线性矩阵不等式约束下矩阵值函数秩和惯量的最优化问题的解析解
Analytical solutions to some optimization problems on ranks and inertias of matrix-valued functions subject to linear matrix inequalities
摘要：
矩阵秩和惯性最优化问题是一类不连续的最优化问题，其中决策变量是在一定可行矩阵集上运行的矩阵，而变量矩阵的秩和惯性是整值目标函数。本文建立了Löwner偏序中埃尔米特矩阵表达式$A_1 - B_1XB_1^{*}$满足线性矩阵不等式$B_2XB_2^{*} \succcurlyeq A_2$ $(B_2XB_2^{*} \preccurlyeq A_2)$的秩目标函数和惯性目标函数的最大值和最小值的一组显式计算公式，并给出了这些公式在某些约束矩阵值函数的性质刻画中的应用.
英文摘要：
Matrix rank and inertia optimization problems are a class of discontinuous optimization problems, in which the decision variables are matrices running over certain feasible 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 expression $A_1 - B_1XB_1^{*}$ subject to the linear matrix inequality $B_2XB_2^{*} \succcurlyeq A_2$ $(B_2XB_2^{*} \preccurlyeq A_2)$ in the Löwner partial ordering, and give applications of these formulas in characterizing behaviors of some constrained matrix-valued functions.
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