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Yongge Tian(yongge.tiangmail.com) Abstract: The rank of a matrix and the inertia of a square matrix are two of the most generic concepts in matrix theory for describing the dimension of the row/column vector space and the sign distribution of the eigenvalues of the matrix. Matrix rank and inertia optimization problems are a class of discontinuous optimization problems, in which decision variables are matrices running over certain matrix sets, while the ranks and inertias of the variable matrices are taken as integervalued objective functions. In this paper, we first establish several groups of explicit formulas for calculating the maximal and minimal ranks and inertias of matrix expression $A + X$ subject to a Hermitian matrix $X$ that satisfies a fixedrank and semidefiniteness restrictions by using some discrete and matrix decomposition methods. We then derive formulas for calculating the maximal and minimal ranks and inertias of matrix expression $A + BXB^*$ subject to a Hermitian matrix $X$ that satisfies a fixedrank and semidefiniteness restrictions and use the formulas obtained to characterize behaviors of $A + BXB^{*}$. Keywords: Hermitian matrix; matrixvalued function; rank; inertia; maximization; minimization; MoorePenrose inverse; equality; inequality; L\"owner partial ordering Category 1: Global Optimization Category 2: Combinatorial Optimization Citation: Download: [PDF] Entry Submitted: 02/15/2013 Modify/Update this entry  
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