Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Delhi)


Parthasarthy, T. (TSMU-Delhi; ISI)

Abstract (Summary of the Work)

This dissertation deals with a number of questions related to the linear complementarity problem (LCP). Given A ∈ Rn*n and q ∈ Rnthe LCP is to find a vector z ∈ R" such that Az+q ≥0,≥ and 2'(Az + 9) = 0. There is a vast literature on LCP developed during the last four decades. LCP plays a crucial role in the study of Mathematical Progranming from the point of view of algorithms as well as applications. The questions on existence and multiplicity of solutions in LCP has led researchers to introduce and study a variety of matrix classes. Most of the work of this dissertation pertains to LCP within the class of Lipschitzian matrices. Besides, various interesting results on INS, adequate and connected ma- trices are also presented. The gist of the dissertation is presented belo in a chapter-wise summary.In Chapter 1, we introduce LCP along with some fundamental concepts and results related to it. In Section 1.1 we state the LCP with a brief historical background and discuss its importance in the applied field. In Section 1.2 we present some basic concepts and results. fundamental to the study of matrix classes in LCP. In Section 1.3 we briefiy discuss the basics of game theory, as a prerequisite for Chapter 5. In Chapter 2 results pertaining to Lipschitzian matrices are presented. Mangasarian and Shiau [25] were the first to consider Lipschitz continuity of solution maps of LCP. In this chapter, we present a number of interest- ing consequences of property (**) introduced by Murthy, Parthasarathy and Sriparna (36). In [35), Murthy, Parthasarathy and Sabatini showed that Lipschitzian Q.º matrices satisfy property (**). In Section 2.3 it is shown that property (**) is sufficient for a Lipschitzian matrix to be in Qº.. Further if A has this property, then A and all its PPTS must be completely Qº and for any q, the linear complementarity problem (q. A) can be processed by a simple principal pivoting method. We deduce, that property (**) characterizes negative N matrices and P matrices, with positive value. In Section 2.4 we study the properties of Lipschitzian ma- trices in general. We establish, that the Lipschitzian property is inherited by all the principal submatrices. We prove, that Lipschitzian matrices, subject to a principal rearrangement, has the block structure. In Chapter 3 we mainly discuss the INS class of matrices, introduced by Stone (53], and using results on this topic we obtain some important results concerning Lipschitzian matrices. In (36), Murthy, Parthasarathy and Sriparna con jectured that nondegenerate matrices satisfying property (**) are Lipschitzian. We settle this conjecture affirmatively and deduce some necessary conditions on the nondegenerate INS class. In answer to a question raised by Stone in [54), we prove that the class of nondegen- crate INS matrices is complete. We conjecture that the block property is sufficient for a matrix to be Lipschitzian and prove it for some special cases.


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