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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).

Definition

The mixed complementarity problem is defined by a mapping $F (x) : ℝ^{n} \to ℝ^{n}$ , lower values $ℓ_{i} \in ℝ \cup {- \infty}$ and upper values $u_{i} \in ℝ \cup {\infty}$ .

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:

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

References

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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).
+==Definition==
+The mixed complementarity problem is defined by a mapping <math>F(x): \mathbb{R}^n \to \mathbb{R}^n</math>, lower values <math>\ell_i \in \mathbb{R} \cup \{-\infty\}</math> and upper values <math>u_i \in \mathbb{R}\cup\{\infty\}</math>.
+The '''solution''' of the MCP is a vector <math>x \in \mathbb{R}^n</math> such that for each index <math>i \in \{1, \ldots, n\}</math> one of the following alternatives holds:
+* <math>x_i = \ell_i, \; F_i(x) \ge 0</math>;
+* <math>\ell_i < x_i < u_i, \; F_i(x) = 0</math>;
+* <math>x_i = u_i, \; F_i(x) \le 0</math>.
+Another definition for MCP is: it is a [[variational inequality]] on the [[parallelepiped]] <math>[\ell, u]</math>.
+== See also ==
+* [[Complementarity theory]]
+== References ==
+* {{cite paper|author=Stephen C. Billups|title=Algorithms for complementarity problems and generalized equations|date=1995|
+url=ftp://ftp.cs.wisc.edu/math-prog/tech-reports/95-14.ps|
+format=[[Adobe Photoshop|PS]]|accessdate=2006-08-14}}
+* {{cite book|author=Francisco Facchinei, Jong-Shi Pang|title=Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume I|date=2003}}
+{{Mathematical programming}}
+[[Category:Mathematical optimization]]

Ovoid (polar space): Difference between revisions

Revision as of 07:47, 6 October 2012

Definition

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