title: Mixed complementarity problem
text: # 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 $[\ell ,u]$.
  
  