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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).

The mixed complementarity problem is defined by a mapping \( {\displaystyle F(x):\mathbb {R} ^{n}\to \mathbb {R} ^{n}} \), lower values \( {\displaystyle \ell _{i}\in \mathbb {R} \cup \{-\infty \}} \) and upper values \( {\displaystyle u_{i}\in \mathbb {R} \cup \{\infty \}}. \)

The solution of the MCP is a vector x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} x \in \mathbb{R}^n such that for each index \( {\displaystyle i\in \{1,\ldots ,n\}} \) one of the following alternatives holds:

\( {\displaystyle x_{i}=\ell _{i},\;F_{i}(x)\geq 0}; \)
\( {\displaystyle \ell _{i}<x_{i}<u_{i},\;F_{i}(x)=0}; \)
\( {\displaystyle x_{i}=u_{i},\;F_{i}(x)\leq 0}. \)

Another definition for MCP is: it is a variational inequality on the parallelepiped \( {\displaystyle [\ell ,u]}. \)
See also

Complementarity theory


Stephen C. Billups (1995). "Algorithms for complementarity problems and generalized equations" (PS). Retrieved 2006-08-14.
Francisco Facchinei, Jong-Shi Pang (2003). Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume I.

Complementarity problems and algorithms
Complementarity Problems

Linear programming (LP) Quadratic programming (QP) Linear complementarity problem (LCP) Mixed linear (MLCP) Mixed (MCP) Nonlinear(NCP)

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