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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).
Definition

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 $${\displaystyle [\ell ,u]}.$$

Complementarity theory

References

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)

Basis-exchange algorithms

Simplex (Dantzig) Revised simplex Criss-cross Lemke