Natural Evolution Strategy for Black-Box Function Optimization with Implicit Constraint

This paper proposes a new natural evolution strategy (NES) for implicitly constrained black-box function optimization (ICBBFO). ICBBFO problems are known to be a difficult class of problems because its objective function is not given explicitly, and only information about whether a solution is feasible or infeasible is given. FM-NES is one of the most promising methods for ICBBFO problems. FM-NES uses a multivariate Gaussian distribution (MGD) for solution generation and updates the mean vector and covariance matrix of the MGD using the natural gradient method to improve the expected value of the solutions generated from the MGD. However, when FM-NES is applied to ICBBFO problems where the optimal solution exists near or on a constraint boundary, it is observed that the performance of FM-NES deteriorates. We believe that the deterioration is caused by two problems of FM-NES. The first problem is that FM-NES performs many unnecessary updates of the MGD on a constraint boundary near the optimal solution because the MGD repeatedly moves into and out of the infeasible region. The second problem is that the shape reset of the MGD may not work properly, resulting in an increase in the number of evaluations required to find the optimal solution. To address the two problems of FM-NES, we propose the Enhanced Fast Moving Natural Evolution Strategy for Implicit Constraint (eFM-NES-IC). We conducted numerical experiments using ICBBFO benchmark problems and an 8-element standard lens system design problem that is a difficult real-world problem with implicit constraints to show the effectiveness of eFM-NES-IC. As a result, eFM-NES-IC outperformed FM-NES, DX-NES-IC, xNES, xNES with the resampling technique, CMA-ES, and CMA-ES with the resampling technique on all the benchmark problems and the 8-element standard lens system design problem.