Distance-weighted Exponential Natural Evolution Strategy for Implicitly Constrained Black-Box Function Optimization

This paper presents a natural evolution strategy for implicitly constrained black-box function optimization. The black-box function optimization is challenging because explicit representations of objective functions are not given, and only evaluation values of solutions can be used. In implicitly constrained black-box function problems, constraints are not explicitly given, and only the feasibility of a solution is obtained when the objective function is evaluated. In other words, the amount of constraint violation cannot be obtained, which makes the optimization difficult. Natural Evolution Strategies (NES) is one of the promising frameworks for black-box function optimization. DX-NES is an improved version of xNES which is a promising NES using a multivariate normal distribution as the probability distribution. DX-NES has been reported to show good performance on unconstrained black-box function optimization problems. However, DX-NES has a serious problem in that its performance degrades when applied to implicitly constrained problems. In order to address the problem, we propose DX-NES taking account of Implicit Constraint (DX-NES-IC). In experiments using benchmark problems and a lens system design problem, DX-NES-IC showed better performance than DX-NES, xNES, CMA-ES, and those with the resampling technique in terms of the number of successful trials and that of evaluations, where the resampling technique is a constraint handling method which can be used for implicitly constrained problems.