An Introduction to A Class of Matrix Optimization Problems

Abstract

In this thesis, we study a class of matrix optimization programming (MOP) problems, which involve minimizing the sum of a linear function and a proper closed convex function subject to an affine constraint. MOP is a broad framework, which includes many important optimization problems involving matrices arising from diverse areas such as scientific computing, finance, engineering, and control. In order to make the defined MOP tractable, as an initial step, we study the first and second order properties of the Moreau-Yosida regularization of the related convex functions. More specifically, we conduct a thorough study on a class of matrix valued functions so-called spectral operators. In the first part of the thesis, several fundamental properties of the spectral operator are studied systematically. In the second part of this thesis, we study the sensitivity analysis of some MOP problems. In particular, we mainly focus on the linear matrix cone programming (MCP) problems involving the Ky Fan $k$-norm epigraph cone ${\cal K}$. For such linear MCP problem, we state the constraint nondegeneracy, the second order necessary condition and the (strong) second order sufficient condition. Furthermore, for the local solution of the linear MCP problem involving the Ky Fan $k$-norm, we establish the equivalent links among the strong regularity of the KKT point, the strong second order sufficient condition and constraint nondegeneracy, and the non-singularity of both the B-subdifferential and Clarke's generalized Jacobian of the non-smooth system at a KKT point.