Limited memory BFGS algorithm for the matrix approximation problem in Frobenius norm

• Published in: Computational & applied mathematics Vol. 39; no. 2
• Main Authors: Shen, Chungen, Fan, Changxing, Wang, Yunlong, Xue, Wenjuan
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.05.2020; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Applied physics; Approximation; Computational geometry; Computational mathematics; Computational Mathematics and Numerical Analysis; Convex analysis; Convexity; Mathematical Applications in Computer Science; Mathematical Applications in the Physical Sciences; Mathematics; Mathematics and Statistics; Matrix methods; Newton methods; Nonlinear programming; Optimization; Matrix approximation; Dual problem; Global convergence; 65K05; 90C30; Active set; L-BFGS; 95C55
• ISSN: 2238-3603; 1807-0302
• DOI: 10.1007/s40314-020-1089-9

This paper proposes an L-BFGS algorithm based on the active set technique to solve the matrix approximation problem: given a symmetric matrix, find a nearest approximation matrix in the sense of Frobenius norm to make it satisfy some linear equalities, inequalities and a positive semidefinite constraint. The problem is a convex optimization problem whose dual problem is a nonlinear convex optimization problem with non-negative constraints. Under the Slater constraint qualification, it has zero duality gap with the dual problem. To handle large-scale dual problem, we make use of the active set technique to estimate the active constraints, and then the L-BFGS method is used to accelerate free variables. The global convergence of the proposed algorithm is established under certain conditions. Finally, we conduct some preliminary numerical experiments, and compare the L-BFGS method with the inexact smoothing Newton method, the projected BFGS method, the alternating direction method and the two-metric projection method based on the L-BFGS. The numerical results show that our algorithm has some advantages in terms of CPU time when a large number of linear constraints are involved.