- Variational inequality
Variational inequality is a
mathematical theoryintended for the study of equilibrium problems. Guido Stampacchiaput forth the theory in 1964to 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" : K → R"n", the finite-dimensional variational inequality problem associated with K is
where <·,·> is the standard
inner producton 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
Consider the problem of finding the minimal value of a continuous differentiable function "f" over a closed interval . Let "x" be the point in "I" where the minimum occurs. Three cases can occur:
:1) if then ′(x)=0;:2) if then ′(x) ≥ 0;:3) if then ′(x) ≤ 0.
These conditions can be summarized as the problem of
*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.
projected dynamical system
differential variational inequality
Mathematical programming with equilibrium constraints
Wikimedia Foundation. 2010.
Look at other dictionaries:
Differential variational inequality — In mathematics, a differential variational inequality (DVI) is a dynamical system that incorporates ordinary differential equations and variational inequalities or complementarity problems. DVIs are useful for representing models involving both… … Wikipedia
Obstacle problem — The obstacle problem is a classic motivating example in the mathematical study of variational inequalities and free boundary problems. The problem is to find the equilibrium position of an elastic membrane whose boundary is held fixed, and which… … Wikipedia
List of mathematics articles (V) — NOTOC Vac Vacuous truth Vague topology Valence of average numbers Valentin Vornicu Validity (statistics) Valuation (algebra) Valuation (logic) Valuation (mathematics) Valuation (measure theory) Valuation of options Valuation ring Valuative… … Wikipedia
Projected dynamical system — Projected dynamical systems is a mathematical theory investigating the behaviour of dynamical systems where solutions are restricted to a constraint set. The discipline shares connections to and applications with both the static world of… … Wikipedia
Complementarity theory — This article is related to mathematical programming. For other uses see complementarity. A complementarity problem is a type of mathematical optimization problem. It is the problem of optimizing (minimizing or maximizing) a function of two vector … Wikipedia
Optimization (mathematics) — In mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an… … Wikipedia
Anna Nagurney — (born ??) is a Ukrainian American mathematician, economist, educator and author, and Professor of Operations Management at the University of Massachusetts at Amherst. Biography Anna Nagurney attended Brown University in Providence, Rhode Island… … Wikipedia
List of mathematics articles (D) — NOTOC D D distribution D module D D Agostino s K squared test D Alembert Euler condition D Alembert operator D Alembert s formula D Alembert s paradox D Alembert s principle Dagger category Dagger compact category Dagger symmetric monoidal… … Wikipedia
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… … Wikipedia
Optimisation (mathématiques) — L optimisation est une branche des mathématiques, cherchant à analyser et à résoudre analytiquement ou numériquement les problèmes qui consistent à déterminer le meilleur élément d un ensemble, au sens d un critère quantitatif donné. Ce mot vient … Wikipédia en Français