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In [[mathematics]], a '''variational inequality''' is an [[inequality (mathematics)|inequality]] involving a [[Functional (mathematics)|functional]], which has to be [[Inequality (mathematics)#Solving Inequalities|solved]] for all possible values of a given [[Variable (mathematics)|variable]], belonging usually to a [[convex set]]. The [[mathematical]] [[theory]] of variational inequalities was initially developed to deal with [[Equilibrium point|equilibrium]] problems, precisely the [[Signorini problem]]: in that model problem, the functional involved was obtained as the [[first variation]] of the involved [[Signorini_problem#The_potential_energy|potential energy]] therefore it has a [[Calculus of variation|variational origin]], recalled by the name of the general abstract problem. The applicability of the theory has since been expanded to include problems from [[economics]], [[finance]], [[Optimization (mathematics)|optimization]] and [[game theory]].

== History ==
The first problem involving a variational inequality was the [[Signorini problem]], posed by [[Antonio Signorini]] in 1959 and solved by [[Gaetano Fichera]] in 1963, according to the references {{Harv|Antman|1983|pp=282–284}} and {{Harv|Fichera|1995}}: the first papers of the theory were {{Harv|Fichera|1963}} and {{Harv|Fichera|1964a}}, {{Harv|Fichera|1964b}}. Later on, [[Guido Stampacchia]] proved his generalization to the [[Lax–Milgram theorem]] in {{Harv|Stampacchia|1964}} in order to study the [[regularity problem]] for [[partial differential equation]]s and [[coin]]ed the name "variational inequality" for all the problems involving [[inequality (mathematics)|inequalities]] of this kind. [[Georges Duvaut]] encouraged his [[graduate student]]s to study and expand on Fichera's work, after attending a conference in [[Brixen]] on 1965 where Fichera presented his study of the Signorini problem, as {{Harvnb|Antman|1983|p=283}} reports: thus the theory become widely known throughout [[France]]. Also in 1965, Stampacchia and [[Jacques-Louis Lions]] extended earlier results of {{Harv|Stampacchia|1964}}, announcing them in the paper {{Harv|Lions|Stampacchia|1965}}: full proofs of their results appeared later in the paper {{Harv|Lions|Stampacchia|1967}}.

== Definition ==
Following {{Harvtxt|Antman|1983|p=283}}, the formal definition of a variational inequality is the following one.

{{EquationRef|1|Definition 1.}} Given a [[Banach space]] '''<math>E</math>''', a [[subset]] '''<math>K</math>''' of '''<math>E</math>''', and a functional <math>\scriptstyle F:\boldsymbol{K}\rightarrow\boldsymbol{E}^\ast</math> from '''<math>K</math>''' to the [[dual space]] '''<math>E^*</math>''' of the space '''<math>E</math>''', the variational inequality problem is the problem of [[Inequality (mathematics)#Solving Inequalities|solving]] for the [[variable (mathematics)|variable]] ''<math>x</math>'' belonging to '''<math>K</math>''' the following [[inequality (mathematics)|inequality]]:

:<math>\langle F(x), y-x \rangle \geq 0\qquad\forall y \in \boldsymbol{K}</math>

where <math>\scriptstyle\langle\cdot,\cdot\rangle: \boldsymbol{E}^*\times\boldsymbol{E}\rightarrow\mathbb{R}</math> is the [[Dual space|duality pairing]]. 

In general, the variational inequality problem can be formulated on any [[Finite set|finite]] – or [[Infinite set|infinite]]-[[dimension]]al [[Banach space]]. The three obvious steps in the study of the problem are the following ones:
#Prove the existence of a solution: this step implies the ''mathematical correctness'' of the problem, showing that there is at least a solution.
#Prove the uniqueness of the given solution: this step implies the ''physical correctness'' of the problem, showing that the solution can be used to represent a physical phenomenon. It is a particularly important step since most of the problems modeled by variational inequalities are of physical origin.
#Find the solution.

