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 $F(x):{ℝ}^n\to {ℝ}^n$, lower values ${\ell }_i\in ℝ\cup \{-\mathrm{\infty }\}$ and upper values $u_i\in ℝ\cup \{\mathrm{\infty }\}$, with $i\in \{1,\dots ,n\}$.

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

• $x_i={\ell }_i,F_i(x)\ge 0$;
• ${\ell }_i<x_i<u_i,F_i(x)=0$;
• $x_i=u_i,F_i(x)\le 0$.

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

See also

• Complementarity theory