CMA-ES and its problem

$$ f(x) = x_0^2 + x_1^2 \text{ , where } x_0 \in \mathbb{R}, x_1 \in \mathbb{Z} \text{.} $$

CMA-ES	CMA-ESwM

Comparison

DX-NES-ICI and LMI-NES (proposed).

$$g(x)=\sum_{j=1}^{N_\text{int}}(1000^{\frac{j-1}{N-1}}\left[\bar{\textbf{x}}_{\text{int}}\right]_j)^2$$

$$h(x)=\sum_{j=1}^{N_{\text{co}}}(1000^{\frac{N_{\text{int}}+j-1}{N-1}}[\bar{\textbf{x}}_{\text{co}}]_j)^2$$

$$f(x)=g(x)+h(x)$$

DX-NES-ICI	LMI-NES

Ryoki Hamano, Shota Saito, Masahiro Nomura, and Shinichi Shirakawa. 2022. CMA-ES with Margin: Lower-Bounding Marginal Probability for Mixed-Integer Black-Box Optimization. In Proceedings of the Genetic and Evolutionary Computation Conference (GECCO ’22). 639–647. https://doi.org/10.1145/3512290.3528827
Koki Ikeda and Isao Ono. 2023. Natural Evolution Strategy for Mixed-Integer Black-Box Optimization. In Proceedings of the Genetic and Evolutionary Computation Conference (GECCO ’23). 8 pages. https://doi.org/10.1145/3583131.3590518