Nonlinearly Constrained Optimization Using Heuristic PenaltyMethods and Asynchronous Parallel Generating Set Search

摘要

Many optimization problems are characterized by expensive objective and/or constraintfunction evaluations paired with a lack of derivative information. Direct search methodssuch as generating set search (GSS) are well understood and efficient for derivative-freeoptimization of unconstrained and linearly constrained problems. This paper presents astudy of heuristic algorithms that address the more difficult problem of general nonlin-ear constraints where derivatives for objective, or constraint functions are unavailable.We focus on penalty methods that use GSS to solve a sequence of linearly constrainedproblems, numerically comparing different penalty functions. A classical choice forpenalizing constraint violations is q, the squared 2 norm, which has advantages forderivative-based optimization methods. In our numerical tests, however, we show thatexact penalty functions based on the el , 2, and eonorms converge to good approximatesolutions more quickly and thus are attractive alternatives. Unfortunately, exact penaltyfunctions are nondifferentiable and consequently degrade the final solution accuracy,so we also consider smoothed variants. Smoothed-exact penalty functions are attrac-tive because they retain the differentiability of the original problem.