- Variational inequality
**Variational inequality**is amathematical theory intended for the study of equilibrium problems.Guido Stampacchia put forth the theory in1964 to studypartial differential equation s. The applicability of the theory has since been expanded to include problems fromeconomics ,finance , optimization andgame theory .The problem is commonly restricted to

**R**^{"n"}. Given a subset**K**of**R**^{"n"}and a mapping "F" :**K**→**R**^{"n"}, the finite-dimensional variational inequality problem associated with**K**is

$mbox\{finding\; \}x\; in\; mathbf\{K\}\; mbox\{\; so\; that\; \}\; langle\; F(x),\; y-x\; angle\; geq\; 0\; mbox\{\; for\; all\; \}\; y\; in\; mathbf\{K\}$where <·,·> is the standard

inner product on**R**^{"n"}.In general, the variational inequality problem can be formulated on any finite- or infinite-dimensional

Banach space . Given a Banach space**E**, a subset**K**of**E**, and a mapping "F" :**K**→**E***, the variational inequality problem is the same as above where <·,·> :**E***x**E**→**R**is the duality pairing.Fact|definition or link required|date=August 2007**Examples**Consider the problem of finding the minimal value of a continuous differentiable function "f" over a closed interval $I\; =\; [a,b]$. Let "x" be the point in "I" where the minimum occurs. Three cases can occur:

:1) if $a\; <\; x\; <\; b$ then $f$ ′(x)=0;:2) if $x\; =\; a$ then $f$ ′(x) ≥ 0;:3) if $x\; =\; b$ then $f$ ′(x) ≤ 0.

These conditions can be summarized as the problem of

$mbox\{finding\; \}\; x\; in\; I\; mbox\{\; so\; that\; \}\; f\text{'}(x)(y-x)\; geq\; 0\; mbox\{\; for\; all\; \}\; y\; in\; I.$**References***Citation | last1=Kinderlehrer | first1=David | last2=Stampacchia | author1-link=David Kinderlehrer| first2=Guido | author2-link=Guido Stampacchia|title=An Introduction to Variational Inequalities and Their Applications | publisher=Academic Press | location=New York | isbn=0-89871-466-4 | year=1980

*G. Stampacchia. "Formes Bilineaires Coercitives sur les Ensembles Convexes", Comptes Rendus de l’Academie des Sciences, Paris, 258, (1964), 4413–4416.**ee also***

projected dynamical system

*differential variational inequality

*Complementarity theory

*Mathematical programming with equilibrium constraints

*Wikimedia Foundation.
2010.*

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