CMA-ES for Discrete and Mixed-Variable Optimization on Sets of Points

TL;DR

This paper tackles optimization over sets of points with mixed discrete-continuous variables, where naive CMA-ES struggles with premature convergence. It introduces CMA-ES-SoP, which encodes samples to nearest points via a Voronoi-inspired scheme, and applies margin correction to the covariance to keep neighboring points’ generation probabilities above a threshold, plus an adaptive margin mechanism to prevent over-adjustment. Experimental results on discrete and mixed-variable problems show that CMA-ES-SoP markedly improves success rates and robustness, especially in higher dimensions, compared to standard CMA-ES. The method enables effective optimization over arbitrary sets of points, with potential real-world applications such as site selection and component design, and points to future work on benchmarks and adaptive step-size strategies.

Abstract

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.

Figures (9)

• Figure 1: Illustration of the search space and the optimization process of CMA-ES-SoP on the two-dimensional Ellipsoid function. The acceptable solutions are depicted as blue points. The red and magenta points are the optimum and closest points, respectively. The CMA-ES-SoP uses the Voronoi diagram to encode the samples and adjust the margin value.
• Figure 2: Before correction
• Figure 3: A step of correction
• Figure 4: After correction
• Figure 6: Transitions of the best evaluation values on the discrete optimization problems with $(N_k, L_k) = (2,10)$. We plot the median and interquartile ranges over 25 independent trials.