Recent improvements in the capabilities of complementarity solvers have led to an increased interest in using the complementarity problem framework to address practical problems arising in mathematical programming, economics, engineering, and the sciences. As a result, increasingly more difficult problems are being proposed that exceed the capabilities of even the best algorithms currently available. There is, therefore, an immediate need to improve the capabilities of complementarity solvers. This thesis addresses this need in two significant ways. First, the thesis proposes and develops a proximal perturbation strategy that enhances the robustness of Newton-based complementarity solvers. This strategy enables algorithms to reliably find solutions even for problems whose natural merit functions have strict local minima that are not solutions. Based upon this strategy, three new algorithms are proposed for solving nonlinear mixed complementarity problems that represent a significant improvement in robustness over previous algorithms. These algorithms have local Q-quadratic convergence behavior, yet depend only on a pseudo-monotonicity assumption to achieve global convergence from arbitrary starting points. Using the MCPLIB and GAMSLIB test libraries, we perform extensive computational tests that demonstrate the effectiveness of these algorithms on realistic problems. Second, the thesis extends some previously existing algorithms to solve more general problem classes. Specifically, the NE/SQP method of Pang & Gabriel (1993), the semismooth equations approach of De Luca, Facchinei & Kanz...

Three generalizations of the criss-cross method for quadratic programming are presented here. Tucker's, Cottle's and Dantzig's principal pivoting methods are specialized as diagonal and exchange pivots for the linear complementarity problem obtained from a convex quadratic program. A finite criss-cross method, based on least-index resolution, is constructed for solving the LCP. In proving finiteness, orthogonality properties of pivot tableaus and positive semidefiniteness of quadratic matrices are used. In the last section some special cases and two further variants of the quadratic criss-cross method are discussed. If the matrix of the LCP has full rank, then a surprisingly simple algorithm follows, which coincides with Murty's `Bard type schema' in the P matrix case.

A combinatorial abstraction of the linear complementarity theory in the setting of oriented matroids was rst considered by M.J. Todd. In this paper, we take a fresh look at this abstraction, and attempt to give a simple treatment of the combinatorial theory of linear complementarity. We obtain new theorems, proofs and algorithms in oriented matroids whose specializations to the linear case are also new. For this, the notion of suciency of square matrices, introduced by Cottle, Pang and Venkateswaran, is extended to oriented matroids. Then, we prove a sort of duality theorem for oriented matroids, which roughly states: exactly one of the primal and the dual system has a complementary solution if the associated oriented matroid satisfies "weak" sufficiency. We give two different proofs for this theorem, an elementary inductive proof and an algorithmic proof using the criss-cross method which solves one of the primal or dual problem by using surprisingly simple pivot rules (without any pertur...

Specially structured Linear Complementarity Problems (LCP's) and their solution by the criss-cross method are examined in this paper. The criss-cross method is known to be finite for LCP's with positive semidefinite bisymmetric matrices and with P-matrices. It is also a simple finite algorithm for oriented matroid programming problems. Recently Cottle, Pang and Venkateswaran identified the class of (column, row) sufficient matrices. They showed that sufficient matrices are a common generalization of P- and PSD-matrices. Cottle also showed that the principal pivoting method (with a clever modification) can be applied to row sufficient LCP's. In this paper the finiteness of the criss-cross method for sufficient LCP's is proved. Further it is shown that a matrix is sufficient if and only if the criss-cross method processes all the LCP's defined by this matrix and all the LCP's defined by the transpose of this matrix and any parameter vector.

A splitting method for solving LCP based models of dry frictional contact problems in rigid multibody systems based on box MLCP solver is presented. Since such methods rely on fast and robust box MLCP solvers, several methods are reviewed and their performance is compared both on random problems and on simulation data. We provide data illustrating the convergence rate of the splitting method which demonstrates that they present a viable alternative to currently available methods.

A general framework of nite algorithms is presented here for quadratic programming. This algorithm is a direct generalization of Van der Heyden's algorithm for the linear complementarity problem and Jensen's `relaxed recursive algorithm', which was proposed for solution of Oriented Matroid programming problems. The validity of this algorithm is proved the same way as the finiteness of the criss-cross method is proved. The second part of this paper contains a generalization of Edmonds-Fukuda pivoting rule for quadratic programming. This generalization can be considered as a finite version of Van de Panne - Whinston algorithm and so it is a simplex method for quadratic programming. These algorithms uses general combinatorial type ideas, so the same methods can be applied for oriented matroids as well. The generalization of these methods for oriented matroids is a subject of another paper.

In this chapter we discuss several methods for solving the LCP based on principal pivot steps. One common feature of these methods is that they do not introduce any arti cial variable. These methods employ either single or double principal pivot steps, and are guaranteed to process LCPs associated with P-matrices or PSD-matrices or both. We consider the LCP (q � M) oforder n, which is the following in tabular form. w z q I;M q w � z> 0 � w T z =0 (4:1)