Recently, it has been shown that Nonlinear Programming solvers can successfully solve a range of Mathematical Programs with Equilibrium Constraints (MPECs).

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...

We present polynomial-time interior-point algorithms for solving the Fisher and Arrow-Debreu competitive market equilibrium problems with linear utilities and n players. Both of them have the arithmetic operation complexity bound of O(n 4 log(1/ɛ)) for computing an ɛ-equilibrium solution. If the problem data are rational numbers and their bit-length is L, then the bound to generate an exact solution is O(n 4 L) which is in line with the best complexity bound for linear programming of the same dimension and size. This is a significant improvement over the previously best bound O(n 8 log(1/ɛ)) for approximating the two problems using other methods. The key ingredient to derive these results is to show that these problems admit convex optimization formulations, efficient barrier functions and fast rounding techniques. We also present a continuous path leading to the set of the Arrow-Debreu equilibrium, similar to the central path developed for linear programming interior-point methods. This path is derived from the weighted logarithmic utility and barrier functions and the Brouwer fixed-point theorem. The defining equations are bilinear and possess some primal-dual structure for the application of the Newton-based path-following method.

Variational inequalities over sets defined by systems of equalities and inequalities are considered. A continuously differentiable merit function is proposed whose unconstrained minima coincide with the KKT-points of the variational inequality. A detailed study of its properties is carried out showing that under mild assumptions this reformulation possesses many desirable features. A simple algorithm is proposed for which it is possible to prove global convergence and a fast local convergence rate. Preliminary numerical results showing viability of the approach are reported.

QPCOMP is an extremely robust algorithm for solving mixed nonlinear complementarity problems that has fast local convergence behavior. Based in part on the NE/SQP method of Pang and Gabriel[14], this algorithm represents a significant advance in robustness at no cost in efficiency. In particular, the algorithm is shown to solve any solvable Lipschitz continuous, continuously differentiable, pseudo-monotone mixed nonlinear complementarity problem. QPCOMP also extends the NE/SQP method for the nonlinear complementarity problem to the more general mixed nonlinear complementarity problem. Computational results are provided, which demonstrate the effectiveness of the algorithm. 1 Introduction This paper describes a new algorithm for solving the mixed nonlinear complementarity problem (MCP), which provides a significant improvement in robustness over previous superlinearly or quadratically convergent algorithms, while preserving these fast local convergence properties. The MCP is defined in...

Conjectured supply function (CSF) models of competition among power generators on a linearized DC network are presented. As a detailed survey of the power market modeling literature shows, CSF models differ from previous approaches in that they represent each GenCo's conjectures regarding how rival firms will adjust sales in response to price changes. The CSF approach is a more realistic and flexible framework for modeling imperfect competition than other models for three reasons. First, the models include as a special case the Cournot conjecture that rivals will not change production if prices change; thus, the CSF framework is more general. Second, Cournot models cannot be used when price elasticity of demand is zero, but the proposed models can. Third, unlike supply function equilibrium models, CSF equilibria can be calculated for large transmission networks. Existence and uniqueness properties for prices and profits are reported. An application shows how transmission limits and strategic interactions affect equilibrium prices under forced divestment of generation.

In this paper we take a new look at smoothing Newton methods for solving the nonlinear complementarity problem (NCP) and the box constrained variational inequalities (BVIs). Instead of using an infinite sequence of smoothing approximation functions, we use a single smoothing approximation function and Robinson's normal equation to reformulate the NCP and the BVIs as an equivalent nonsmooth equation H(u, x) = 0, where H : # 2n # # 2n , u # # n is a parameter variable and x # # n is the original variable. The central idea of our smoothing Newton methods is that we construct a sequence {z k = (u k , x k )} such that the mapping H(·) is continuously di#erentiable at each z k and may be non-di#erentiable at the limiting point of {z k }. We prove that three most often used Gabriel-More smoothing functions can generate strongly semismooth functions, which play a fundamental role in establishing superlinear and quadratic convergence of our new smoothing Newton methods....

A new algorithm for the solution of large-scale nonlinear complementarity problems is introduced. The algorithm is based on a nonsmooth equation reformulation of the complementarity problem and on an inexact Levenberg-Marquardt-type algorithm for its solution. Under mild assumptions, and requiring only the approximate solution of a linear system at each iteration, the algorithm is shown to be both globally and superlinearly convergent, even on degenerate problems. Numerical results for problems with up to 10000 variables are presented. 1 Introduction We consider the complementarity problem NCP(F ), which is to find a vector in IR n satisfying the conditions x 0; F (x) 0; x T F (x) = 0; where F : IR n ! IR n is a continuously differentiable function. Nonlinear complementarity problems have important applications, see, e.g., [11,19], which often call for the solution of large-scale problems. During the last few years many methods have been developed for the solution of the non...

. This paper provides a means for comparing various computer codes for solving large scale mixed complementarity problems. We discuss inadequacies in how solvers are currently compared, and present a testing environment that addresses these inadequacies. This testing environment consists of a library of test problems, along with GAMS and MATLAB interfaces that allow these problems to be easily accessed. The environment is intended for use as a tool by other researchers to better understand both their algorithms and their implementations, and to direct research toward problem classes that are currently the most challenging. As an initial benchmark, eight different algorithm implementations for large scale mixed complementarity problems are briefly described and tested with default parameter settings using the new testing environment. Keywords: complementarity problems, variational inequalities, computation, algorithms 1. Introduction In recent years, a considerable number of new algori...

A variational inequality problem with a mapping g : ! n ! ! n and lower and upper bounds on variables can be reformulated as a system of nonsmooth equations F (x) = 0 in ! n . Recently, several homotopy methods, such as interior-point and smoothing methods, have been employed to solve the problem. All of these methods use parametric functions and construct perturbed equations to approximate the problem. The solution to the perturbed system constitutes a smooth trajectory leading to the solution of the original variational inequality problem. The methods generate iterates to follow the trajectory. Among these methods Chen-Mangasarian and Gabriel-Mor'e proposed a class of smooth functions to approximate F . In this paper, we study several properties of the trajectory defined by solutions of these smooth systems. We propose a homotopy-smoothing method for solving the variational inequality problem, and show that the method converges globally and superlinearly under mild conditions. ...