Analyzing design principles for competitive evolution strategies in constrained search spaces

TL;DR

The paper tackles the challenge of understanding why the $\varepsilon$MAg-ES performs well on constrained optimization benchmarks. It conducts a thorough, component-level empirical study of the algorithm, including a Jacobian-based repair, back-calculation, epsilon-level ordering, and matrix adaptation, across the constrained CEC2017 suite. By comparing seven variants and applying Wilcoxon significance testing, the work identifies when each component helps or hinders performance, and it highlights nuanced effects across problem features such as separability, equality constraints, and rotations. The findings show that matrix adaptation is broadly beneficial, epsilon-level handling with equality constraints helps in many cases, but some components may slow progress in high dimensions; these insights guide principled algorithm design and parameter tuning for constrained evolution strategies.

Abstract

In the context of the 2018 IEEE Congress of Evolutionary Computation, the Matrix Adaptation Evolution Strategy for constrained optimization turned out to be notably successful in the competition on constrained single objective real-parameter optimization. Across all considered instances the so-called $ε$MAg-ES achieved the second rank. However, it can be considered to be the most successful participant in high dimensions. Unfortunately, the competition result does not provide any information about the modus operandi of a successful algorithm or its suitability for problems of a particular shape. To this end, the present paper is concerned with an extensive empirical analysis of the $ε$MAg-ES working principles that is expected to provide insights about the performance contribution of specific algorithmic components. To avoid rankings with respect to insignificant differences within the algorithm realizations, the paper additionally introduces significance testing into the ranking process.

Figures (13)

• Figure 1: Median ranking approach according to the technical report of the constrained CEC2017 benchmarks CEC2017. The green parts display the newly integrated aspect of statistical testing into the ranking flow. To this end, the Wilcoxon Signed Rank (WSR) test has been used: a value of $h=1$ is indicating that both algorithm realizations are significantly different at a 5% significance level and $h=0$ is indicating the opposite, respectively.
• Figure 2: Fitness plus constraint violation dynamics on problem $C03$ in dimensions $N=10$ and $N=100$. All curves represent the mean values of 25 independent algorithm runs. The circular markers display the number of function evaluations needed to satisfy all constraints on average.
• Figure 3: Mutation strength dynamics on problem $C03$ in dimensions $N=10$ and $N=100$.
• Figure 4: Fitness plus constraint violation dynamics on problem $C07$ in dimensions $N=10$ and $N=100$. All curves represent the mean values of 25 independent algorithm runs. The circular markers display the number of function evaluations needed to satisfy all constraints on average.
• Figure 5: Fitness plus constraint violation dynamics on problem $C07$ in dimensions $N=10$ and $N=100$. All curves represent the mean values of 25 independent algorithm runs. The circular markers display the number of function evaluations needed to satisfy all constraints on average.