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

摘要

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