Since their introduction, modern evolution strategies, such as the Covariance Matrix Adaptation Evolution Strategy (CMA-ES), have become established as powerful methods for continuous black-box optimization. This success has led to a wide range of proposed modifications, each designed to improve performance or behavior in specific optimization scenarios. However, because these developments have largely been introduced and studied in isolation, their interactions remain comparatively underexplored. In this paper, we present the Modular CMA-ES (ModCMA), a configurable framework that integrates a wide range of mechanisms from modern evolution strategies within a single implementation. By decomposing CMA-ES into modules with interchangeable options for sampling, selection and recombination, step-size adaptation, matrix adaptation, and restarting, ModCMA enables systematic exploration of a large design space of modern evolution strategies and facilitates the construction, comparison, and automated configuration of new algorithm variants. We illustrate the benefits of this modular approach through two example studies. First, we compare several matrix-adaptation mechanisms in terms of their computational cost and optimization performance. Second, we use automated algorithm configuration to specialize ModCMA to individual benchmark problems and analyze the resulting configurations. Together, these examples demonstrate how the framework can be used both to study individual algorithmic design choices and to explore their combinations in a systematic and reproducible manner.
Table 1: Implemented modules and configuration options available in ModCMA. The default option is provided in boldface. The component affected by the algorithm is shown alongside each module.
Name
Type
LB
Default
UB
Description
λ0
Z
1
4+⌊3logd⌋
50d
Initial offspring population size.
μ0
Z
1
⌊λ0/2⌋
λ0
Initial number of selected parents.
σ0
R
10−15
2.0
1015
Initial global step size.
cσ
R
0
d+μeff+3μeff+2
1
Learning rate of the step-size evolution path.
cc
R
0
d+4+2μeff/d4+μeff/d
1
Learning rate of the covariance evolution path.
cμ
R
0
min(1−c1,(d+2)2+μeff2(μeff−2+1/μeff))
1
Learning rate of the rank- μ covariance update.
Table 2: Numerical parameters exposed by the ConfigSpace interface of ModCMA under the default module configuration. Here d is the search-space dimension, λ0 is the initial offspring population size, μ0 is the number of selected parents, and μeff=(∑iμwi2)−1 is computed from the normalized positive recombination weights. The bounds for λ0 and σ0 are practical numerical limits rather than theoretical constraints; extreme values, particularly for σ0 , are generally better handled through appropriate scaling or normalization of the search space.
Figure 1: Running time in seconds vs. search space dimensionality d . The left figure shows the total time to perform 1 000 generations of the algorithm, using the default population size of 4+⌊3logd⌋ . The median time over 15 independent runs is shown. The right figure shows the average time per matrix update.
Figure 2: log10 Expected Running Time (ERT) divided by the dimensionality d for the benchmark functions from the BBOB suite, for six different values of d . All matrix adaptation methods are compared with the official Python CMA-ES implementation Pycma . The ERT is computed over 100 independent runs, each with a maximum budget of 104d , to achieve a target value of 10−9 . Lower values are better.
Figure 3: Expected running time (ERT) of default and tuned ModCMA configurations on the 24 BBOB functions in dimension 5 , compared to the best-performing solver per function from the bbob2009 portfolio. Lower values indicate better performance. The best (lowest) value per function is shown in bold. For some functions, configurations fail to reach the target within the evaluation budget, resulting in large or infinite ERT values.
Figure 4: Global contribution of module options based on SHAP values. Each bar shows the median SHAP value of a configuration option, aggregated over all evaluated configurations from the SMAC runs. Options are grouped by module (separated by vertical lines), with module names shown as annotations. Negative values indicate options that tend to improve performance, while positive values indicate options that tend to degrade performance, according to the surrogate model.
Figure 5: Per-function SHAP analysis for the Matrix Adaptation module. Rows correspond to the available matrix adaptation strategies, and columns to the 24 BBOB functions. Each cell shows the median SHAP value over all evaluated configurations, indicating whether a given option tends to improve (negative values) or degrade (positive values) performance for that function.
Appendix figures & tables13 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 6: Expected running time (ERT) of the default and tuned ModCMA configurations on the 24 BBOB functions in dimension 5 , together with the individual algorithms from the bbob2009 benchmark portfolio. Lower values indicate better performance. The best (lowest) value per function is shown in bold. For some functions, algorithms fail to reach the target within the evaluation budget, resulting in large or infinite ERT values.
Figure 7: Per-function SHAP analysis for the Active Update module. Rows correspond to the available module options, and columns to the 24 BBOB functions. Each cell shows the median SHAP value over all evaluated configurations, indicating whether a given option tends to improve (negative values) or degrade (positive values) performance for that function.
Figure 8: Per-function SHAP analysis for the Elitism module. Rows correspond to the available module options, and columns to the 24 BBOB functions. Each cell shows the median SHAP value over all evaluated configurations, indicating whether a given option tends to improve (negative values) or degrade (positive values) performance for that function.
Figure 9: Per-function SHAP analysis for the Mirrored Sampling module. Rows correspond to the available module options, and columns to the 24 BBOB functions. Each cell shows the median SHAP value over all evaluated configurations, indicating whether a given option tends to improve (negative values) or degrade (positive values) performance for that function.
Figure 10: Per-function SHAP analysis for the Orthogonal Sampling module. Rows correspond to the available module options, and columns to the 24 BBOB functions. Each cell shows the median SHAP value over all evaluated configurations, indicating whether a given option tends to improve (negative values) or degrade (positive values) performance for that function.
