Mathematical programming with equilibrium constraints
Contents |
Introduction
Mathematical programming with equilibrium constraints (MPEC) is a type of nonlinear programming with constrained optimization. Constraints must satisfy an equilibrium condition, of which the simplest form is given by the critical point:
φ
Therefore, an equilibrium constrained optimization model is given by:
φ
φ
MPEC plays a central role in the modeling of transportation problems, economics, and engineering design.
Problem formulation
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φ
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φ
Feasible set
For a feasible set, the conditions for convexity and closedness are as follows. If Y(x) is convex, and functions f, g, and ∅ are concave, then the mathematical program is convex. Furthermore, if the Mangasarian-Fromovitz constraint qualification holds at all z ∈ Y(x), then Y(x) is the lower semi-continuous bound, and the mathematical program is closed.
KKT transformation
Complementarity constrained optimization
By applying the Karush-Kuhn-Tucker (KKT) approach to solving an equilibrium constraint problem (EC), a program with complementarity constraints can be obtained (CC):
The complementarity constraints can be written equivalently as:
perp-to
A mathematical program with complementarity constraints (MPCC) is a relaxed MPEC.
Linear constrained optimization
The KKT approach may also lead to an MPCC with only linear functions:
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λ
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Conclusion
References
[1] G.B. Allende. Mathematical programs with equilibrium constraints: solution techniques from parametric optimization (1977).
[2] M.C. Ferris, S.P. Dirkse, A. Meeraus. Mathematical programs with equilibrium constraints: automatic reformation and solution via constrained optimization. Northwestern University (2002).
[3] H. Pieper. Algorithms for mathematical programs with equilibrium constraints with applications to deregulated electricity markets. Stanford University (2001).
[4] R. Andreani, J.M. Martinez. On the solution of mathematical programming problems with equilibrium constraints (2008).