==Examples==
===The problem of finding the minimal value of a real-valued function of real variable===
This is a standard example problem, reported by {{Harvtxt|Antman|1983|p=283}}: consider the problem of finding the [[minimum|minimal value]] of a  [[differentiable function]] <math>f</math> over a [[closed interval]] <math>I = [a,b]</math>. Let <math>\scriptstyle x^*</math> be a point in ''<math>I</math>'' where the minimum occurs. Three cases can occur:

# if <math>\scriptstyle a<x^*< b</math> then <math>\scriptstyle f'(x^*) = 0;</math>
# if <math>\scriptstyle x^*=a</math> then <math>\scriptstyle f'(x^*) \ge 0;</math>
# if <math>\scriptstyle x^*=b</math> then <math>\scriptstyle f'(x^*) \le 0.</math>

These necessary conditions can be summarized as the problem of finding  <math>\scriptstyle x^*\in I</math> such that

:<math>f'(x^*)(y-x^*) \geq 0\qquad\forall y \in I</math>

The absolute minimum must be searched between the solutions (if more than one) of the preceding [[inequality (mathematics)|inequality]]: note that the solution is a [[real number]], therefore this is a finite [[Dimension (mathematics)|dimensional]] variational inequality.

===The general finite dimensional variational inequality===
A formulation of the general problem in <math>\scriptstyle\mathbb{R}^n</math> is the following: given a [[subset]] <math>K</math> of <math>\scriptstyle\mathbb{R}^n</math> and a [[Map (mathematics)|mapping]] <math>\scriptstyle F:K\rightarrow\mathbb{R}^n</math>, the [[Finite set|finite]]-[[dimension]]al variational inequality problem associated with <math>K</math> consist of finding a [[Dimension|<math>n</math>-dimensional]] [[Euclidean vector|vector]] <math>x</math> belonging to <math>K</math> such that 

:<math>\langle F(x), y-x \rangle \geq 0\qquad\forall y \in K</math>

where <math>\scriptstyle\langle\cdot,\cdot\rangle:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}</math> is the standard [[inner product]] on the [[vector space]] <math>\scriptstyle\mathbb{R}^n</math>.

=== The variational inequality for the Signorini problem ===
[[File:Classical Signorini problem.svg|thumb|right|400px|The classical [[Signorini problem]]: what will be the [[Linear elasticity#Elastostatics|equilibrium]] [[Continuum mechanics#Mathematical modeling of a continuum|configuration]] of the orange spherically shaped [[Physical body|elastic body]] resting on the blue [[Rigid body|rigid]] [[friction]]less [[Plane (geometry)|plane]]?]]
In the historical survey {{Harv|Fichera|1995}}, [[Gaetano Fichera]] describes the genesis of his solution to the [[Signorini problem]]: the problem consist in finding the [[Linear elasticity#Elastostatics|elastic equilibrium]] [[Continuum mechanics#Mathematical modeling of a continuum|configuration]] <math>\scriptstyle\boldsymbol{u}(\boldsymbol{x})=\left(u_1(\boldsymbol{x}),u_2(\boldsymbol{x}),u_3(\boldsymbol{x})\right)</math> of an [[Anisotropy#Material science and engineering|anisotropic]] [[Homogeneous media|non-homogeneous]] [[Physical body|elastic body]] that lies in a [[subset]] <math>A</math> of the three-[[dimension]]al [[euclidean space]] whose [[boundary (topology)|boundary]] is <math>\scriptstyle\partial A</math>, resting on a [[Rigid body|rigid]] [[frictionless]] [[surface]] and subject only to its [[Weight|mass force]]s. The solution '''<math>u</math>''' of the problem exists and is unique (under precise assumptions) in the [[set (mathematics)|set]] of '''admissible displacements''' <math>\scriptstyle\mathcal{U}_\Sigma</math> i.e. the set of [[displacement vector]]s satisfying the system of [[Signorini problem#The ambiguous boundary conditions|ambiguous boundary conditions]] if and only if

