The Analytic Solutions of a Class of Constrained Matrix Minimization and Maximization Problems with Applications
Weiwei Xu, Wen Li, Lei Zhu, Xueping Huang · SIAM Journal on Optimization · 2019
In this paper we present the analytic solutions of the following constrained matrix determinant and trace minimization and maximization problems: $\min_{U_{1},\ldots, U_{m}\in\mathbb{U}_{n}}\vert\det (cI_{n}\pm \prod_{j=1}^{m} A_{j} U_{j})\vert$, $\max_{U_{1},\ldots, U_{m}\in\mathbb{U}_{n}}\vert\det (cI_{n}\pm \prod_{j=1}^{m} A_{j} U_{j})\vert$ and $\min_{U_{1},\ldots, U_{m}\in\mathbb{U}_{n}}\vert\operatorname{tr}(cI_{n}\pm \prod_{j=1}^{m} A_j U_j)\vert$, $\max_{U_{1},\ldots, U_{m}\in\mathbb{U}_{n}}\vert\operatorname{tr} (cI_{n}\pm \prod_{j=1}^{m} A_j U_j)\vert$, where $c\in\mathbb{R}$, $A_{1},\ldots,A_{m}$ are $n\times n$ complex matrices, $I_{n}$ is the $n\times n$ identity matrix, $\mathbb{U}_{n}$ is the set of $n\times n$ unitary matrices, and $\det(\cdot)$ and $\operatorname{tr}(\cdot)$ denote the matrix determinant function and the trace function, respectively. The given results improve on the corresponding ones in Marshall, Olkin, and Arnold [ Inequalities: Theory of Majorization and Its Applications, Springer, New York, 2009], Lu [ Acta Math. Sinica, 13 (1963), pp. 49--62], and Sun [ SIAM J. Matrix Anal. Appl., 20 (1983), pp. 611--625]. Some theoretical and practical applications are presented. In particular, some examples of applications to test signals of mechanical systems and aero engine fault diagnosis are given to show the efficiency of the proposed theoretical results.