Figure 11: Per-function SHAP analysis for the Repelling module. Rows correspond to the available module options, and columns to the 24 BBOB functions. Each cell shows the median SHAP value over all evaluated configurations, indicating whether a given option tends to improve (negative values) or degrade (positive values) performance for that function.
Figure 12: Per-function SHAP analysis for the Restart Strategy module. Rows correspond to the available module options, and columns to the 24 BBOB functions. Each cell shows the median SHAP value over all evaluated configurations, indicating whether a given option tends to improve (negative values) or degrade (positive values) performance for that function.
Figure 13: Per-function SHAP analysis for the Sample Transformation module. Rows correspond to the available module options, and columns to the 24 BBOB functions. Each cell shows the median SHAP value over all evaluated configurations, indicating whether a given option tends to improve (negative values) or degrade (positive values) performance for that function.
Figure 14: Per-function SHAP analysis for the Base Sampler module. Rows correspond to the available module options, and columns to the 24 BBOB functions. Each cell shows the median SHAP value over all evaluated configurations, indicating whether a given option tends to improve (negative values) or degrade (positive values) performance for that function.
Figure 15: Per-function SHAP analysis for the Sequential Selection module. Rows correspond to the available module options, and columns to the 24 BBOB functions. Each cell shows the median SHAP value over all evaluated configurations, indicating whether a given option tends to improve (negative values) or degrade (positive values) performance for that function.
Figure 16: Per-function SHAP analysis for the Step-Size Adaptation module. Rows correspond to the available module options, and columns to the 24 BBOB functions. Each cell shows the median SHAP value over all evaluated configurations, indicating whether a given option tends to improve (negative values) or degrade (positive values) performance for that function.
Figure 17: Per-function SHAP analysis for the Threshold Convergence module. Rows correspond to the available module options, and columns to the 24 BBOB functions. Each cell shows the median SHAP value over all evaluated configurations, indicating whether a given option tends to improve (negative values) or degrade (positive values) performance for that function.
Figure 18: Per-function SHAP analysis for the Recombination Weights module. Rows correspond to the available module options, and columns to the 24 BBOB functions. Each cell shows the median SHAP value over all evaluated configurations, indicating whether a given option tends to improve (negative values) or degrade (positive values) performance for that function.
This paper proposes RCMAES, a novel variant of the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) for CEC benchmark optimization. RCMAES integrates a dimension-dependent nonlinear population-size reduction strategy with an adaptive restart mechanism within a pure CMA-ES framework. RCMAES is evaluated on three benchmark suites (CEC2017, CEC2020, and CEC2022) and compared with state-of-the-art DE algorithms as well as its closely related counterpart, BIPOP-aCMAES. Experimental results show that RCMAES achieves competitive and robust performance across all benchmarks.
Khoirul Faiq Muzakka, Sören Möller, Martin Finsterbusch
Institute of Energy Materials and Devices (IMD-2), Forschungszentrum Jülich GmbH, Germany
Multimodal optimization aims to locate multiple globally optimal or near-optimal solutions in a single run. This paper presents \emph{S-CARD-CMSA}, a score-aware candidate-archive and density-filtered reporting framework built on the covariance matrix self-adaptation evolution strategy with repelling subpopulations (RS-CMSA-ESII). The method is developed for the IEEE CEC 2026 Competition on Benchmarking Niching Methods for Multimodal Optimization. Rather than modifying the core search dynamics of RS-CMSA-ESII, S-CARD-CMSA preserves its sampling, covariance adaptation, taboo-region update, restart, and termination mechanisms. Two conservative extensions are introduced. First, a passive secondary candidate archive records the restart-level best candidates without influencing the search trajectory. Second, a score-aware density-filtered reporting rule constructs the final solution set by balancing robust peak ratio and precision-driven F1-score. Development experiments show that the density-filtered rule preserves the peak coverage obtained by a medium score-aware rule while reducing redundant reports. On a broader validation subset, it maintains the same mean RPR while improving mean precision, F1-score, and the official-score-oriented average. The method does not use true global-minimum locations during optimization; such information is used only for offline development analysis and post-run scoring. The source code of S-CARD-CMSA is available at https://github.com/ChauhanDikshit.
Dikshit Chauhan
Department of Electrical and Computer Engineering, National University of Singapore
Covariance matrix adaptation evolution strategy (CMA-ES) is a state-of-the-art black-box optimization algorithm. In general, CMA-ES uses a portfolio of multiple stopping criteria to automatically determine when to stop the search. This mechanism aims to avoid unnecessary consumption of the function evaluation budget during stagnation. Stopping criteria play an important role in CMA-ES, particularly when restart strategies are employed. However, the effectiveness of stopping criteria in CMA-ES remains poorly understood. To address this issue, this paper investigates how the 11 stopping criteria in CMA-ES behave on the noiseless BBOB function set. The performance of the stopping criteria is quantitatively evaluated based on the optimal stopping point in terms of the number of function evaluations in a single run of CMA-ES. Our results show that, although which stopping criterion is triggered first depends significantly on the sample size λ and the dimension n, \texttt{tolflatfitness} and \texttt{tolfun} are frequently the first criteria to be triggered among the portfolio of 11 stopping criteria. We also demonstrate that \texttt{tolfunhist} and the portfolio achieve the highest stopping accuracy in most cases. In addition, our results show that the \texttt{tolfun} and \texttt{tolfunhist} criteria are frequently triggered before CMA-ES reaches complete stagnation.