Direct algorithms for complementarity problems

Abstract

This thesis deals with three types of complementarity problems: linear complementarity problems, nonlinear complementarity problems and generalized nonlinear complementarity problems. The emphasis is on algorithms for solving complementarity problems using a "directll approach. In this direct approach a finite sequence of systems of equations are solved approximately. The final system of equations considered represents the complementarity condition at the solution The algorithms gain efficiency because: 1) intermediate systems of equations are of reduced dimensionality and 2) high accuracy solutions to intermediate systems are unnecessary.

A set of numerical experiments are presented and performance of direct algorithms is assayed.

Next considered is the relationship between direct algorithms and certain systems of equations which are equivalent to a complementarity problem. In special cases, the application of Newton's Method to these equations produces the same sequence of points as a direct algorithm. Those considered are the algorithms of Bard, Chandrasekaran and Murty.

Pigeonholing of complementary points is used to introduce a family of direct algorithms which are capable of solving the generalized nonlinear complementarity problem for the case when the cone K is an orthant of Rⁿ. The algorithm known as Murty's Bard-Type Scheme is extended to solve the same problem. This is accomplished by considering a class of functions denoted by P combinatorial homeomorphisms. These functions-generalize the affine P-functions.

A property of certain nonlinear functions (a subclass of the P-functions) is introduced and denoted by "pigeon holing of complementary points". Pigeonholing occurs in two classes of functions which arise in the literature: differentiable functions having a positively bounded Jacobian matrix and uniform P-functions on Rⁿ. In the case of an affine function, pigeonholing occurs if and only if the coefficient matrix has positive principal minors. A characterization of P-matrices first presented by Murty is discussed and refined.