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Mathematical Program with Equilibrium Constraints

Mathematically, a Mathematical Program with Equilibrium Constraints (MPEC) model looks like:


      Minimize    f(x,y)
                  h(x)   < 0
            st    g(x,y) < 0
                  Lx < x < Ux
                  F(x,y)  perp to   Ly < y < Uy,

where perp to indicates the usual complementarity relationship,

                  F_i > 0   requires  y_i = Ly_i 
                  F_i < 0   requires  y_i = Uy_i 
                  F_i = 0   otherwise

and where the vector x represents the upper level or control variables and the vector y represents the lower level or state variables. f(x,y) is the objective function, and h(x) and g(x,y) represent the set of upper level constraints. The linking constraints g are upper-level constraints that involve the lower-level variables y; since some proposed solution methods for MPEC do not allow for such constraints, we distinguish them from the constraints h. Lx and Ux are vectors of lower and upper bounds on the variables x. Together with y, the function F(x,y) defines the equilibrium or lower-level constraints for the problem.

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