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Mixed complementarity problem

Formulation in mathematical programming

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Source updated April 2, 2025

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).

01Definition

The mixed complementarity problem is defined by a mapping $F(x):\mathbb {R} ^{n}\to \mathbb {R} ^{n}$ , lower values $\ell _{i}\in \mathbb {R} \cup \{-\infty \}$ and upper values $u_{i}\in \mathbb {R} \cup \{\infty \}$ , with $i\in \{1,\ldots ,n\}$ .

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

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

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

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Mixed complementarity problem

01Definition

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Related topics

Complementarity theory