Quadratic growth conditions for convex matrix optimization problems associated with spectral functions

TL;DR

This paper establishes sufficient conditions for quadratic growth in convex matrix optimization problems with spectral functions, ensuring fast convergence of augmented Lagrangian methods, supported by numerical experiments.

Contribution

It introduces two novel sufficient conditions for quadratic growth based on spectral function properties and applies them to improve convergence guarantees of ALM.

Findings

Quadratic growth conditions ensure faster convergence rates.

Spectral function properties are key to establishing these conditions.

Numerical experiments confirm the practical relevance of the theoretical results.

Abstract

In this paper, we provide two types of sufficient conditions for ensuring the quadratic growth conditions of a class of constrained convex symmetric and non-symmetric matrix optimization problems regularized by nonsmooth spectral functions. These sufficient conditions are derived via the study of the $\mathcal{C}^2$-cone reducibility of spectral functions and the metric subregularity of their subdifferentials, respectively. As an application, we demonstrate how quadratic growth conditions are used to guarantee the desirable fast convergence rates of the augmented Lagrangian methods (ALM) for solving convex matrix optimization problems. Numerical experiments on an easy-to-implement ALM applied to the fastest mixing Markov chain problem are also presented to illustrate the significance of the obtained results.