approximation. As baseline comparison, we provide results of CMA-ES up to dimension 320.

We propose a computationally efficient limited memory Covariance Matrix Adaptation Evolution Strategy for large scale optimization, which we call the LM-CMA-ES. The LM-CMA-ES is a stochastic, derivative-free algorithm for numerical optimization of non-linear, non-convex optimization problems in continuous domain. Inspired by the limited memory BFGS method of Liu and Nocedal (1989), the LM-CMA-ES samples candidate solutions according to a covariance matrix reproduced from m direction vectors selected during the optimization process. The decomposition of the covariance matrix into Cholesky factors allows to reduce the time and memory complexity of the sampling to O(mn), where $n$ is the number of decision variables. When $n$ is large (e.g., n > 1000), even relatively small values of $m$ (e.g., m=20,30) are sufficient to efficiently solve fully non-separable problems and to reduce the overall run-time.

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The Increasing Population Covariance Matrix Adaptation Evolution Strategy (IPOP‐CMA‐ES) algorithm is a reference stochastic optimizer dedicated to blackbox optimization, where no prior knowledge about the underlying problem structure is available. This paper aims to accelerate IPOP‐CMA‐ES thanks to high‐performance computing and parallelism when solving large optimization problems. We first show how BLAS and LAPACK routines can be introduced in linear algebra operations, and we then propose two strategies for deploying IPOP‐CMA‐ES efficiently on large‐scale parallel architectures with up to thousands of CPU cores. The first parallel strategy processes the multiple searches in the same ordering as the sequential IPOP‐CMA‐ES, while the second one processes concurrently these multiple searches. These strategies are implemented in MPI+OpenMP and compared on 6144 cores of the supercomputer Fugaku. We manage to obtain substantial speedups (up to several thousand) and even super‐linear ones, and we provide an in‐depth analysis of our results to understand precisely the superior performance of our second strategy. These results are finally confirmed on a local compute cluster with 512 cores.

Cooperative co-evolution (CC) is a practical and efficient evolutionary framework for solving large-scale global optimization problems (LSGOPs). The performance of CC depends on how variables are being grouped and can be improved through guided variable decomposition for various optimization problems. However, achieving a proper variable decomposition is computationally expensive. This article proposes an effective yet efficient differential grouping (EDG) method to reduce the associated computational cost. Our method exploits the historical interrelationship information of previous variable groups to examine interactions between the remnant variable groups. This allows us to spend less computing resources without compromising the accuracy of the final grouping result. Our proposal utilizes the covariance matrix adaptation evolution strategy (CMA-ES) algorithm, in conjunction with EDG, to solve LSGOPs. Further, to reduce time complexity and improve the stability of CMA-ES, we substitute the complex matrix decomposition step with simpler matrix operations to compute the square root of the covariance matrix. Results from our experiments and analysis indicate that EDG is a competitive method to solve LSGOPs and improve the performance of CC. The proposed schemes significantly enhance the searchability of CMA-ES compared to the other large-scale variants of CMA-ES and state-of-the-art large-scale optimizers. Moreover, our EDG could be integrated with evolutionary optimizers of different flavors like differential evolution (DE).

In this article, we introduce a novel learning mechanism based on evolutionary algorithm to train subspace optical neural networks (SONNs). To demonstrate the effectiveness of our proposed covariance matrix adaptation evolution strategy (CMA-ES), the trained SONNs are applied in a typical MNIST dataset classification task. The calculated results demonstrate that the accuracy and stability of the training method are competitive with the traditional gradient algorithm, but the SONNs trained by the CMA-ES can achieve 1.5 times acceleration.

In the post-Moore era, main performance gains of black-box optimizers are increasingly depending on parallelism, especially for large-scale optimization (LSO). Here we propose to parallelize the well-established covariance matrix adaptation evolution strategy (CMA-ES) and in particular its one latest LSO variant called limited-memory CMA-ES (LM-CMA). To achieve efficiency while approximating its powerful invariance property, we present a multilevel learning-based meta-framework for distributed LM-CMA. Owing to its hierarchically organized structure, Meta-ES is well-suited to implement our distributed meta-framework, wherein the outer-ES controls strategy parameters while all parallel inner-ESs run the serial LM-CMA with different settings. For the distribution mean update of the outer-ES, both the elitist and multi-recombination strategy are used in parallel to avoid stagnation and regression, respectively. To exploit spatiotemporal information, the global step-size adaptation combines Meta-ES with the parallel cumulative step-size adaptation. After each isolation time, our meta-framework employs both the structure and parameter learning strategy to combine aligned evolution paths for CMA reconstruction. Experiments on a set of large-scale benchmarking functions with memory-intensive evaluations, arguably reflecting many data-driven optimization problems, validate the benefits (e.g., effectiveness w.r.t. solution quality, and adaptability w.r.t. second-order learning) and costs of our meta-framework.

