A Progressive Barrier for Derivative-Free Nonlinear Programming

Abstract

We propose a new constraint-handling approach for general constraints that is applicable to a widely used class of constrained derivative-free optimization methods. As in many methods that allow infeasible iterates, constraint violations are aggregated into a single constraint violation function. As in filter methods, a threshold, or barrier, is imposed on the constraint violation function, and any trial point whose constraint violation function value exceeds this threshold is discarded from consideration. In the new algorithm, unlike the filter method, the amount of constraint violation subject to the barrier is progressively decreased adaptively as the iteration evolves. We test this progressive barrier (PB) approach versus the extreme barrier (EB) with the generalized pattern search (Gps) and the lower triangular mesh adaptive direct search (LTMads) methods for nonlinear derivative-free optimization. Tests are also conducted using the Gps-filter, which uses a version of the Fletcher–Leyffer filter approach. We know that Gps cannot be shown to yield kkt points with this strategy or the filter, but we use the Clarke nonsmooth calculus to prove Clarke stationarity of the sequences of feasible and infeasible trial points for LTMads-PB. Numerical experiments are conducted on three academic test problems with up to 50 variables and on a chemical engineering problem. The new LTMads-PB method generally outperforms our LTMads-EB in the case where no feasible initial points are known, and it does as well when feasible points are known. which leads us to recommend LTMads-PB. Thus the LTMads-PB is a useful practical extension of our earlier LTMads-EB algorithm, particularly in the common case for real problems where no feasible point is known. The same conclusions hold for Gps-PB versus Gps-EB.