Abstract
A mathematical program with complementarity constraints is an optimization problem with equality/inequality constraints in which a complementarity type constraint is considered in addition. This complementarity condition modifies the feasible region so as to remove many of those properties that are usually important to obtain the standard optimality conditions, e.g., convexity and constraint qualifications. In this paper, in the linear case, we introduce a decomposition method of the given problem in a sequence of parameterized problems, that aim to force complementarity. Once we obtain a feasible solution, by means of duality results, we are able to eliminate a set of parameterized problems which are not worthwhile to be considered. Furthermore, we provide some bounds for the optimal value of the objective function and we present an application of the proposed technique in a non trivial example and some numerical experiments.
Highlights
In the field of equilibrium models, mathematical programs with complementarity constraints (MPCC) form a class of important, but extremely difficult, problems
Exploiting the classic tools of the duality theory, we propose an iterative method which explores the set of parameters, excluding at each step a subset of them, by means of a suitable cut; the optimal values of the linear problems associated with such a subset are proved to be greater than or equal to the optimal value related to the current parameter
The method that we propose is different from the classic relaxation and penalty methods for mathematical programs with equilibrium constraints (MPEC), and allows us to define an algorithm which can be implemented in an interactive way taking advantage of some devices that speed up the solving procedure, owing to the decomposition of the given problem in a sequence of parameterized problems
Summary
In the field of equilibrium models, mathematical programs with complementarity constraints (MPCC) form a class of important, but extremely difficult, problems. MPCC’s constitute a subclass of the well-known mathematical programs with equilibrium constraints (MPEC), widely studied in recent years Their relevance comes from many applications which arise, for example, in economics and structural engineering [3, 6]. Their difficulty is due to the presence of the complementarity constraints, because the feasible region may not enjoy some fundamental properties: it may be not convex, even not connected and such that many of the standard constraint qualifications are violated at any feasible point.
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