Numerical Algorithms For Matrix Optimization Problems

Matrix optimization problem are optimization problems whose objective or constraint func-tion are function of matrix variables or matrix functions. This type of problems has many ap-plications in engineering computing, financial analysis, machine learning, data mining, high dimensional statistics. As the big data era arrives, matrix optimization has become an important branch of optimization.This dissertation is devoted to the study on numerical algorithms for three types of matrix optimization problems, including a smoothing majorization method for ι22-ιpp matrix minimiza-tion, an alternating direction method for solving a class of inverse semi-definite quadratic pro-gramming problems and an augmented Lagrangian method based on the accelerated proximal gradient strategy for an inverse damped gyroscopic eigenvalue problem. The main results of this dissertation are summarized as follows:1. In Chapter 3, a smoothing majorization method for ι22-ιpp matrix minimization is pro-posed, where the ι22-ιpp model is a type of non-convex regularization for matrix rank minimization problems. Based on the first-order and second-order necessary optimality conditions, we first present lower bounds for nonzero singular values at the local solutions of ι22-ιpp matrix minimiza-tion problem. Then, we use the smoothing technique and the majorization approach to tackle the term ιpp matrix quasi-norm. A smoothing model is constructed and the corresponding smoothing majorization method is proposed. The convergence theorem shows that any accumulation point of the sequence generated by the proposed approach satisfies the first-order necessary optimality condition of the ι22-ιpp problem. Finally, we use the proposed method and the results of lower bounds for nonzero singular values to solve matrix completion problems.2. In Chapter 4, we focus on solving a class of inverse semi-definite quadratic problems and propose an alternating direction method. In our method, one direction problem can be obtained explicitly. The other direction problem is essentially a strictly convex semi-definite quadratic programming problem over lower dimensional positive semi-definite matrix cone. Consequent-ly, we present a spectral projected gradient type subroutine to solve this matrix subproblem and prove its convergence. Numerical experiments indicate that the proposed method is easy to im-plement and code compared with the Newton-type methods, and can solve the test problems efficiently.3. In Chapter 5, based on the framework of augmented Lagrangian method, the disserta-tion is devoted to the study of numerical algorithms for an inverse damped gyroscopic eigen-value problem. In our approach, we use an accelerated proximal gradient (APG) to solve the corresponding subproblems. Under the mild condition, the global convergence of the proposed method is proved. Without any additional regularity condition, the iteration-complexity analysis of our method indicates that the proposed approach needs at most O(log(ε-1)) iterations and at most O(ε-1) APG calls to obtain an ε-feasible and ε-optimal solution of the inverse damped gyroscopic eigenvalue problem.