2020/05/17 by K. C. Sivakumar, Sivakumar, K. C., P. Sushmitha +3
Mathematics · #15A09 #90C33 #FOS: Mathematics #Optimization and Control (math.OC) #math.OC #msc:15A09 #msc:90C33
paper · pdf · doi:10.48550/arxiv.2005.08171
arxiv created 2021/01/16 · arxiv updated 2021/01/19
A real square matrix A is called a Q-matrix if the linear complementarity problem LCP(A,q) has a solution for all q ∈ ℝn. This means that for every vector q there exists a vector x such that x ≥ 0, y=Ax+q≥ 0 and xTy=0. A well known result of Karamardian states that if the problems LCP(A,0) and LCP(A,d) for some d∈ ℝn, d >0 have only the zero solution, then A is a Q-matrix. By relaxing the condition on d and imposing a condition on the solution vector x in the two problems as above, the authors introduce a new class of matrices called Karamardian matrices, requiring that these two modified problems have only zero as a solution. In this article, a systematic treatment of Karamardian matrices is undertaken. Among other things, it is shown how Karamardian matrices have properties that are analogous to those of Q-matrices. A subclass of a recently introduced notion of P#-matrices is shown to possess the Karamardian property, and for this reason we undertake a thorough study of P#-matrices and make some fundamental contributions.