MPEC

MPEC s (Mathematical Programs with Equilibrium Constraints), in German about 'Mathematical Optimization Problems with Equilibrium Constraints', represent a special problem class of mathematical optimization . MPECs are closely related to optimal control problems and are characterized by the fact that the essential constraints in the form of a variation inequality or an equivalent complementarity system are formulated. Numerous applications can be found in the engineering world or in business, such as in robotics , in game theory or in the calculation of options .

Problem formulation

In the problem class of MPECs, the objective function to be minimized depends on two variables and . Furthermore, let us not be empty and closed and be a set-valued function with convex function values. The MPEC in its most general form is defined by: ${\displaystyle f}$ ${\displaystyle x\in \mathbb {R} ^{n}}$ ${\displaystyle y\in \mathbb {R} ^{m}}$ ${\displaystyle F:\mathbb {R} ^{n+m}\rightarrow \mathbb {R} ^{m},\;Z\subset \mathbb {R} ^{n+m}}$ ${\displaystyle C:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{m}}$

Minimize , under the constraint ${\displaystyle f(x,y)}$

${\displaystyle (x,y)\in Z,\,y\in S(x).}$

Here is the solution set of variational: ${\displaystyle S(x)}$

${\displaystyle y\in C(x),\,(v-y)^{T}F(x,y)\geq 0\;\forall \,v\in C(x).}$

particularities

Some special features of the problem class of MPECs are:

• The set of admissible points is not necessarily closed , connected, or convex . The results and methods of convex optimization cannot therefore be used.
• The reduced problem is i. General not Fréchet-differentiable .
• Classic constraint qualifications are not met.
• There is no clear concept of stationarity (see necessary optimality conditions ), but a whole hierarchy of concepts.

literature

• Z.-Q. Luo, J.-S. Pang and D. Ralph: Mathematical Programs with Equilibrium Constraints . Cambridge University Press, 1996, ISBN 0-521-57290-8 .