:<math>B(\boldsymbol{u},\boldsymbol{v}) - F(\boldsymbol{v}) \geq 0 \qquad \forall \boldsymbol{v} \in \mathcal{U}_\Sigma </math>

where <math>\scriptstyle B(\boldsymbol{u},\boldsymbol{v}) </math> and <math>\scriptstyle F(\boldsymbol{v}) </math> are the following [[Functional (mathematics)|functionals]], written using the [[Einstein notation]]

:<math>B(\boldsymbol{u},\boldsymbol{v}) = -\int_A \sigma_{ik}(\boldsymbol{u})\varepsilon_{ik}(\boldsymbol{v})\mathrm{d}x</math> {{spaces|6}} <math>F(\boldsymbol{v}) = \int_A v_i f_i\mathrm{d}x + \int_{\partial A\setminus\Sigma}\!\!\!\!\! v_i g_i \mathrm{d}\sigma \qquad \boldsymbol{u},\boldsymbol{v} \in \mathcal{U}_\Sigma </math>

where, for all <math>\scriptstyle\boldsymbol{x}\in A</math>, 
*<math>\Sigma</math> is the [[Contact (mechanics)|contact]] [[surface]] (or more generally a contact [[set (mathematics)|set]]),
*<math>\scriptstyle\boldsymbol{f}(\boldsymbol{x}) = \left( f_1(\boldsymbol{x}), f_2(\boldsymbol{x}), f_3(\boldsymbol{x}) \right)</math> is the ''[[body force]]'' applied to the body,
*<math>\scriptstyle\boldsymbol{g}(\boldsymbol{x})=\left(g_1(\boldsymbol{x}),g_2(\boldsymbol{x}),g_3(\boldsymbol{x})\right)</math> is the [[surface force]] applied to <math>\scriptstyle\partial A\setminus\Sigma</math>,
*<math>\scriptstyle\boldsymbol{\varepsilon}=\boldsymbol{\varepsilon}(\boldsymbol{u})=\left(\varepsilon_{ik}(\boldsymbol{u})\right)=\left(\frac{1}{2} \left( \frac{\partial u_i}{\partial x_k} + \frac{\partial u_k}{\partial x_i} \right)\right)</math> is the [[Infinitesimal strain#Infinitesimal strain tensor|infinitesimal strain tensor]],
*<math>\scriptstyle\boldsymbol{\sigma}=\left(\sigma_{ik}\right)</math>is the [[Cauchy stress tensor]], defined as
::<math>\sigma_{ik}= - \frac{\partial W}{\partial \varepsilon_{ik}} \qquad\forall i,k=1,2,3</math>

:where <math>\scriptstyle W(\boldsymbol{\varepsilon})=a_{ikjh}(\boldsymbol{x})\varepsilon_{ik}\varepsilon_{ik}</math> is the [[elastic potential energy]] and <math>\scriptstyle\boldsymbol{a}(\boldsymbol{x})=\left(a_{ikjh}(\boldsymbol{x})\right)</math> is the [[elasticity tensor]].

==See also==
*[[Complementarity theory]]
*[[Differential variational inequality]]
*[[Mathematical programming with equilibrium constraints]]
*[[Obstacle problem]]
*[[Projected dynamical system]]
*[[Signorini problem]]

