Numerical Algorithms for a Class of Matrix Norm Approximation Problems

This thesis focuses on designing robust and efficient algorithms for a class of matrix norm approximation (MNA) problems that are to find an affine combination of given matrices having the minimal spectral norm subject to some prescribed linear equality and inequality constraints. These problems arise often in numerical algebra, network, control, engineering and other areas, such as finding the Chebyshev polynomials of matrices and fastest mixing Markov chain models. In this thesis, we first apply the popular first-order algorithm alternating direction method (ADM) to solve such problems. At each iteration of the algorithm, the subproblems involved can either be solved by a fast algorithm or admit closed form solutions, which allows us to implement the ADM easily and simply. Unfortunately, numerical experiments on MNA problems reveal that the ADM performs unstably, and it may fail to achieve satisfactory accuracy in reasonable cpu time for some tested examples, especially for the constrained cases. To overcome this difficulty, we also introduce an inexact dual proximal point algorithm (in short SNDPPA) for solving the MNA problems. At each iteration, the inner problem, rewritten as a system of semismooth equations, is solved by an inexact