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 ${\displaystyle F(x):\mathbb {R} ^{n}\to \mathbb {R} ^{n}}${\displaystyle F(x):\mathbb {R} ^{n}\to \mathbb {R} ^{n}}, lower values ${\displaystyle \ell _{i}\in \mathbb {R} \cup \{-\infty \}}${\displaystyle \ell _{i}\in \mathbb {R} \cup \{-\infty \}} and upper values ${\displaystyle u_{i}\in \mathbb {R} \cup \{\infty \}}${\displaystyle u_{i}\in \mathbb {R} \cup \{\infty \}}, with ${\displaystyle i\in \{1,\ldots ,n\}}${\displaystyle i\in \{1,\ldots ,n\}}.

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

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

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

• Complementarity theory

• Stephen C. Billups (1995). "Algorithms for complementarity problems and generalized equations" (PS). Retrieved 2006年08月14日.
• Francisco Facchinei, Jong-Shi Pang (2003). Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume I.

• Mathematical optimization

• Articles with short description
• Short description matches Wikidata