== Bibliography ==
*{{Citation
| last = Antman
| first = Stuart
| author-link = 
| title = The influence of elasticity in analysis: modern developments
| journal = [[Bulletin of the American Mathematical Society]]
| volume = 9
| issue = 3
| pages = 267–291
| date = 
| year = 1983
| url = http://www.ams.org/bull/1983-09-03/S0273-0979-1983-15185-6/home.html?pagingLink=%3Ca+href%3D%22%2Fjoursearch%2Fservlet%2FDoSearch%3Fco1%3Dand%26co2%3Dand%26co3%3Dand%26cperpage%3D50%26csort%3Dd%26endmo%3D00%26f1%3Dmsc%26f2%3Dtitle%26f3%3Danywhere%26f4%3Dauthor%26format%3Dstandard%26jrnl%3Done%26sendit22%3DSearch%26sperpage%3D30%26ssort%3Dd%26startmo%3D00%26timingString%3DQuery%2Btook%2B71%2Bmilliseconds.%26v4%3DAntman%26onejrnl%3Dbull%26startRec%3D1%22%3E
| doi = 10.1090/S0273-0979-1983-15185-6 
| mr = 714990 
| zbl = 0533.73001
}}. An historical paper about the fruitful interaction of [[elasticity theory]] and [[mathematical analysis]]: the creation of the theory of [[variational inequalities]] by [[Gaetano Fichera]] is described in paragraph 5, pages 282&ndash;284.
*{{Citation
| first = Gaetano
| last = Fichera
| author-link = Gaetano Fichera
| editor-last = 
| editor-first = 
| editor2-last = 
| editor2-first = 
| contribution = La nascita della teoria delle diseguaglianze variazionali ricordata dopo trent'anni (The birth of the theory of variational inequalities remembered thirty years later)
| contribution-url = 
| title = Incontro scientifico italo-spagnolo. Roma, 21 ottobre 1993
| url = http://www.lincei.it/pubblicazioni/catalogo/volume.php?rid=36500 
| year = 1995
| pages = 47–53
| place = [[Rome|Roma]]
| series = Atti dei Convegni Lincei
| volume = 114
| publisher = [[Accademia Nazionale dei Lincei]]
}} (in [[Italian language|Italian]]).

