##### Author

Kostreva, Michael M.

##### Other Contributors

Habetler, George Joseph; Lemke, Carlton E.; Rogers Edwin H.; Arden, Dean N.;

##### Date Issued

1976-08

##### Subject

Mathematics

##### Degree

PhD;

##### Terms of Use

This electronic version is a licensed copy owned by Rensselaer Polytechnic Institute, Troy, NY. Copyright of original work retained by author.;

##### 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.;

##### Description

August 1976; School of Science

##### Department

Dept. of Mathematical Sciences;

##### Publisher

Rensselaer Polytechnic Institute, Troy, NY

##### Relationships

Rensselaer Theses and Dissertations Online Collection;

##### Access

Restricted to current Rensselaer faculty, staff and students. Access inquiries may be directed to the Rensselaer Libraries.;