1 research outputs found
Marginal Probability-Based Integer Handling for CMA-ES Tackling Single-and Multi-Objective Mixed-Integer Black-Box Optimization
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 integer
handling for the CMA-ES does not address this stagnation. To overcome these
limitations, we propose a simple integer handling for 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.
Furthermore, in order to demonstrate the generality of the idea of the proposed
method, in addition to the single-objective optimization case, we incorporate
it into multi-objective CMA-ES and verify its performance on bi-objective
mixed-integer benchmark problems.Comment: Camera-ready version for ACM Transactions on Evolutionary Learning
and Optimization (TELO). This paper is an extended version of the work
presented in arXiv:2205.1348