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

*(English)*Zbl 1449.65123Summary: 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.

##### MSC:

65K05 | Numerical mathematical programming methods |

90C55 | Methods of successive quadratic programming type |

90C30 | Nonlinear programming |

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\textit{C. Shen} et al., Comput. Appl. Math. 39, No. 2, Paper No. 43, 25 p. (2020; Zbl 1449.65123)

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