An implementable proximal-type method for computing critical points to minimization problems with a nonsmooth and nonconvex constraint

TL;DR

This paper introduces a flexible, implementable proximal-type method for solving broad classes of nonsmooth, nonconvex constrained optimization problems, effectively computing critical points with practical numerical validation.

Contribution

It presents a novel approach using nonconvex models formed by the minimum of convex models, unifying analysis for various composite problems, and demonstrating practical effectiveness.

Findings

Method computes points satisfying necessary optimality conditions.

Framework covers multiple subclasses of composite optimization.

Numerical experiments show practical performance on stochastic problems.

Abstract

This work proposes an implementable proximal-type method for a broad class of optimization problems involving nonsmooth and nonconvex objective and constraint functions. In contrast to existing methods that rely on an ad hoc model approximating the nonconvex functions, our approach can work with a nonconvex model constructed by the pointwise minimum of finitely many convex models. The latter can be chosen with reasonable flexibility to better fit the underlying functions' structure. We provide a unifying framework and analysis covering several subclasses of composite optimization problems and show that our method computes points satisfying certain necessary optimality conditions, which we will call model criticality. Depending on the specific model being used, our general concept of criticality boils down to standard necessary optimality conditions. Numerical experiments on some stochastic reliability-based optimization problems illustrate the practical performance of the method.