Operations Research Transactions ›› 2025, Vol. 29 ›› Issue (3): 160-178.doi: 10.15960/j.cnki.issn.1007-6093.2025.03.008

Special Issue: 第九届中国运筹学会科学技术奖获奖者专辑

• Research Article • Previous Articles     Next Articles

Real pairwise completely positive matrices

Anwa ZHOU1,*(), Jiayi HE1   

  1. 1. College of Sciences, Shanghai University, Shanghai 200444, China
  • Received:2025-03-07 Online:2025-09-15 Published:2025-09-09
  • Contact: Anwa ZHOU E-mail:zhouanwa@shu.edu.cn

Abstract:

In this paper, we introduce the real pairwise completely positive (RPCP) matrices with one of them is necessarily positive semidefinite while the other one is necessarily entrywise nonnegative, which has a real pairwise completely positive (RPCP) decomposition. We study the properties of RPCP matrices and give some necessary and sufficient conditions for a matrix pair to be RPCP. First, we give an equivalent decomposition for the RPCP matrices, which is different from the RPCP-decomposition and show that the matrix pair (X, X) is RPCP if and only if X is completely positive. Besides, we also prove that the RPCP matrices checking problem is equivalent to the separable completion problem. A semidefinite algorithm is also proposed for detecting whether or not a matrix pair is RPCP. The asymptotic and finite convergence of the algorithm are also discussed. If it is RPCP, we can further give a RPCP-decomposition for it; if it is not, we can obtain a certificate for this.

Key words: real pairwise completely positive matrices, truncated moment problem, semidefinite relaxation

CLC Number: