Advances in Operations Research
Volume 2012 (2012), Article ID 357954, 15 pages
http://dx.doi.org/10.1155/2012/357954
Research Article

Solving the Matrix Nearness Problem in the Maximum Norm by Applying a Projection and Contraction Method

M. H. Xu1 and H. Shao2

1School of Mathematics and Physics, Changzhou University, Jiangsu Province, Changzhou 213164, China
2Department of Mathematics, School of Sciences, China University of Mining and Technology, Xuzhou 221116, China

Received 11 April 2012; Accepted 17 June 2012

Academic Editor: Abdellah Bnouhachem

Copyright © 2012 M. H. Xu and H. Shao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let S be a closed convex set of matrices and C be a given matrix. The matrix nearness problem considered in this paper is to find a matrix X in the set S at which max { | 𝑥 𝑖 𝑗 − 𝑐 𝑖 𝑗 | } reaches its minimum value. In order to solve the matrix nearness problem, the problem is reformulated to a min-max problem firstly, then the relationship between the min-max problem and a monotone linear variational inequality (LVI) is built. Since the matrix in the LVI problem has a special structure, a projection and contraction method is suggested to solve this LVI problem. Moreover, some implementing details of the method are presented in this paper. Finally, preliminary numerical results are reported, which show that this simple algorithm is promising for this matrix nearness problem.