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-Kolkata)


Arthanari, T. S.

Abstract (Summary of the Work)

This dissertation deals with a number of problems related to linear comple- mentarity problem (LCP). Given a real square matrix A of order n and a real n-vector q, the LCP is to find a nonnegative n-vector z such that Az + q 2 0 and zt(Az + 9) = 0. There is vast literature on LCP, evolved during the last four decades. LCP plays a crucial role in the study of mathematical program- ming from the view point of algorithms as well as applications. The inherent nature of the problem has led the researchers to introduce and study a variety of matrix classes in connection with LCP. Most of the work of this dissertation pertains to LCP within the class of semimonotone matrices (E.) introduced by Eaves. Besides, results on the matrix classes Q and Qo, which are of fundamen- tal interest in the theory of LCP, are presented. The usefulness of these results is demonstrated through a number of applications. The gist of the dissertation is presented below in a chapter-wise summary. Chapter 1 is introduciory in nature and presents LCP and related material that is needed for the discussion in the subsequent chapters. In each,of the chapters from 2 to 5, the first section introduces the background of the contents of that chapter while the second section presents the related results from the literature; the subsequent sections are devoted to our work pertaining to that chapter.Al-Khayyal (1991) specified a condition (through polyhedral sets) for a matrix to belong to P. and raised the question whether the same could be sufficient for membership in Q.. In Chapter 2, while answering his question in the affirmative, the author further relaxed the condition and obtained a new one which is sufficient for membership in Q.. It has also been established that, if A ϵ RnXn satisfies the relaxed condition and g is such that (g, A) has a feasible solution, then the solutions of (g, A) can be obtained by solving a suitable linear programming problem (LPP). Examples of matrices which satisfy the relaxed condition have been provided. The problem of solving LCP as LPP has been studied at length by several authors. It is known that when LPPS have solutions, they have solutions with bases. However, this is not so with LCP. A linear complementarity problem may have a solution but may not have a complementary basis. The author analyses this aspect in Section 2.4 and provides a sufficient condition under which the existence of complementary bases can be guaranteed. In Chapter 3, results pertaining to the matrix classes Q and Q. are pre- sented. Though a number of subelasses of Q and Q, have been identified, the problem of testing whether a given matrix A belongs to Q or Q., in general, remains complex. In this regard, we present some elementary propositions providing sufficient conditions under which the membership in Q can be as- serted. The usefulness of these propositions is demonstrated through a number of applications. Section 3.4 is devoted to our results on Q.-matrices. Many of these results are analogies of known results on Q-matrices. The main re- sults are : characterization of nonnegative Q.-matrices, necessary and sufficient conditions, in some special cases, for a matrix to be completely Q., sufficient conditions for principal submatrices of order (n- 1) of n x n to be in Q., and necessary condlitions on Q.-matrices in some special cases. As an application of characterization of nonnegative Q. matrices, it is established that, if A+ A' is a nonnegative Q,-matrix, both A and At are Q.-matrices. Results in Chapter 4 are concerned with the class of semimonotone matrices introduced by Eaves (1971). In Section 4.3, we settle a conjecture initially posed by Pang (1979) and later modified by Jeter and Pye (1984) and Gowda (1990). The conjecture in its modified version states that every copositive Q-matrix is an R,-matrix. A counter example is constructed to show that the conjecture is false. We derive conditions under which copositive (semimonotone) Q-matrices belong to R. It is well known that symmetric semimonotone Q-matrices are completely Q. We show, in Section 4.6, that symmetric semimonotone Q.- matrices are completely Qo. Another important result in this section is the extension of Pang's result on E,nQ-matrices which states that if A is in E,nQ., then every nontrivial solution of (0, A) has at least two nonzero coordinates. It is established that if A is a E, n Q.-matrix and if every row of A has a positive entry, then every nontrivial solution of (0, A) has at least two nonzero coordinates.The results in Chapter 5 pertain to the class of fully semimonotone (E) matrices introduced by Cottle and Stone (1983). Stone (1981) proved that, within the class of Q.-matrices, the U-matrices are P,-matrices, and conjec- tured that the same must be true for Ef. It is established that this conjecture is true for matrices of order upto 4 x 4, and in a number of special cases (of any order). The special cases include Ef-matrices which are either symmetric or nonnegative or copositive-plus or Z-martrices or E-matrices. In the sequel we introduce a subclass of E, the class of fully copositive (C!) matrices, and show that Cf ∩Q.S P..


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