Perturbation Analysis Of Two Classes Of Matrix Optimization Problems

Matrix optimization problems are optimization problems whose objective or constrain-t functions are functions of matrix variables or that contain matrix constraints. This type of problems has many applications in financial analysis and engineering computing. Since the the-ory of perturbation analysis of these problems plays an important role in algorithm design for solving them, especially in the termination criterion and convergence analysis. Therefore, it is significant to investigate perturbation analysis of matrix optimization problems. This disserta-tion is mainly devoted to the research of two classes of matrix optimization problems, including the matrix optimization problems induced by the epigraph of spectral norm and the semidefinite matrix optimization problems.The main results of this dissertation are summarized as follows:1. Chapter 3 focuses on the optimality conditions of matrix optimization problems induced by the epigraph of spectral norm (MOSN). We first give the variational geometrical properties of the cone defined by the epigraph of spectral norm and characterization of the critical cone. Since the constraint condition of MOSN can be written equivalently as a semidefinite matrix constraint, which makes MOSN can be reformulated as a semidefinite programming (SDP) problem. Then the relationships between MOSN and its SDP form are considered with respect to the constrain-t nondegeneracy conditions and strong second-order sufficient conditions. We prove that the strong second-order sufficient conditions of two problems are equivalent, but for the constraint nondegeneracy conditions, MOSN's is weaker than SDP's, and an example is illustrated.2. Chapter 4 is devoted to the study of the perturbation analysis of matrix optimization problems induced by the epigraph of spectral norm. First, we reformulate the necessary opti-mality conditions of MOSN as a nonsmooth equation, and then obtain a smoothing equation by smoothing the projection operator in this nonsmooth equation. We investigate the differential properties of the smoothing operator, and finally establish the equivalence among the constraint nondegeneracy condition and strong second-order sufficient condition, the nonsingularity of the Clarke's generalized subdifferential of the smoothing equation at its solution point and others. For applications, we provide the convergence result of the smoothing Newton method for solving this class of problems.3. Chapter 5 focuses on the semidefinite matrix optimization problems, including the semidef-inite matrix constrained generalized equation and the Euclidean distance matrix optimization problem. First, under the partial nondegeneracy constraint condition and strict complemen-tary condition, an exact representation of the limiting coderivative of the solution mapping to a semidefinite matrix generalized equation is established. Applying this explicit representation, an equivalent condition for the Aubin property of a solution mapping and a sufficient condi-tion for the strong regularity of the Karush-Kuhn-Tucker (KKT) point for a nonlinear convex semidefinite programming problem are provided. Then, we demonstrate that the KKT mapping of the Euclidean distance matrix optimization problem is isolated calm at a KKT point, if the strict Robinson's constraint qualification and second-order sufficient optimality conditions hold.