Augmented Lagrangian algorithms are very popular tools for solving nonlinear programming problems. At each outer iteration of these methods a simpler optimization problem is solved, for which ecient algorithms can be used, especially when the problems are large. The most famous Augmented Lagrangian algorithm for minimization with inequality constraints is known as Powell-Hestenes-Rockafellar (PHR) method. The main drawback of PHR is that the objective function of the subproblems is not twice continuously dierentiable. This is the main motivation for the introduction of many alternative Augmented Lagrangian methods.

Mathematical programming problems with equilibrium constraints (MPEC) are nonlinear programming problems where the constraints have a form that is analogous to first-order optimality conditions of constrained optimization. We prove that, under reasonable sufficient conditions, stationary points of the sum of squares of the constraints are feasible points of the MPEC. In usual formulations of MPEC all the feasible points are nonregular in the sense that they do not satisfy the Mangasarian-Fromovitz constraint qualification of nonlinear programming. Therefore, all the feasible points satisfy the classical Fritz-John necessary optimality conditions. In principle, this can cause serious difficulties for nonlinear programming algorithms applied to MPEC. However, we show that most feasible points do not satisfy a recently introduced stronger optimality condition for nonlinear programming. This is the reason why, in general, nonlinear programming algorithms are successful when applied to MPEC. Keywords. Mathematical programming with equilibrium constraints, optimality conditions, minimization algorithms, reformulation. AMS: 90C33, 90C30

v Acknowledgements vii List of Tables ix List of Figures x Glossary of Symbols xi 1 Notation, Problem Definition and Review of Classical Techniques 1 1.1 Notation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.2 Nonlinear Programming : : : : : : : : : : : : : : : : : : : : : : : 3 1.3 Block Angular Programs : : : : : : : : : : : : : : : : : : : : : : : 6 1.4 Dantzig-Wolfe Decomposition : : : : : : : : : : : : : : : : : : : : 10 1.5 Simplicial Decomposition : : : : : : : : : : : : : : : : : : : : : : : 12 1.6 Resource Directed Decomposition : : : : : : : : : : : : : : : : : : 15 1.7 Barrier Function Methods : : : : : : : : : : : : : : : : : : : : : : 17 2 Interior Point Methods for Block-Angular Problems 24 2.1 Shifting Barriers to Obtain Feasibility : : : : : : : : : : : : : : : : 25 2.1.1 Properties of Shifted Barriers : : : : : : : : : : : : : : : : 27 2.1.2 Computing a Feasible Point : : : : : : : : : : : : : : : : : 31 2.2 Convergence of Barrier Minimizers :...