Difference between revisions of "Mathematical programming with equilibrium constraints"
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<math>u_i </math> = <math>y_i</math><math>p(T)</math> - <math>f_i</math><math>(y_i)</math> | <math>u_i </math> = <math>y_i</math><math>p(T)</math> - <math>f_i</math><math>(y_i)</math> | ||
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+ | In this case, if company 1 were to product a product, their profitability would be based on the amount of other products in the market. Therefore, the portion of the profit that is a summation, is based on another optimization problem. This case is known as a bi-level problem. | ||
=References= | =References= |
Revision as of 23:13, 24 May 2015
Author: Alexandra Rodriguez (ChE 345 Spring 2015) and Brandon Muncy (ChE345 Spring 2015)
Stewards: Dajun Yue and Fengqi You
Contents |
Introduction
Mathematical programming with equilibrium constraints (MPEC) is a type of nonlinear programming with constrained optimization. Constraints must satisfy an equilibrium condition, which can be an equilibrium inequality or a complimentarity condition, of which the simplest form is given by the critical point:
φ
Therefore, an equilibrium constrained optimization model is given by:
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φ
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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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λ![]()
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Applications
As mentioned above, there are several applications of MPEC problems. Two published examples deal with economics and mathematical physics.
Economics
For this application, consider that n companies product the same product. We will introduce an integer variable y to denote the number of units a company will sell. We will further denote yi as the number of items that company i decides to sell. The total price of the product on the market will be notated P(T), where T = . The total cost of production for a company will be given by fi(yi). With this notation, the profit can then be given by the expression:
=
-
In this case, if company 1 were to product a product, their profitability would be based on the amount of other products in the market. Therefore, the portion of the profit that is a summation, is based on another optimization problem. This case is known as a bi-level problem.
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).