## Doctoral Theses

### Properties of Some Matrix Classes In Linear Complementarity Theory.

11-22-2005

11-22-2006

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

SQC and OR Unit (Delhi)

#### Supervisor

Neogy, S. K. (SQCOR-Delhi; ISI)

#### Abstract (Summary of the Work)

The linear complementarity problem is a fundamental problem that arises in optimization, game theory, economics, and engineering. It can be stated as follows:Given a square matrix A of order n with real entries and an n dimensional vector q, find n dimensional vectors w and z satisfying w âˆ’ Az = q, w â‰¥ 0, z â‰¥ 0 (1.1.1) w t z = 0. (1.1.2)This problem is denoted as LCP(q, A). The name comes from the condition (1.1.2), the complementarity condition which requires that at least one variable in the pair (wj , zj ) should be equal to 0 in the solution of the problem, for each j = 1, 2, . . . , n. This pair is therefore known as the jth complementary pair in the problem, and for each j, the variable wj is known as the complement of zj and vice versa. If a pair of vectors (w, z) satisfies (1.1.1), then the problem LCP(q, A) is said to have a feasible solution. A pair (w, z) of vectors satisfying (1.1.1) and (1.1.2) is called a solution to the LCP(q, A). The problem has undergone several name changes, from composite problem to complementary pair in the problem. The current name linear complementarity problem was proposed by Cottle [7, p. 37]. The LCP is normally identified as a problem of mathematical programming and provides a unifying framework for several optimization problems like linear programming, linear fractional programming, convex quadratic programming and the bimatrix game problem. More specifically, the LCP models the optimality conditions of these problems. It is well studied in the literature on mathematical programming and a number of applications are reported in operations research [29], multiple objective programming problem [50], mathematical economics [78], geometry and engineering ([12], [26] and [79]). Some new applications of the linear complementarity problem have been reported in the area of stochastic games. For details, see the survey paper by Mohan, Neogy and Parthasarathy [53] and the references cited therein. This sort of applications and the potential for future applications have motivated the study of the LCP, especially the study of the algorithms for the LCP and the study of matrix classes. In fact, much of linear complementarity theory and algorithms are based on the assumption that the matrix A belongs to a particular class of matrices. The early motivation for studying the linear complementarity problem was that the KKT optimality conditions for linear and quadratic programs reduce to an LCP of the form given by (1.1.1) and (1.1.2). The algorithm presented by Lemke and Howson [42] to compute an equilibrium pair of strategies to a bimatrix game, later extended by Lemke [41] (known as Lemkeâ€™s algorithm) to solve an LCP(q, A), contributed significantly to the development of the linear complementarity theory. In fact, the study of the LCP really came into prominence only when Lemke and Howson [42] and Lemke [41] showed that the problem of computing a Nash equilibrium General Introduction and Some Basic Concepts 3 point of a bimatrix game can be posed as an LCP following the publication by Cottle [1].

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842822

ISILib-TH345