A (1+1)-CMA-ES for constrained optimisation

Abstract

This paper introduces a novel constraint handling approach for covariance matrix adaptation evolution strategies (CMA-ES). The key idea is to approximate the directions of the local normal vectors of the constraint boundaries by accumulating steps that violate the respective constraints, and to then reduce variances of the mutation distribution in those directions. The resulting strategy is able to approach the boundary of the feasible region without being impeded in its ability to search in directions tangential to the boundaries. The approach is implemented in the (1 + 1)-CMA-ES and evaluated numerically on several test problems. The results compare very favourably with data for other constraint handling approaches applied to unimodal test problems that can be found in the literature.

Quantile-based optimization under uncertainties using adaptive Kriging surrogate models

“…Here we consider that the first stage of enrichment has been already performed. The selected optimization algorithm is the (1+1)-CMA-ES (Covariance matrix adaptation -evolution strategy) for constrained problems (Arnold and Hansen, 2012). This is a stochastic global search algorithm which relies on multivariate normal distributions to search candidates with increased fitness as iterations grows.…”

Modified Covariance Matrix Adaptation – Evolution Strategy algorithm for constrained optimization under uncertainty, application to rocket design

“…To incorporate the handling of constraints, Arnold and Hansen [1] proposed to reduce the covariance of the distribution of the offspring candidate in the approximated directions of the normal vectors of the constraint boundaries in the Offspring infeasible Offspring feasible Figure 2. The dot is the parent, the cross is the generated offspring, the solid line circle is characterized by A (k) defining the ellipsoid delimiting an iso-probability research hypervolume.…”

“…Experimental evaluations have been performed highlighting its efficiency for unimodal constrained optimization problems. However, as (1+1)-CMA-ES, it is not able to optimize multimodal functions and becomes inefficient in high dimensions [1].…”

“…An adaptive penalty function has been proposed to update the penalty coefficient as a function of the sum of the violated constraint values [6]. Arnold and Hansen [1] proposed a new approach to handle the constraints for a simplified version of CMA-ES: (1+1)-CMA-ES which involves one offspring candidate generated from one parent. Modified (1+1)-CMA-ES approximates the normal vector directions of the constraint boundaries in the vicinity of the current candidate solution.…”

“…A control of the covariance of the distribution of the offspring candidate allows to get closer to the boundary of the feasible regions without violating them. The approach provides interesting results but is limited to (1+1)-CMA-ES which becomes inefficient in high dimensions (local convergence, large number of calls to the functions) [1].…”