Abstract. Some properties of regularized and penalized nonlinear programming formulations of mathematical programs with equilibrium constraints (MPECs) are described. The focus is on the properties of these formulations near a local solution of the MPEC at which strong stationarity and a second-order sufficient condition are satisfied. In the regularized formulations, the complementarity condition is replaced by a constraint involving a positive parameter that can be decreased to zero. In the penalized formulation, the complementarity constraint appears as a penalty term in the objective. Existence and uniqueness of solutions for these formulations are investigated, and estimates are obtained for the distance of these solutions to the MPEC solution under various assumptions.

Abstract. Generalized stationary points of the mathematical program with equilibrium constraints (MPEC) are studied to better describe the limit points produced by interior point methods for MPEC. A primal-dual interior-point method is then proposed, which solves a sequence of relaxed barrier problems derived from MPEC. Global convergence results are deduced without assuming strict complementarity or the linear independence constraint qualification for MPEC (MPEC-LICQ). Under certain general assumptions, the algorithm can always find some point with strong or weak stationarity. In particular, it is shown that every limit point of the generated sequence is a strong stationary point of MPEC if the penalty parameter of the merit function is bounded. Otherwise, a certain point with weak stationarity can be obtained. Preliminary numerical results are reported, which include a case analyzed by Leyffer for which the penalty interior-point algorithm failed to find a stationary point. Key words: Global convergence, interior-point methods, mathematical programming with equilibrium constraints, stationary point

Abstract. The elastic-mode formulation of the problem of minimizing a nonlinear function subject to equilibrium constraints has appealing local properties in that, for a finite value of the penalty parameter, local solutions satisfying first- and second-order necessary optimality conditions for the original problem are also first- and second-order points of the elastic-mode formulation. Here we study global convergence properties of methods based on this formulation, which involve generating an (exact or inexact) first- or second-order point of the formulation, for nondecreasing values of the penalty parameter. Under certain regularity conditions on the active constraints, we establish finite or asymptotic convergence to points having a certain stationarity property (such as strong stationarity, M-stationarity, or C-stationarity). Numerical experience with these approaches is discussed. In particular, our analysis and the numerical evidence show that exact complementarity can be achieved finitely even when the elastic-mode formulation is solved inexactly. Key words. Nonlinear programming, equilibrium constraints, complementarity constraints, elastic-mode formulation, strong stationarity, C-stationarity, Mstationarity. AMS subject classifications 49M30, 49M37, 65K05, 90C30, 90C33 1.

Abstract. An interior-point method for solving mathematical programs with equilibrium constraints (MPECs) is proposed. At each iteration of the algorithm, a single primal-dual step is computed from each subproblem of a sequence. Each subproblem is defined as a relaxation of the MPEC with a nonempty strictly feasible region. In contrast to previous approaches, the proposed relaxation scheme preserves the nonempty strict feasibility of each subproblem even in the limit. Local and superlinear convergence of the algorithm is proved even with a less restrictive strict complementarity condition than the standard one. Moreover, mechanisms for inducing global convergence in practice are proposed. Numerical results on the MacMPEC test problem set demonstrate the fast-local convergence properties of the algorithm. Key words. nonlinear programming, mathematical programs with equilibrium constraints, constrained minimization, interior-point methods, primal-dual methods, barrier methods

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. To this end, we first introduce an active set identification technique. Then, by applying this technique to a smoothing continuation method presented by Fukushima and Pang (1999), we propose a hybrid method for solving MPCC. Under reasonable assumptions, the hybrid algorithm is shown to possess a finite termination property. Numerical experience shows that the proposed approach is quite e#ective.

This paper presents a parameter-free integer-programming based algorithm for the global resolution of a linear program with linear complementarity constraints (LPEC). The cornerstone of the algorithm is a minimax integer program formulation that characterizes and provides certificates for the three outcomes—infeasibility, unboundedness, or solvability—of an LPEC. An extreme point/ray generation scheme in the spirit of Benders decomposition is developed, from which valid inequalities in the form of satisfiability constraints are obtained. The feasibility problem of these inequalities and the carefully guided linear programming relaxations of the LPEC are the workhorse of the algorithm, which also employs a specialized procedure for the sparsification of the satifiability cuts. We establish the finite termination of the algorithm and report computational results using the algorithm for solving randomly generated LPECs of reasonable sizes. The results establish that the algorithm can handle infeasible, unbounded, and solvable LPECs effectively.

Support vector machines and related classification models require the solution of convex optimization problems that have one or more regularization hyper-parameters. Typically, the hyper-parameters are selected to minimize cross validated estimates of the out-of-sample classification error of the model. This cross-validation optimization problem can be formulated as a bilevel program in which the outer-level objective minimizes the average number of misclassified points across the cross-validation folds, subject to inner-level constraints such that the classification functions for each fold are (exactly or nearly) optimal for the selected hyper-parameters. Feature selection is included in the bilevel program in the form of bound constraints in the weights. The resulting bilevel problem is converted to a mathematical program with linear equilibrium constraints, which is solved using state-of-the-art optimization methods. This approach is significantly more versatile than commonly used grid search procedures, enabling, in particular, the use of models with many hyper-parameters. Numerical results demonstrate the practicality of this approach for model selection in machine learning.

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

Abstract. In this paper we propose a merit function piecewise SQP al-gorithm for solving mathematical programs with equilibrium constraints (MPECs) formulated as mathematical programs with complementarity constraints. Under some mild conditions, the new algorithm is globally convergent to a piecewise stationary point. Moreover if the partial MPEC-LICQ is satisfied at the accumulation point then the accumulation point is a S-stationary point.

We propose to solve generalized bilevel programs by a trust region approach where the "model" takes the form of a bilevel program involving a linear program at the upper level and a linear variational inequality at the lower level. By coupling the concepts of trust region and linesearch in a novel way, we obtain an implementable algorithm that converges to a strong stationary point of the original bilevel program.