Summary Oilfield development related decisions such as well placement and production control settings are crucial for commercial success of any project. Planning is done to maximize return on investment or fluid recovery, and involves reservoir simulation studies. Large scale field planning involves many variables including well location and design, as well as bottom hole controls. Reservoir simulation workflows employ optimization algorithms to search for well settings which maximizes our objective. This work presents efficacy of a hybrid evolutionary optimization algorithm for solving high dimensional well placement and control optimization problem, and demonstrates its application using Olympus benchmark. Particle swarm optimization (PSO) is first used in standalone mode to solve joint optimization problem. Two different objective functions, weighted sum of cumulative fluid (WCF) and net present value (NPV) of discounted cash-flow, are used for rigorous analysis and comparison. Next, PSO algorithm is used with another popular optimization algorithm, covariance matrix adaptation – evolution strategy (CMA-ES), in hybrid mode. Hybrid optimization run is made by transferring the best result from PSO algorithm to CMA-ES as its starting point for further improvement. Hybrid PSO-CMA-ES algorithm is more effective in handling high-dimensional and multi-modal optimization problem.

In the context of real-world path planning applications for Unmanned Aerial Vehicles (UAVs), aspects such as handling of multiple objectives (e.g., minimizing risk, path length, travel time, energy consumption, or noise pollution), generation of smooth trajectories in 3D space, and the ability to deal with urban environments have to be taken into account jointly by an optimization algorithm to provide practically feasible solutions. Since the currently available methods do not allow for that, in this paper, we propose a holistic approach for solving a Multi-Objective Path Planning (MOPP) problem for UAVs in a three-dimensional, large-scale urban environment. For the tackled optimization problem, we propose an energy model and a noise model for a UAV, following a smooth 3D path. We utilize a path representation based on 3D Non-Uniform Rational B-Splines (NURBS). As optimizers, we use a conventional version of an Evolution Strategy (ES), two standard Multi-Objective Evolutionary Algorithms (MOEAs) - NSGA2 and MO-CMA-ES, and a gradient-based L-BFGS-B approach. To guide the optimization, we propose hybrid versions of the mentioned algorithms by applying an advanced initialization scheme that is based on the exact bidirectional Dijkstra algorithm. We compare the different algorithms with and without hybrid initialization in a statistical analysis, which considers the number of function evaluations and quality features of the obtained Pareto fronts indicating convergence and diversity of the solutions. We evaluate the methods on a realistic 3D urban path planning scenario in New York City, based on real-world data exported from OpenStreetMap. The examination's results indicate that hybrid initialization is the main factor for the efficient identification of near-optimal solutions.

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The Cooperative Coevolution (CC) framework is the state of the art for solving large scale global optimization (LSGO) problems. A particular challenge in using CC lies in the decomposition of variables and resource allocation. In this work, the decomposition phase of the framework is performed in two stages to address both variable interaction and efficient resource allocation. The algorithm starts with differential analysis followed by differential grouping. The differential analysis allows efficient resource allocation while the differential grouping will detect variable interactions. The differential grouping will act on a small number of variables and will not consume as much computational budget as a single-stage grouping. While not all variable interactions will be detected, separable variables will be recognized hence specialized solvers for separable problems can be employed on these subproblems. In this work, the twostage grouping CC (TSCC) is paired with a hybrid algorithm where sep-CMA-ES is used to solve the separable subproblem and CMA-ES is used to solve the non-separable subproblems, the algorithm is referred as TSCC-CMAES. A comparison between TSCC and CC with groups based on either differential analysis or differential grouping is carried out. In general, TSCC could outperform the two single-stage grouping methods. Additionally, the TSCC-CMAES shows a competitive advantage on a number of more complex problems against state-of-the-art algorithms and it is shown that the effect of the population size and group size is crucial in achieving these results.

