tailieunhanh - Extensions of P-roperty, r0-property and semidefinite linear complementarity problems

In this manuscript, we present some new results for the semidefinite linear complementarity problem in the context of three notions for linear transformations, viz., pseudo w-P property, pseudo Jordan w-P property and pseudo SSM property. | Yugoslav Journal of Operations Research 27 (2017), Number 2, 135–151 DOI: EXTENSIONS OF P-PROPERTY, R0 -PROPERTY AND SEMIDEFINITE LINEAR COMPLEMENTARITY PROBLEMS I. JEYARAMAN Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Surathkal - 575 025, India i jeyaraman@ Kavita BISHT Department of Mathematics,Indian Institute of Technology Madras, Chennai - 600 036, India kavitabishtiitm2512@ . SIVAKUMAR Department of Mathematics,Indian Institute of Technology Madras, Chennai - 600 036, India kcskumar@ Received: January 2017 / Accepted: May 2017 Abstract: In this manuscript, we present some new results for the semidefinite linear complementarity problem in the context of three notions for linear transformations, viz., pseudo w-P property, pseudo Jordan w-P property, and pseudo SSM property. Interconnections with the P# -property (proposed recently in the literature) are presented. We also study the R# -property of a linear transformation, extending the rather well known notion of an R0 -matrix. In particular, results are presented for the Lyapunov, Stein, and the multiplicative transformations. Keywords: Linear Complementarity Problem, P-property, R-property, Semidefinite Linear Complementarity Problem, w-P properties, Jordan w-P property, Moore-Penrose Inverse. MSC: 90C33, 15A09. 136 , , / Extensions of P-property 1. INTRODUCTION Let Sn denote the vector space of all n × n real symmetric matrices and Sn+ be the set of all symmetric positive semidefinite matrices in Sn . Given a linear transformation L : Sn → Sn and a matrix Q ∈ Sn , the semidefinite linear complementarity problem, denoted by SDLCP(L, Q), is to find an X ∈ Sn such that X ∈ Sn+ , Y = L(X) + Q ∈ Sn+ , and hX, Yi = tr(XY) = 0, where tr(A) denotes the trace of the square matrix A. If such an X exists, we call X, a solution of SDLCP(L, Q). The set of all

• Toán học
• Linear complementarity problem
• P property
• R property
• Semidefinite linear complementarity problem
• W P properties
• Jordan w P property
• Moore penrose inverse
• Scarf’s complementarity problem
• Vertical linear complementarity problem
• CottleDantzig’s algorithm
• Lemke’s algorithm
• Neural network approach
• Horizontal linear complementarity problem (HLCP)
• Infeasible interiorpoint method
• Central path
• Horizontal linear