Augmented and exact Lagrangian approaches to continuous constrained optimization with evolution strategies

Abstract

We consider variations on Lagrangian approaches to constraint-handling in the context of stochastic black-box optimization. The well-known augmented Lagrangian function transforms a constrained problem into a sequence of unconstrained problems, and has been adapted for use with evolution strategies. Existing adaptations are compared analytically and experimentally, and a new weakness highlighted. This leads to proposing a new algorithm for constrained optimization that adapts an exact Lagrangian approach for use with evolution strategies. The approach is distinguished by framing the multipliers as dependent on position in the search space rather than as separate parameters and by approaching a solution through solving implicit quadratic subproblems with identical optimal multipliers. Efficacy of the EL-ES algorithm is justified by a single-step analysis along with experimental comparisons on selected benchmark results from the literature and a range of archetypal test problems evaluated against implementations using the augmented Lagrangian approach.