Abstract Covariance Matrix Adaptation Evolution Strategy (CMA-ES) has shown great performance on nonseparable optimization problems largely due to its rotation-invariant feature. However, as the computational cost of the self-adaption operation is sensitive to the scale of problems, the performance of CMA-ES heavily suffers from the well-known curse of dimensionality, which makes it impractical to many Large Scale Global Optimization (LSGO) problems. In this paper, a correlation coefficient based grouping (CCG) strategy is proposed to detect the correlations between variables in a simple yet efficient way. Then coupled with a model complexity control (MCC) framework, a new variant of CMA-ES, named MCC-CCG-CMAES, is presented for LSGO problems, which suffers less from curse of dimensionality and significantly reduces the computational cost compared with the standard CMA-ES. To the best of our knowledge, this work is the first attempt at enhancing CMA-ES with the MCC framework rather than the cooperative coevolution (CC) framework. Experimental results on the CEC′2010 large-scale global optimization (LSGO) benchmark functions show that the performance of MCC-CCG-CMAES outperforms the state-of-the-art counterparts.

This work provides an efficient sampling method for the covariance matrix adaptation evolution strategy (CMA-ES) in large-scale settings. In contract to the Gaussian sampling in CMA-ES, the proposed method generates mutation vectors from a mixture model, which facilitates exploiting the rich variable correlations of the problem landscape within a limited time budget. We analyze the probability distribution of this mixture model and show that it approximates the Gaussian distribution of CMA-ES with a controllable accuracy. We use this sampling method, coupled with a novel method for mutation strength adaptation, to formulate the mixture model-based evolution strategy (MMES)—a CMA-ES variant for large-scale optimization. The numerical simulations show that, while significantly reducing the time complexity of CMA-ES, MMES preserves the rotational invariance, is scalable to high dimensional problems, and is competitive against the state-of-the-arts in performing global optimization.

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Covariance matrix adaptation evolution strategy (CMA-ES) is a successful gradient-free optimization algorithm. Yet, it can hardly scale to handle high-dimensional problems. In this paper, we propose a fast variant of CMA-ES (Fast CMA-ES) to handle large-scale black-box optimization problems. We approximate the covariance matrix by a low-rank matrix with a few vectors and use two of them to generate each new solution. The algorithm achieves linear internal complexity on the dimension of search space. We illustrate that the covariance matrix of the underlying distribution can be considered as an ensemble of simple models constructed by two vectors. We experimentally investigate the algorithm’s behaviors and performances. It is more efficient than the CMA-ES in terms of running time. It outperforms or performs comparatively to the variant limited memory CMA-ES on large-scale problems. Finally, we evaluate the algorithm’s performance with a restart strategy on the CEC’2010 large-scale global optimization benchmarks, and it shows remarkable performance and outperforms the large-scale variants of the CMA-ES.

The covariance matrix adaptation evolution strategy (CMA-ES) is a popular method to deal with nonconvex and/or stochastic optimization problems when gradient information is not available. Being based on the CMA-ES, the recently proposed matrix adaptation evolution strategy (MA-ES) establishes the rather surprising result that the covariance matrix and all associated operations (e.g., potentially unstable eigen decomposition) can be replaced by an iteratively updated transformation matrix without any loss of performance. In order to further simplify MA-ES and reduce its <inline-formula> <tex-math notation="LaTeX">$\boldsymbol {\mathcal {O}}({n}^{ { {2}}})$ </tex-math></inline-formula> time and storage complexity to <inline-formula> <tex-math notation="LaTeX">$\boldsymbol {\mathcal {O}}({{mn}})$ </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">$\boldsymbol {m}~{\boldsymbol \ll }~{n}$ </tex-math></inline-formula> such as <inline-formula> <tex-math notation="LaTeX">$\boldsymbol {m}~{\boldsymbol \in }~\boldsymbol {\mathcal {O}}{(1)}$ </tex-math></inline-formula> or <inline-formula> <tex-math notation="LaTeX">${m} {\in } \boldsymbol {\mathcal {O}}({\log {({n})}})$ </tex-math></inline-formula>, we present the limited-memory MA-ES for efficient zeroth order large-scale optimization. The algorithm demonstrates state-of-the-art performance on a set of established large-scale benchmarks.

For large-scale optimization, CMA-ES has the disadvantages of high complexity and premature stagnation. An improved CMA-ES algorithm called GI-ES was proposed in this paper. For the problem of high complexity, the method in this paper replaces the calculation of a covariance matrix with the modeling of expected fitting degrees for a given covariance matrix. At the same time, to solve the problem of premature stagnation, this paper replaces the historical information of elite individuals with the historical information of all individuals. The information can be seen as approximate gradients. The parameters of the next generation of individuals are generated based on the approximate gradients. The experimental results were tested using CEC 2010 and CEC2013 LSGO benchmark test suite, and the experimental results verified the effectiveness of the algorithm on a number of different tasks.

