Multi-stage Constraint Surrogate Models for Evolution Strategies

Real-parameter blackbox optimization using evolution strategies (ES) is often applied when the fitness function or its characteristics are not explicitly given. The evaluation of fitness and feasibility might be expensive. In the past, different surrogate model (SM) approaches have been proposed to address this issue. In our previous work, local feasibility SM have been proposed, which are trained with already evaluated individuals. This tightly coupled interdependency with the optimization process leads to a highly dynamic complexity when it is applied to optimized meta-heuristic approaches like the covariance matrix adaption ES (CMA-ES). The objective of this paper is to propose a new type of constraint surrogate model, which uses the concept of active learning in multiple stages for the estimation of the constraint boundary for a stage-depending accuracy. The underlying linear model of the constraint boundary is estimated in every stage with binary search. In the optimization process the pre-selection scheme is employed to save constraint function calls. The SM is evaluated on a simple adaptive (1 + 1)-ES as well as on the complex (1 + 1)-CMA-ES for constrained optimization. The results of both ES on a linearly-constrained test bed look promising.