Variational inequality

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Variational inequality

Variational inequality is a mathematical theory intended for the study of equilibrium problems. Guido Stampacchia put forth the theory in 1964 to study partial differential equations. The applicability of the theory has since been expanded to include problems from economics, finance, optimization and game theory.

The problem is commonly restricted to R"n". Given a subset K of R"n" and a mapping "F" : KR"n", the finite-dimensional variational inequality problem associated with K is

$mbox\left\{finding \right\}x in mathbf\left\{K\right\} mbox\left\{ so that \right\} langle F\left(x\right), y-x angle geq 0 mbox\left\{ for all \right\} y in mathbf\left\{K\right\}$

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" : KE*, the variational inequality problem is the same as above where <·,·> : E* x ER 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 $I = \left[a,b\right]$. Let "x" be the point in "I" where the minimum occurs. Three cases can occur:

:1) if $a < x < b$ then $f$ &prime;(x)=0;:2) if $x = a$ then $f$ &prime;(x) &ge; 0;:3) if $x = b$ then $f$ &prime;(x) &le; 0.

These conditions can be summarized as the problem of

$mbox\left\{finding \right\} x in I mbox\left\{ so that \right\} f\text{'}\left(x\right)\left(y-x\right) geq 0 mbox\left\{ for all \right\} y in I.$

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

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