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2014 Some optimization problems on ranks and inertias of matrix-valued functions subject to linear matrix equation restrictions
Yongge Tian
Banach J. Math. Anal. 8(1): 148-178 (2014). DOI: 10.15352/bjma/1381782094

Abstract

Matrix rank and inertia optimization problems are a class of discontinuous optimization problems in which the decision variables are matrices running over certain matrix sets, while the ranks and inertias of the variable matrices are taken as integer-valued objective functions. In this paper, we establish a group of explicit formulas for calculating the maximal and minimal values of the rank and inertia objective functions of the Hermitian matrix-valued function $A_1 - B_1XB_1^{*}$ subject to the common Hermitian solution of a pair of consistent matrix equations $B_2XB^{*}_2 = A_2$ and $B_3XB_3^{*} = A_3$, and Hermitian solution of the consistent matrix equation $B_4X= A_4$, respectively. Many consequences are obtained, in particular, necessary and sufficient conditions are established for the triple matrix equations $B_1XB^{*}_1 =A_1$, $B_2XB^{*}_2 = A_2$ and $B_3XB^{*}_3 = A_3$ to have a common Hermitian solution, as well as necessary and sufficient conditions for the two matrix equations $B_1XB^{*}_1 =A_1$ and $B_4X = A_4$ to have a common Hermitian solution.

Citation

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Yongge Tian. "Some optimization problems on ranks and inertias of matrix-valued functions subject to linear matrix equation restrictions." Banach J. Math. Anal. 8 (1) 148 - 178, 2014. https://doi.org/10.15352/bjma/1381782094

Information

Published: 2014
First available in Project Euclid: 14 October 2013

zbMATH: 1278.15019
MathSciNet: MR3161689
Digital Object Identifier: 10.15352/bjma/1381782094

Subjects:
Primary: ‎15A24‎
Secondary: 15B57, 47A62, 49K30, 65K10, 90C11, 90C22

Rights: Copyright © 2014 Tusi Mathematical Research Group

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Vol.8 • No. 1 • 2014
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