A constraint dissolving method for generally constrained optimization over a nonconvex set

Abstract

This manuscript studies a constraint dissolving method for solving optimization problems with general constraints, comprising an abstract constraint G(x) \in C and a potentially nonconvex geometric constraint x \in D, which, to the best of our knowledge, is the first constraint dissolving type method applicable to this general setting. More specifically, we introduce an auxiliary variable s to decouple G(x) \in C into G(x) = s and s \in C, and construct the constraint dissolving mapping acting jointly on both variables. We then propose a penalty-based constraint dissolving method, in which the penalty parameter is updated adaptively according to the feasibility error. Under mild conditions, every accumulation point is Mordukhovich-stationary. By a specific choice of the constraint dissolving mapping, every accumulation point is guaranteed to be feasible, which stands in contrast to standard penalty-type methods in nonconvex settings. Numerical experiments on Rosenbrock, portfolio, and MAXCUT problems demonstrate the effectiveness of the proposed method against the augmented Lagrangian and the penalty decomposition methods.