  This paper proposes a natural evolution strategy (NES) for mixed-integer
black-box optimization (MI-BBO) that appears in real-world problems such as
hyperparameter optimization of machine learning and materials design. This
problem is difficult to optimize because plateaus where the values do not
change appear when the integer variables are relaxed to the continuous ones.
CMA-ES w. Margin that addresses the plateaus reportedly showed good performance
on MI-BBO benchmark problems. However, it has been observed that the search
performance of CMA-ES w. Margin deteriorates when continuous variables
contribute more to the objective function value than integer ones. In order to
address the problem of CMA-ES w. Margin, we propose Distance-weighted
eXponential Natural Evolution Strategy taking account of Implicit Constraint
and Integer (DX-NES-ICI). We compare the search performance of DX-NES-ICI with
that of CMA-ES w. Margin through numerical experiments. As a result, DX-NES-ICI
was up to 3.7 times better than CMA-ES w. Margin in terms of a rate of finding
the optimal solutions on benchmark problems where continuous variables
contribute more to the objective function value than integer ones. DX-NES-ICI
also outperformed CMA-ES w. Margin on problems where CMA-ES w. Margin
originally showed good performance.
