Mixed complementarity problem
# Mixed complementarity problem

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 $F(x):\mathbb {R} ^{n}\to \mathbb {R} ^{n}$, lower values $\ell _{i}\in \mathbb {R} \cup \{-\infty \}$ and upper values $u_{i}\in \mathbb {R} \cup \{\infty \}$.

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 $[\ell ,u]$.