==References==
*{{Citation
| last1=Facchinei 
| first1=Francisco 
| author1-link=
| last2=Pang 
| first2=Jong-Shi 
| author2-link=
| title=Finite Dimensional Variational Inequalities and Complementarity Problems, Vol. 1
| series = Springer Series in Operations Research
| publisher=[[Springer-Verlag]] 
| location= [[Berlin]]-[[Heidelberg]]-[[New York]] 
| isbn=0-387-95580-1 
| year=2003
| zbl=1062.90001
}}
*{{Citation
| last1=Facchinei 
| first1=Francisco 
| author1-link=
| last2=Pang 
| first2=Jong-Shi 
| author2-link=
| title=Finite Dimensional Variational Inequalities and Complementarity Problems, Vol. 2
| series = Springer Series in Operations Research
| publisher=[[Springer-Verlag]] 
| location= [[Berlin]]-[[Heidelberg]]-[[New York]] 
| isbn=0-387-95581-X 
| year=2003
| zbl=1062.90001
}}
*{{citation
| last = Fichera
| first = Gaetano
| title = Sul problema elastostatico di Signorini con ambigue condizioni al contorno (On the elastostatic problem of Signorini with ambiguous boundary conditions)
| journal = Rendiconti della [[Accademia Nazionale dei Lincei]], Classe di Scienze Fisiche, Matematiche e Naturali
| volume = 34 
| series =  8
| issue = 2
| year = 1963
| pages=138&ndash;142
| zbl = 0128.18305
}} (in [[Italian language|Italian]]). A short paper describing briefly the approach to the solution of the [[Signorini problem]].
*{{citation
| last = Fichera
| first = Gaetano
| title = Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno (Elastostatic problems with unilateral constraints: the Signorini problem with ambiguous boundary conditions)
| journal = Memorie della [[Accademia Nazionale dei Lincei]], Classe di Scienze Fisiche, Matematiche e Naturali
| volume = 7
| series =  8
| issue = 2
| year = 1964a
| pages=91&ndash;140
| zbl = 0146.21204
}} (in [[Italian language|Italian]]). The paper containing the [[Existence theorem|existence]] and [[uniqueness theorem]] for the [[Signorini problem]].
* {{citation
| last = Fichera
| first = Gaetano
| contribution = Elastostatic problems with unilateral constraints: the Signorini problem with ambiguous boundary conditions
| title = Seminari dell'istituto Nazionale di Alta Matematica 1962&ndash;1963
| year = 1964b
| publisher = Edizioni Cremonese
| place = [[Rome]]
| pages=613&ndash;679
}}. An English translation of the paper {{Harv|Fichera|1964a}}.
*{{Citation
| last = Glowinski
| first = Roland 
| author-link = Roland Glowinski
| last2 = Lions
| first2 = Jacques-Louis 
| author2-link = Jacques-Louis Lions
| last3 = Trémolières
| first3 = Raymond 
| author3-link=
| title = Numerical analysis of variational inequalities. Translated from the French
| place = [[Amsterdam]]-[[New York]]-[[Oxford]]
| publisher = [[Elsevier|North-Holland]]
| year = 1981
| series = Studies in Mathematics and its Applications
| volume = 8
| mr = 635927 
| isbn = 0-444-86199-8
| zbl = 0463.65046
| pages = xxix+776 }}
*{{Citation
| last1=Kinderlehrer 
| first1=David 
| author1-link=David Kinderlehrer
| last2=Stampacchia 
| first2=Guido 
| author2-link=Guido Stampacchia
| title=An Introduction to Variational Inequalities and Their Applications 
| publisher=[[Academic Press]] 
| series= Pure and Applied Mathematics
| url = http://books.google.com/?id=B1cPRJ3qiw0C&printsec=frontcover&dq=An+Introduction+to+Variational+Inequalities+and+Their+Applications
| volume = 88
| location=[[Boston]]-[[London]]-[[New York]]-[[San Diego]]-[[Sydney]]-[[Tokyo]]-[[Toronto]]
| isbn=0-89871-466-4
| year=1980
| zbl=0457.35001}}.
*{{Citation
| last = Lions
| first = Jacques-Louis
| author-link = Jacques-Louis Lions
| last2 = Stampacchia
| first2 = Guido
| author2-link = Guido Stampacchia
| title = Inéquations variationnelles non coercives
| journal = Comptes rendus hebdomadaires des séances de l'Académie des sciences
| volume = 261
| pages = 25–27
| year = 1965
| url = http://gallica.bnf.fr/ark:/12148/bpt6k4022z.image.r=Comptes+Rendus+Academie.langEN.f26.pagination
| zbl = 0136.11906
}}, available at [[Gallica]]. Announcements of the results of paper {{Harv|Lions|Stampacchia|1967}}.
*{{Citation
| last = Lions
| first = Jacques-Louis
| author-link = Jacques-Louis Lions
| last2 = Stampacchia
| first2 = Guido
| author2-link = Guido Stampacchia
| title = Variational inequalities
| journal = [http://www3.interscience.wiley.com/journal/29240/home?CRETRY=1&SRETRY=0 Communications on Pure and Applied Mathematics]
| volume = 20
| pages = 493–519 
| year = 1967
| url = http://www3.interscience.wiley.com/journal/113397217/abstract
| doi = 10.1002/cpa.3160200302
| zbl = 0152.34601
| issue = 3
}}. An important paper, describing the abstract approach of the authors to the theory of variational inequalities.
*{{Citation
| last = Stampacchia
| first = Guido
| author-link = Guido Stampacchia
| title = Formes bilineaires coercitives sur les ensembles convexes
| journal = Comptes rendus hebdomadaires des séances de l'Académie des sciences
| volume = 258
| pages = 4413–4416
| year = 1964
| url = http://gallica.bnf.fr/ark:/12148/bpt6k4012p.image.r=Comptes+Rendus+Academie.f20.langEN
| doi = 
| zbl = 0124.06401
}}, available at [[Gallica]]. The paper containing Stampacchia's generalization of the [[Lax–Milgram theorem]].

== External links ==
*{{springer
 | title= Variational inequalities
 | id= V/v120010
 | last= Panagiotopoulos
 | first= P.D. 
 | author-link= 
}}

{{DEFAULTSORT:Variational Inequality}}
[[Category:Partial differential equations]]
[[Category:Calculus of variations]]