Penalty and Augmented Lagrangian Methods for Constrained DC Programming

TL;DR

This paper introduces two infeasible penalty and augmented Lagrangian methods with strong convergence guarantees for structured nonsmooth DC constrained problems, overcoming limitations of existing approaches.

Contribution

It proposes novel infeasible penalty and augmented Lagrangian methods with convergence guarantees for a class of nonsmooth DC constrained programs.

Findings

Both methods find approximate D-stationary points under weak assumptions.

Feasible accumulation points satisfy B-stationarity and KKT conditions.

Numerical experiments demonstrate the efficiency of the proposed methods.

Abstract

In this paper we consider a class of structured nonsmooth difference-of-convex (DC) constrained DC program in which the first convex component of the objective and constraints is the sum of a smooth and nonsmooth functions while their second convex component is the supremum of finitely many convex smooth functions. The existing methods for this problem usually have a weak convergence guarantee or require a feasible initial point. Inspired by the recent work (Math Oper. Res. 42(1):95--118, 2017 by Pang et al.), in this paper we propose two infeasible methods with strong convergence guarantee for the considered problem. The first one is a penalty method that consists of finding an approximate D-stationary point of a sequence of penalty subproblems. We show that any feasible accumulation point of the solution sequence generated by such a penalty method is a B-stationary point of the problem under a weakest possible assumption that it satisfies a pointwise Slater constraint qualification (PSCQ). The second one is an augmented Lagrangian (AL) method that consists of finding an approximate D-stationary point of a sequence of AL subproblems. Under the same PSCQ condition as for the penalty method, we show that any feasible accumulation point of the solution sequence generated by such an AL method is a B-stationary point of the problem, and moreover, it satisfies a KKT type of optimality condition for the problem, together with any accumulation point of the sequence of a set of auxiliary Lagrangian multipliers. We also propose an efficient successive convex approximation method for computing an approximate D-stationary point of the penalty and AL subproblems. Finally, some numerical experiments are conducted to demonstrate the efficiency of our proposed methods.