We present our asynchronous implementation of the LM-CMA-ES algorithm, which is a modern evolution strategy for solving complex large-scale continuous optimization problems. Our implementation brings the best results when the number of cores is relatively high and the computational complexity of the fitness function is also high. The experiments with benchmark functions show that it is able to overcome its origin on the Sphere function, reaches certain thresholds faster on the Rosenbrock and Ellipsoid function, and surprisingly performs much better than the original version on the Rastrigin function.

The covariance matrix adaptation evolution strategy (CMA-ES) is a powerful evolutionary algorithm for single-objective real-valued optimization. However, the time and space complexity may preclude its use in high-dimensional decision space. Recent studies suggest that putting sparse or low-rank constraints on the structure of the covariance matrix can improve the efficiency of CMA-ES in handling large-scale problems. Following this idea, this paper proposes a search direction adaptation evolution strategy (SDA-ES) which achieves linear time and space complexity. SDA-ES models the covariance matrix with an identity matrix and multiple search directions, and uses a heuristic to update the search directions in a way similar to the principal component analysis. We also generalize the traditional 1/5th success rule to adapt the mutation strength which exhibits the derandomization property. Numerical comparisons with nine state-of-the-art algorithms are carried out on 31 test problems. The experimental results have shown that SDA-ES is invariant under search-space rotational transformations, and is scalable with respect to the number of variables. It also achieves competitive performance on generic black-box problems, demonstrating its effectiveness in keeping a good tradeoff between solution quality and computational efficiency.

The increasing complexity of modern power systems has led to the emergence of large-scale dynamic economic dispatch (DED) problems. To solve a large-scale DED problem with high-dimensional decision variables and various constraints is still a challenge using most existing evolutionary algorithms. In this paper, we propose a covariance matrix adaptation evolution strategy based on cooperative co-evolutionary framework (CC-CMA-ES) using delta grouping for solving large-scale DED problem. The experiment results suggest that the CC-CMA-ES is a fast and accurate approach for large-scale DED problems in terms of computation time, solution quality and convergence speed. Integrating CMA-ES into CC the framework can reduce the computation time by 97.5%, compared with basic CMA-ES, revealing the great potential of CC-CMA-ES for solving more difficult large-scale DED problems.

The Covariance Matrix Adaptation Evolution Strategy (CMA-ES) is a popular method to deal with nonconvex and/or stochastic optimization problems when the gradient information is not available. Being based on the CMA-ES, the recently proposed Matrix Adaptation Evolution Strategy (MA-ES) provides a rather surprising result that the covariance matrix and all associated operations (e.g., potentially unstable eigendecomposition) can be replaced in the CMA-ES by a updated transformation matrix without any loss of performance. In order to further simplify MA-ES and reduce its $\mathcal{O}\big(n^2\big)$ time and storage complexity to $\mathcal{O}\big(n\log(n)\big)$, we present the Limited-Memory Matrix Adaptation Evolution Strategy (LM-MA-ES) for efficient zeroth order large-scale optimization. The algorithm demonstrates state-of-the-art performance on a set of established large-scale benchmarks. We explore the algorithm on the problem of generating adversarial inputs for a (non-smooth) random forest classifier, demonstrating a surprising vulnerability of the classifier.

The cooperative coevolution (CC) algorithm features a “divide-and-conquer” problem-solving process. This feature has great potential for large-scale global optimization (LSGO) while inducing some inherent problems of CC if a problem is improperly decomposed. In this work, a novel CC named selective multiple population- (SMP-) based CC (CC-SMP) is proposed to enhance the cooperation of subproblems by addressing two challenges: finding informative collaborators whose fitness and diversity are qualified and adapting to the dynamic landscape. In particular, a CMA-ES-based multipopulation procedure is employed to identify local optima which are then shared as potential informative collaborators. A restart-after-stagnation procedure is incorporated to help the child populations adapt to the dynamic landscape. A biobjective selection is also incorporated to select qualified child populations according to the criteria of informative individuals (fitness and diversity). Only selected child populations are active in the next evolutionary cycle while the others are frozen to save computing resource. In the experimental study, the proposed CC-SMP is compared to 7 state-of-the-art CC algorithms on 20 benchmark functions with 1000 dimensionality. Statistical comparison results figure out significant superiority of the CC-SMP. In addition, behavior of the SMP scheme and sensitivity to the cooperation frequency are also analyzed.

