A semismooth Newton-CG based dual PPA for matrix spectral norm approximation problems

• Published in: Mathematical programming Vol. 155; no. 1-2; pp. 435 - 470
• Main Authors: Chen, Caihua, Liu, Yong-Jin, Sun, Defeng, Toh, Kim-Chuan
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2016; Springer Nature B.V
• Subjects: Accuracy; Algorithms; Approximation; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Engineering; Full Length Paper; Markov analysis; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical models; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Norms; Numerical Analysis; Spectra; Spectral lines; Studies; Theoretical; Spectral operator; PPA; 65F10; 90C25; Spectral norm approximation; 90C06; Semismooth Newton-CG method
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-014-0853-2

We consider a class of matrix spectral norm approximation problems for finding 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, engineering and other areas, such as finding Chebyshev polynomials of matrices and fastest mixing Markov chain models. Based on classical analysis of proximal point algorithms (PPAs) and recent developments on semismooth analysis of nonseparable spectral operators, we propose a semismooth Newton-CG based dual PPA for solving the matrix norm approximation problems. Furthermore, when the primal constraint nondegeneracy condition holds for the subproblems, our semismooth Newton-CG method is proven to have at least a superlinear convergence rate. We also design efficient implementations for our proposed algorithm to solve a variety of instances and compare its performance with the nowadays popular first order alternating direction method of multipliers (ADMM). The results show that our algorithm, which is warm-started with an initial point obtained from the ADMM, substantially outperforms the pure ADMM, especially for the constrained cases and it is able to solve the problems robustly and efficiently to a relatively high accuracy.