Mixed complementarity problem

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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 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].

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