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:


 * $$x_i = \ell_i, \; F_i(x) \ge 0$$;
 * $$\ell_i < x_i < u_i, \; F_i(x) = 0$$;
 * $$x_i = u_i, \; F_i(x) \le 0$$.

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