Extending distance-weighted exponential natural evolution strategy for function optimization in uncertain environments

This paper presents an extended variant of the distance-weighted exponential natural evolution strategy (DXNES) that works well in uncertain environments. Since we often face objective functions with uncertain parameters in real-world problems, function optimization in uncertain environments is an important problem. The covariance matrix adaptation evolution strategy (CMA-ES) and DX-NES have been proposed as promising methods for function optimization in deterministic environments. The performance of these methods, however, deteriorates in uncertain environments. The uncertain handling CMA-ES (TIH-CMA-ES) has been proposed as an extended variant of CMA-ES for uncertain environments and has shown relatively good performance on problems with uncertain parameters. In this paper, we propose an extended variant of DX-NES named DX-NES for uncertain environments (DX-NES-TIE). DX-NES-TIE approximates the objective function by a quadratic function. DXNES-TIE uses approximation function values for updating the mutation distribution if the noise is strong; otherwise it uses observed objective function values. The strength of the noise is quantified by using the approximation function and the evolution path. Through numerical experiments on 20-dimensional uncertain benchmark problems, we demonstrate that DX-NES-TIE can find ten to 2,000 times as accurate solutions as TIH-CMA-ES can. We also apply DX-NES-TIE to 80-dimensional problems and confirm that DX-NES-TIE is scalable with respect to problem dimensionality.