Lagrange optimality system for a class of nonsmooth convex optimization

TL;DR

This paper develops a novel Newton-based algorithm for nonsmooth convex optimization by analyzing the Lagrange optimality system of the augmented Lagrangian, establishing its theoretical properties and practical applicability.

Contribution

It introduces the Lagrange optimality system for nonsmooth convex problems and applies a linear Newton method, providing convergence analysis and a new algorithm.

Findings

Proves nonsingularity of the Newton system under certain conditions

Establishes local convergence of the proposed algorithm

Connects the optimality system with standard optimality and saddle point conditions

Abstract

In this paper, we revisit the augmented Lagrangian method for a class of nonsmooth convex optimization. We present the Lagrange optimality system of the augmented Lagrangian associated with the problems, and establish its connections with the standard optimality condition and the saddle point condition of the augmented Lagrangian, which provides a powerful tool for developing numerical algorithms. We apply a linear Newton method to the Lagrange optimality system to obtain a novel algorithm applicable to a variety of nonsmooth convex optimization problems arising in practical applications. Under suitable conditions, we prove the nonsingularity of the Newton system and the local convergence of the algorithm.