Mathematical program with equilibrium constraints (MPEC)

Mathematically, the mathematical program with equilibrium constraints (MPEC) looks like:

 F(x,y) ┴ Ly ≤ y ≤ Uy

 Lx < x < Ux

where

x and y are vectors of variables that are continuous or discrete where the variables x are often called the state variables or upper-level variables, while the variables y are called the control or lower-level variables.

f(x,y) is the objective function.

g(x,y) represents the set of constraints; in some cases, they can only involve the state variables x.

F(x,y) and the bounds Ly and Uy define the equilibrium constraints.

Note: If x is fixed, then F(x,y) and the bounds Ly and Uy define an MCP.  From this definition, we see that the MPEC model type contains NLP and MCP models as special cases of MPEC.

While the MPEC model formulation is very general, it also results in problems that are very difficult to solve.  Work on MPEC algorithms is not nearly so advanced as that for the other model types.  As a result, there is only an experimental MPEC solver included in the GAMS distribution.  For more see http://gamsworld.org/mpec/index.htm and CPNET:Complementarity Problem Net.