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Scheduling is a crucial task in many manufacturing environments. Depending on the type of the manufactured product, a multitude of different properties and constraints complicate the already complex structure of a job-shop scheduling problem. Semiconductor Manufacturing is well known to be one of the most complex production processes, namely for its stochastic nature, diverse product mix, which leads to strict tool dedications, and for the presence of re-entrant flows in the production line. In this paper, we propose an approach for finding a dispatching policy through the use of (NN), whose weights are optimized through an Evolution Strategies method called (CMA-ES), able to minimize the tardiness, throughput and other relevant metrics within a digital twin of a real-world, stochastic, large-scale semiconductor manufacturing facility.

Dexterous hand manipulation increasingly relies on large-scale motion datasets with precise hand-object trajectory data. However, existing resources such as DexYCB and HO3D are primarily optimized for visual alignment but often yield physically implausible interactions when replayed in physics simulators, including penetration, missed contact, and unstable grasps. We propose a simulation-in-the-loop refinement framework that converts these visually aligned trajectories into physically executable ones. Our core contribution is to formulate this as a tractable black-box optimization problem. We parameterize the hand's motion using a low-dimensional, spline-based representation built on sparse temporal keyframes. This allows us to use a powerful gradient-free optimizer, CMA-ES, to treat the high-fidelity physics engine as a black-box objective function. Our method finds motions that simultaneously maximize physical success (e.g., stable grasp and lift) while minimizing deviation from the original human demonstration. Compared to MANIPTRANS-recent transfer pipelines, our approach achieves lower hand and object pose errors during replay and more accurately recovers hand-object physical interactions. Our approach provides a general and scalable method for converting visual demonstrations into physically valid trajectories, enabling the generation of high-fidelity data crucial for robust policy learning.

Particulate matter (PM10) poses a serious threat to public health by increasing the risk of respiratory issues like asthma and bronchitis, as well as cardiovascular problems such as heart attacks and strokes. In Makkah, Saudi Arabia, the combined impact of dense vehicular traffic, large-scale construction projects, and an arid climate contributes to elevated PM10 concentrations, posing substantial challenges to air quality management and urban sustainability. This study utilizes the TabNet model to estimate PM10 concentrations, taking advantage of its ability to perform sparse feature selection and sequential decision-making to uncover complex relationships among different environmental variables. To improve the predictive accuracy of proposed model, hyperparameter tuning was carried out using the Covariance Matrix Adaptation Evolution Strategy (CMA-ES). The dataset containing meteorological and atmospheric parameters was collected from the Haram station in Makkah over the period from January 2016 to December 2018. TabNet outperformed other machine learning models, achieving a Mean Absolute Error (MAE) of 8.27 and coefficient of determination (R2) of 0.872 on the training set, while attaining an MAE of 9.05 and an R2 of 0.805 on the testing set. Afterwards, SHAP analysis illustrated the relative contributions of various features to PM10 concentrations, identifying atmospheric pressure as the most significant factor, closely followed by humidity and temperature. Lower to medium atmospheric pressure was found to substantially elevate PM10 levels, whereas medium to high humidity and elevated temperatures were likewise associated with increased PM10 concentrations. Furthermore, SHAP interaction plots revealed a moderating effect of atmospheric pressure on the influence of temperature on PM10 levels. These insights highlight the importance of considering both individual and interactive environmental factors when developing air quality models, leading to a deeper understanding of air pollution dynamics in Makkah and supporting more effective mitigation strategies.

The mobile robot uses the camera to capture the motion scene information to learn the relative relationship between the target pose and the robot pose. In image-based mobile robot visual servoing, model-free reinforcement learning methods' learning efficiency is often reduced due to continuous or large-scale state problems, leading to the curse of dimensionality. We believe that reinforcement learning based on prediction models is an efficient and promising alternative to model-free reinforcement learning. Therefore, we propose an intelligent control method for the mobile robot based on the generative recurrent neural network (RNN) and deep reinforcement learning (DRL) called GRMRL; the main part of the model encodes the image, receives the action vector, decodes the predicted image. This trained model can predict short-term future images based on current observations. For scalability and training cost reasons, we use covariance matrix adaptation evolutionary strategies (CMA-ES) to train the policy network in the control system. Various experiments are conducted to evaluate the proposed algorithm on a simulated mobile robot path planning environment called Carracing. In a low-data area of 10k, the performance of the proposed method exceeds most model-free algorithms and achieves high learning efficiency.