In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a variational inequality is an
inequality involving a
functional
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...
, which has to be
solved for all possible values of a given
variable, belonging usually to a
convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
. The
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
theory
A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may ...
of variational inequalities was initially developed to deal with
equilibrium problems, precisely the
Signorini problem
The Signorini problem is an elastostatics problem in linear elasticity: it consists in finding the elastic equilibrium configuration of an anisotropic non-homogeneous elastic body, resting on a rigid frictionless surface and subject only to i ...
: in that model problem, the functional involved was obtained as the
first variation In applied mathematics and the calculus of variations, the first variation of a functional ''J''(''y'') is defined as the linear functional \delta J(y) mapping the function ''h'' to
:\delta J(y,h) = \lim_ \frac = \left.\frac J(y + \varepsilon h ...
of the involved
potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potenti ...
. Therefore, it has a
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
Economics () is the social science that studies the production, distribution, and consumption of goods and services.
Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...
,
finance
Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of f ...
,
optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
and
game theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
.
History
The first problem involving a variational inequality was the
Signorini problem
The Signorini problem is an elastostatics problem in linear elasticity: it consists in finding the elastic equilibrium configuration of an anisotropic non-homogeneous elastic body, resting on a rigid frictionless surface and subject only to i ...
, posed by
Antonio Signorini in 1959 and solved by
Gaetano Fichera
Gaetano Fichera (8 February 1922 – 1 June 1996) was an Italian mathematician, working in mathematical analysis, linear elasticity, partial differential equations and several complex variables. He was born in Acireale, and died in Rome.
Biog ...
in 1963, according to the references and : the first papers of the theory were and , . Later on,
Guido Stampacchia
Guido Stampacchia (26 March 1922 – 27 April 1978) was an Italian mathematician, known for his work on the theory of variational inequalities, the calculus of variation and the theory of elliptic partial differential equations..
Life and academ ...
proved his generalization to the
Lax–Milgram theorem Weak formulations are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, equations or c ...
in in order to study the
regularity problem for
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to h ...
s and
coin
A coin is a small, flat (usually depending on the country or value), round piece of metal or plastic used primarily as a medium of exchange or legal tender. They are standardized in weight, and produced in large quantities at a mint in order ...
ed the name "variational inequality" for all the problems involving
inequalities
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
of this kind.
Georges Duvaut Georges may refer to:
Places
*Georges River, New South Wales, Australia
*Georges Quay (Dublin)
*Georges Township, Fayette County, Pennsylvania
Other uses
*Georges (name)
* ''Georges'' (novel), a novel by Alexandre Dumas
* "Georges" (song), a 1977 ...
encouraged his
graduate student
Postgraduate or graduate education refers to academic or professional degrees, certificates, diplomas, or other qualifications pursued by post-secondary students who have earned an undergraduate (bachelor's) degree.
The organization and s ...
s to study and expand on Fichera's work, after attending a conference in
Brixen
Brixen (, ; it, Bressanone ; lld, Porsenù or ) is a town in South Tyrol, northern Italy, located about north of Bolzano.
Geography
First mentioned in 901, Brixen is the third largest city and oldest town in the province, and the artistic an ...
on 1965 where Fichera presented his study of the Signorini problem, as reports: thus the theory become widely known throughout
France
France (), officially the French Republic ( ), is a country primarily located in Western Europe. It also comprises of Overseas France, overseas regions and territories in the Americas and the Atlantic Ocean, Atlantic, Pacific Ocean, Pac ...
. Also in 1965, Stampacchia and
Jacques-Louis Lions
Jacques-Louis Lions (; 3 May 1928 – 17 May 2001) was a French mathematician who made contributions to the theory of partial differential equations and to stochastic control, among other areas. He received the SIAM's John von Neumann Lecture ...
extended earlier results of , announcing them in the paper : full proofs of their results appeared later in the paper .
Definition
Following , the definition of a variational inequality is the following one.
Given a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
, a
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of
, and a functional
from
to the
dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
of the space
,
the variational inequality problem
is the problem of
solving
for the
variable belonging to
the following
inequality:
:
where
is the
duality pairing
Duality may refer to:
Mathematics
* Duality (mathematics), a mathematical concept
** Dual (category theory), a formalization of mathematical duality
** Duality (optimization)
** Duality (order theory), a concept regarding binary relations
** D ...
.
In general, the variational inequality problem can be formulated on any
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
– or
infinite
Infinite may refer to:
Mathematics
* Infinite set, a set that is not a finite set
*Infinity, an abstract concept describing something without any limit
Music
*Infinite (group), a South Korean boy band
*''Infinite'' (EP), debut EP of American m ...
-
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
al
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
. 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 or prove its regularity.
Examples
The problem of finding the minimal value of a real-valued function of real variable
This is a standard example problem, reported by : consider the problem of finding the
minimal value of a
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
over a
closed interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...