Exact Penalty Method for Variationally Coherent Stochastic Programming Problems

TL;DR

This paper develops an exact penalty method combined with mirror descent for stochastic programming with complex constraints, establishing convergence under variational coherence and extending to Clarke's generalized gradients.

Contribution

It introduces a novel approach integrating exact penalty functions with mirror descent for constrained stochastic optimization, extending existing theory to variationally coherent problems.

Findings

Proposed a mirror descent algorithm with stochastic approximations for exact penalty problems.

Established global convergence under variational coherence and Clarke's generalized gradients.

Numerical examples demonstrate the effectiveness of the method.

Abstract

The paper concerns optimization problems with general equality and inequality constraints and with constraints expressed by a convex set. In order to solve these problems, the general constraints are treated by an exact penalty functions while the others by mirror descent approach. The paper introduces a constraint qualification condition under which the solution of the optimization problem with an exact penalty function and constraints defined by the convex set is a solution of the original problem with constraints. The paper extends results on exact penalty functions to the case when together with general equality and inequality constraints additional constraints defined by a convex set are present. In order to solve the optimization problems with exact penalty functions, a mirror descent algorithm is proposed. It is assumed that instead of using gradients of functions defining constrained optimization problems, their stochastic approximations can be applied. The paper establishes global convergence of the proposed method under the assumption that applied exact penalty functions lead to variationally coherent optimization problems. Since exact penalty functions are not differentiable, the concept of variationally coherent problems is extended to the problems defined by functions exhibiting Clarke's generalized gradients. The behavior of the proposed method is illustrated by some numerical examples.