Analytic solutions of a class of extended constrained matrix maximization problems

摘要

In this paper we extend key Lemma 3.1 of Xu et al. (2018) and Theorem 3.5 of Xu et al. (2019) to more general cases and propose analytic solutions of the following extended constrained matrix maximization problems:maxUkUkH=In,VkVkH=Im|det(cIm+∏k=1sΓkUkΔkVk)|,maxUkUkH=In,VkVkH=Im|tr(cIm+∏k=1sΓkUkΔkVk)|,whereΓk=(ΓrkOr×(n−r)O(m−r)×rO(m−r)×(n−r))∈Cm×n,Δk=(ΔrkOr×(m−r)O(n−r)×rO(n−r)×(m−r))∈Cn×mwithΓrk=diag(γ1k,⋯,γrk),Δrk=diag(δ1k,⋯,δrk),r=min{m,n},γik,δik∈R,k=1,…,s;i=1,…,rand c is a complex number, Im denotes the m-order identity matrix, Γk, Δk are m × n and n × m complex matrices, γik,δik are real numbers, and det(·),tr(·) denote the matrix determinant and trace functions. This is a non-convex nonlinear constrained matrix maximization problem. The new results improve the corresponding existing ones in the above two references.