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A mathematical programming problem with affine equilibrium constraints (AMPEC) is a bilevel programming problem where the lower one is a parametric affine variational inequality. We formulate some classes of bilevel programming in forms of MPEC. Then we use a regularization technique to formulate the resulting problem as a mathematical program with an additional constraint defined by the difference of two convex functions (DC function). A main feature of this DC decomposition is that the second component depends upon only the parameter in the lower problem. This property allows us to develop branch-and-bound algorithms for globally solving AMPEC where the adaptive rectangular bisection takes place only in the space of the parameter. As an example, we use the proposed algorithm to solve a bilevel Nash-Cournot equilibrium market model. Computational results show the efficiency of the proposed algorithm.

A class of smoothing methods is proposed for solving mathematical programs with equimibrium constraints. We introduce new and very simple regularizations of the complementarity constraints. Some estimate distance to optimal solution and expansions of the optimal value function are presented. Limited numerical experiments using SNOPT algorithm are presented to verify the efficiency of our approach.

We present a new framework for the solution of mathematical programs with equilibrium constraints (MPECs). In this algorithmic framework, an MPECs is viewed as a concentration of an unconstrained optimization which minimizes the complementarity measure and a nonlinear programming with general constraints. A strategy generalizing ideas of Byrd-Omojokun's trust region method is used to compute steps. By penalizing the tangential constraints into the objective function, we circumvent the problem of not satisfying MFCQ. A trust-funnel-like strategy is used to balance the improvements on feasibility and optimality. We show that, under MPEC-MFCQ, if the algorithm does not terminate in finite steps, then at least one accumulation point of the iterates sequence is an S-stationary point.

We consider optimization problems with a disjunctive structure of the constraints. Prominent examples of such problems are mathematical programs with equilibrium constraints or vanishing constraints. Based on the concepts of directional subregularity and their characterization by means of objects from generalized differentiation, we obtain the new stationarity concept of extended M-stationarity, which turns out to be an equivalent dual characterization of B-stationarity. These results are valid under a very weak constraint qualification of Guignard type which is usually very difficult to verify. We also state a new constraint qualification which is a little bit stronger but verifiable. Further we present second-order optimality conditions, both necessary and sufficient. Finally we apply these results to the special case of mathematical programs with equilibrium constraints and compute explicitly all the objects from generalized differentiation. For this type of problems we also introduce the concept of strong M-stationarity which builds a bridge between S-stationarity and M-stationarity.

In this paper, we study the mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with a parametric generalized equation involving the regular normal cone. Compared with the usual way of formulating MPEC through a KKT condition, this formulation has the advantage that it does not involve extra multipliers as new variables, and it usually requires weaker assumptions on the problem data. Using the so-called first order sufficient condition for metric subregularity, we derive verifiable sufficient conditions for the metric subregularity of the involved set-valued mapping, or equivalently the calmness of the perturbed generalized equation mapping.

The article considers a particular class of optimization problems involving set-valued stochastic equilibrium constraints. A solution procedure is developed by relying on an approximation scheme for the equilibrium constraints, based on regularization, that replaces them by equilibrium constraints involving only single-valued Lipschitz continuous functions. In addition, sampling has the further effect of replacing the simplified equilibrium constraints by more manageable ones obtained by implicitly discretizing the (given) probability measure so as to render the problem computationally tractable. Convergence is obtained by relying, in particular, on the graphical convergence of the approximated equilibrium constraints. The problem of estimating the characteristics of a demand model, a widely studied problem in micro-economics, serves both as motivation and illustration of the regularization and sampling procedure.