# 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 ${\displaystyle F(x):\mathbb {R} ^{n}\to \mathbb {R} ^{n}}$, 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 ${\displaystyle x\in \mathbb {R} ^{n}}$ 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};
• ${\displaystyle x_{i}=u_{i},\;F_{i}(x)\leq 0}$.

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

• Stephen C. Billups (1995). [https:/ftp.cs.wisc.edu/math-prog/tech-reports/95-14.ps "Algorithms for complementarity problems and generalized equations"] (PS). Retrieved 2006-08-14. {{cite web}}: Check |url= value (help)