Exact Penalties for Decomposable Optimization Problems

TL;DR

This paper introduces an exact penalty method for decomposable convex optimization problems, transforming them into smaller problems and solving the master problem efficiently with a two-speed subgradient method, confirmed by computational experiments.

Contribution

It proposes a novel exact non-smooth penalty approach combined with an improved subgradient method for solving decomposable convex optimization problems.

Findings

Efficient solution of decomposable convex problems demonstrated.

Two-speed subgradient method enhances convergence.

Preliminary computational results confirm method's effectiveness.

Abstract

We consider a general decomposable convex optimization problem. By using right-hand side allocation technique, it can be transformed into a collection of small dimensional optimization problems. The master problem is a convex non-smooth optimization problem. We propose to apply the exact non-smooth penalty method, which gives a solution of the initial problem under some fixed penalty parameter and provides the consistency of lower level problems. The master problem is suggested to be solved by a two-speed subgradient projection method, which enhances the step-size selection. Preliminary results of computational experiments confirm its efficiency.