Mathematical programming with equilibrium constraints

MPECs ( Mathematical Programs with Equilibrium Constraints ) to German as ' Mathematical optimization problems with equilibrium constraints ', are a special class of problems of mathematical optimization dar. MPECs are closely related to optimal control problems and are characterized by the fact that the essential constraints in the form of a variational inequality or a equivalent Komplementaritätssystems are formulated. Numerous applications can be found in the engineering world, or in the economy, such as in robotics, game theory, or in the assessment of options.

Problem formulation

In the problem class of MPECs to minimizing the objective function of two variables and depends. Next is non-empty and closed and a set-valued function with convex function values ​​. The MPEC in its most general form being defined by:

Minimize, subject to the constraint

Here is the solution set of the variational inequality:

Special

Some peculiarities of the problem class of MPECs are:

• The set of feasible points is not necessarily complete, coherent or convex. Results and methods of convex optimization can therefore not be applied.
• The reduced problem is i.Allg. not Fréchet - differentiable.
• Classical constraint Qualifications are not met.
• There is no clear Stationaritätsbegriff (see necessary optimality conditions ), but a whole hierarchy of concepts.