On generalizations of positive subdefinite matrices and the linear complementarity problem
TL;DR: The notion of generalized positive subdefinite matrices of level k is introduced by generalizing the definition of generalized Positive Subdefinite Matrices introduced by Crouzeix and Komlósi in [Applied optimization].
Abstract: In this paper, we introduce the notion of generalized positive subdefinite matrices of level k by generalizing the definition of generalized positive subdefinite matrices introduced by Crouzeix and Komlosi in [Applied optimization. Vol. 59, Dordrecht: Kluwer Academic Publications; 2001. p. 45–63]. The main motivation behind this generalization is to identify new subclasses of row sufficient matrices [Cottle, Pang and Venkateswaran, Linear Algebra Appl. 1989;114/115:231–249] which are not a subclass of copositive matrices. We establish new sufficient conditions for a matrix to be a non-copositive matrix and a non-copositive row sufficient matrix using the notion of generalized positive subdefinite matrices of level k. We present a new termination result for Lemke’s algorithm which supplements Jones’ result [Math Program. 1986;35:239–242].
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TL;DR: In this paper, the authors introduce the class of column sufficient matrices, which are the transpose of a matrix M such that for every vector q, the solutions of the linear complementarity problem are identical to the Karush-Kuhn-Tucker points associated with ( q, M ).
131 citations
TL;DR: The convergence conditions of modulus-based matrix splitting and matrix two-stage splitting iteration methods for linear complementarity problems are weakened, and their applied scopes are further extended.
76 citations
TL;DR: In this article, finite criteria for copositive matrices are proposed and compared with existing determinantal tests, including principal pivoting and principal pivot-based matrices, and the basic mathematical tool is principal pivot.
64 citations
TL;DR: In this paper, the authors define a new class of real symmetric matrices, which they call subdefinite, and investigate the properties of these matrices from a practical point of view.
Abstract: Recent results in quasiconvex and pseudoconvex programming [1], [3], [4] call the attention to these classes of functions. Little work has been done, however, in order to make easier the recognition of these types of functions. The definitions of quasiconvexity and pseudoconvexity do not offer a useful test from a practical point of view. In this paper one of the simplest problems of this kind is investigated, that of a quasiconvex, or pseudoconvex quadratic form. The connection of this problem with quadratic programming is obvious. The problem leads to a new class of real symmetric matrices, which we call subdefinite. After introducing the main definitions in the first section, we look for the characteristics of merely subdefinite (not semidefinite) matrices (? 2). The third section deals with the convexity properties of subdefinite quadratic forms. A few numerical examples are attached.
63 citations
"On generalizations of positive subd..." refers background in this paper
...The class of symmetric PSBDmatrices is introduced byMartos [3] while characterizing generalized convex quadratic functions and was further studied by Cottle and Ferland [4]....
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TL;DR: A relaxation modulus-based matrix splitting iteration method is established, which covers the known general modulus based matrix splitting iterations methods and is efficient and accelerate the convergence performance with less iteration steps and CPU times.
Abstract: In this paper, a relaxation modulus-based matrix splitting iteration method is established, which covers the known general modulus-based matrix splitting iteration methods. The convergence analysis and the strategy of the choice of the parameters are given. Numerical examples show that the proposed methods are efficient and accelerate the convergence performance with less iteration steps and CPU times.
53 citations