A version of bundle method with linear programming

TL;DR

This paper introduces a bundle optimization method that exclusively uses linear programming subproblems, enabling convergence for convex and certain nonconvex functions with promising initial numerical results.

Contribution

It presents a novel bundle algorithm that avoids quadratic programming, applicable to convex and prox-regular nonconvex functions, with proven convergence under specific conditions.

Findings

Convergence to a minimizer for convex functions.

Effective minimization over trust regions for nonconvex functions.

Preliminary numerical experiments show promising results.

Abstract

Bundle methods have been intensively studied for solving both convex and nonconvex optimization problems. In most of the bundle methods developed thus far, at least one quadratic programming (QP) subproblem needs to be solved in each iteration. In this paper, we exploit the feasibility of developing a bundle algorithm that only solves linear subproblems. We start from minimization of a convex function and show that the sequence of major iterations converge to a minimizer. For nonconvex functions we consider functions that are locally Lipschitz continuous and prox-regular on a bounded level set, and minimize the cutting-plane model over a trust region with infinity norm. The para-convexity of such functions allows us to use the locally convexified model and its convexity properties. Under some conditions and assumptions, we study the convergence of the proposed algorithm through the outer semicontinuity of the proximal mapping. Encouraging results of preliminary numerical experiments on standard test sets are provided.