Abstract. This paper studies algorithms for equilibrium problems with equilibrium constraints (EPECs). We present a generalization of Scholtes’s regularization scheme for MPECs and extend his convergence results to this new relaxation method. We propose a sequential nonlinear complementarity (SNCP) algorithm to solve EPECs and establish the convergence of this algorithm. We present numerical results comparing the SNCP algorithm and diagonalization (nonlinear Gauss-Seidel and nonlinear Jacobi) methods on randomly generated EPEC test problems. The computational experience to date shows that both the SNCP algorithm and the nonlinear Gauss-Seidel method outperform the nonlinear Jacobi method. 1

Abstract. We consider a mathematical program with complementarity constraints (MPCC). Our purpose is to develop methods that enable us to compute a solution or a point with some kind of stationarity to MPCC by solving a finite number of nonlinear programs. We apply an active set identification technique to a smoothing continuation method (Fukushima and Pang, 1999) and propose a hybrid algorithm for solving MPCC. We also develop two kinds of modifications, one of which makes use of an index addition strategy and the other adopts an index subtraction strategy. We show that, under reasonable assumptions, all the proposed algorithms possess a finite termination property. Further discussions and computational results are given as well. Key Words. mathematical program with complementarity constraints, MPCC-LICQ, weak secondorder necessary condition, (B-, M-, C-) stationarity, asymptotically weak nondegeneracy, identification function. 1

Introduction. The piecewise sequential quadratic programming (PSQP) method is a numerical method for solving certain mathematical programs with equilibrium constraints (MPEC), based on the classical sequential quadratic programming (SQP) method for nonlinear programs (NLP) [2, 12]. This description draws on both [9] and [4], which extend the original proposal for PSQP [13] that was restricted to the case of MPEC with linear complementarity constraints. See [7] for a brief account of an application of PSQP to a problem in civil engineering. It's performance on randomly generated quadratic programs with affine equilibrium constraints is documented in [4] and also in [9, 10]. PSQP can be applied directly to any MPEC whose lower-level problem is a mixed complementarity problem, and indirectly to any MPEC where the lower-level problem is a variational inequality (VI) that can be written via its Karush-Kuhn-Tucker (KKT

We describe several new tools for modeling MPEC problems that are built around the introduction of an MPEC model type into the GAMS language. We develop subroutines that allow such models to be communicated directly to MPEC solvers. This library of interface routines, written in the C language, provides algorithmic developers with access to relevant problem data, including for example, function and Jacobian evaluations. A MATLAB interface to the GAMS MPEC model type has been designed using the interface routines. Existing MPEC models from the literature have been written in GAMS, and computational results are given that were obtained using all the tools described. Keywords Complementarity, Algorithm, MPEC, Modeling 1 Introduction The Mathematical Program with Equilibrium Constraints (MPEC) arises when one seeks to optimize an objective function subject to equilibrium contraints. These equilibrium constraints may take the form of a variational inequality or complementarity problem, o...