Nonsmooth Composite Matrix Optimization: Strong Regularity, Constraint Nondegeneracy and Beyond

TL;DR

This paper characterizes the strong regularity of nonsmooth composite matrix optimization problems, extending existing theories to include nonsmooth objectives with broad applications in numerical linear algebra and engineering.

Contribution

It introduces a generalized strong second order sufficient condition and analyzes constraint nondegeneracy for nonsmooth matrix optimization problems, expanding theoretical understanding beyond smooth cases.

Findings

Provides a new characterization of strong regularity for nonsmooth CMatOPs

Extends existing smooth case results to nonsmooth objectives

Enhances theoretical tools for analyzing nonsmooth matrix optimization problems

Abstract

The nonsmooth composite matrix optimization problem (CMatOP), in particular, the matrix norm minimization problem, is a generalization of the matrix conic programming problem with wide applications in numerical linear algebra, computational statistics and engineering. This paper is devoted to the characterization of the strong regularity for the CMatOP via the generalized strong second order sufficient condition and constraint nondegeneracy for problems with nonsmooth objective functions. The derived result supplements the existing characterization of the strong regularity for the constrained optimization problems with twice continuously differentiable data.