An adaptive proximal safeguarded augmented Lagrangian method for nonsmooth DC problems with convex constraints

TL;DR

This paper introduces an adaptive proximal safeguarded augmented Lagrangian method for solving nonsmooth DC problems with convex constraints, ensuring convergence to a KKT point.

Contribution

It develops a novel method that handles nonsmooth DC functions with convex constraints, providing convergence guarantees under a modified Slater condition.

Findings

Converges to a generalized KKT point under certain conditions

Handles nonsmooth DC functions with convex constraints

Demonstrates effectiveness through numerical experiments

Abstract

A proximal safeguarded augmented Lagrangian method for minimizing the difference of convex (DC) functions over a nonempty, closed and convex set with additional linear equality as well as convex inequality constraints is presented. Thereby, all functions involved may be nonsmooth. Iterates (of the primal variable) are obtained by solving convex optimization problems as the concave part of the objective function gets approximated by an affine linearization. Under the assumption of a modified Slater constraint qualification, both convergence of the primal and dual variables to a generalized Karush-Kuhn-Tucker (KKT) point is proven, at least on a subsequence. Numerical experiments and comparison with existing solution methods are presented using some classes of constrained and nonsmooth DC problems.