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{{Short description|Formulation in mathematical programming}}
'''Mixed Complementarity Problem''' ('''MCP''') is a problem formulation in [[mathematical programming]]. Many well-known problem types are special cases of, or may be reduced to MCP. It is a generalization of [[nonlinear complementarity problem]] (NCP).

== Definition ==
The mixed complementarity problem is defined by a mapping <math>F(x): \mathbb{R}^n \to \mathbb{R}^n</math>, lower values <math>\ell_i \in \mathbb{R} \cup \{-\infty\}</math> and upper values <math>u_i \in \mathbb{R}\cup\{\infty\}</math>, with <math>i \in \{1, \ldots, n\}</math>.

The '''solution''' of the MCP is a vector <math>x \in \mathbb{R}^n</math> such that for each index <math>i \in \{1, \ldots, n\}</math> one of the following alternatives holds:

* <math>x_i = \ell_i, \; F_i(x) \ge 0</math>;
* <math>\ell_i < x_i < u_i, \; F_i(x) = 0</math>;
* <math>x_i = u_i, \; F_i(x) \le 0</math>.

Another definition for MCP is: it is a [[variational inequality]] on the [[parallelepiped]] <math>[\ell, u]</math>.

== See also ==
* [[Complementarity theory]]

== References ==
* {{cite web|author=Stephen C. Billups|title=Algorithms for complementarity problems and generalized equations|date=1995| url=https://ftp.cs.wisc.edu/math-prog/tech-reports/95-14.ps|
format=[[Adobe Photoshop|PS]]|accessdate=2006-08-14}}
* {{cite book|author=Francisco Facchinei, Jong-Shi Pang|title=Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume I|date=2003}}

{{Mathematical programming}}

[[Category:Mathematical optimization]]

Mixed complementarity problem - Revision history

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