# American Institute of Mathematical Sciences

2015, 5(3): 289-326. doi: 10.3934/naco.2015.5.289

## A survey on rank and inertia optimization problems of the matrix-valued function $A + BXB^{*}$

 1 CEMA, Central University of Finance and Economics, Beijing 100081, China

Received  April 2014 Revised  July 2015 Published  August 2015

This paper is concerned with some rank and inertia optimization problems of the Hermitian matrix-valued functions $A + BXB^{*}$ subject to restrictions. We first establish several groups of explicit formula for calculating the maximum and minimum ranks and inertias of matrix sum $A + X$ subject to a Hermitian matrix $X$ that satisfies a fixed-rank and semi-definiteness restrictions by using some discrete and matrix decomposition methods. We then derive formulas for calculating the maximum and minimum ranks and inertias of the matrix-valued function $A + BXB^*$ subject to a Hermitian matrix $X$ that satisfies a fixed-rank and semi-definiteness restrictions, and show various properties $A + BXB^{*}$ from these ranks and inertias formulas. In particular, we give necessary and sufficient conditions for the equality $A + BXB^* = 0$ and the inequality $A + BXB^* \succ 0\, (\succeq 0, \prec 0, \, \preceq 0)$ to hold respectively for these specified Hermitian matrices $X$.
Citation: Yongge Tian. A survey on rank and inertia optimization problems of the matrix-valued function $A + BXB^{*}$. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 289-326. doi: 10.3934/naco.2015.5.289
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