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).

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 ${\displaystyle u_{i}\in \mathbb {R} \cup \{\infty \}}$, with ${\displaystyle i\in \{1,\ldots ,n\}}$.

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:

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

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