an:07195782
Zbl 1449.65123
Shen, Chungen; Fan, Changxing; Wang, Yunlong; Xue, Wenjuan
Limited memory BFGS algorithm for the matrix approximation problem in Frobenius norm
EN
Comput. Appl. Math. 39, No. 2, Paper No. 43, 25 p. (2020).
2238-3603 0101-8205 1807-0302
2020
j
65K05 90C55 90C30
matrix approximation; active set; dual problem; L-BFGS; global convergence
Summary: 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.