Analytical solutions to some optimization problems on ranks and inertias of matrix-valued functions subject to linear matrix inequalities

2013/01/06 by Yongge Tian, Tian, Yongge
Computer Science · Engineering · Mathematics · #15A24 #15A39 #15A45 #15B57 #49K30 #65K10 #90C11 #90C22 #Advanced Optimization Algorithms Research #FOS: Mathematics #Matrix Theory and Algorithms #Optimization and Control (math.OC) #Stability and Control of Uncertain Systems #math.OC #msc:15A24 #msc:15A39 #msc:15A45 #msc:15B57 #msc:49K30 #msc:65K10 #msc:90C11 #msc:90C22

paper · pdf · doi:10.48550/arxiv.1301.0986

22pages

arxiv created 2013/01/06 · openalex publication_date 2013/01/06 · arxiv updated 2013/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 A₁ - B₁XB₁^* subject to the linear matrix inequality B₂XB₂^* \succcurlyeq A₂ (B₂XB₂^* \preccurlyeq A₂) in the Löwner partial ordering, and give applications of these formulas in characterizing behaviors of some constrained matrix-valued functions.

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