Boosted scaled subgradient method for DC programming

TL;DR

This paper introduces a boosted scaled subgradient method for DC programming that improves computational efficiency and convergence properties by leveraging auxiliary points and closed-form solutions for quadratic subproblems.

Contribution

The paper proposes a novel boosted scaled subgradient method that enhances DC programming by using auxiliary points and closed-form quadratic subproblems for better performance.

Findings

Method has similar convergence to classical descent methods.

Quadratic subproblems have closed-form solutions.

Improved computational performance over classical DC algorithms.

Abstract

The purpose of this paper is to present a boosted scaled subgradient-type method (BSSM) to minimize the difference of two convex functions (DC functions), where the first function is differentiable and the second one is possibly non-smooth. Although the objective function is in general non-smooth, under mild assumptions, the structure of the problem allows to prove that the negative scaled generalized subgradient at the current iterate is a descent direction from an auxiliary point. Therefore, instead of applying the Armijo linear search and computing the next iterate from the current iterate, both the linear search and the new iterate are computed from that auxiliary point along the direction of the negative scaled generalized subgradient. As a consequence, it is shown that the proposed method has similar asymptotic convergence properties and iteration-complexity bounds as the usual descent methods to minimize differentiable convex functions employing Armijo linear search. Finally, for a suitable scale matrix the quadratic subproblems of BSSM have a closed formula, and hence, the method has a better computational performance than classical DC algorithms which must solve a convex (not necessarily quadratic) subproblem.