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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 (Antman 1983, pp. 282–284) and (Fichera 1995): the first papers of the theory were (Fichera 1963) and (Fichera 1964a), (Fichera 1964b).
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 is constrained to lie above a given obstacle.
Many free boundary problems can profitably be viewed as variational inequalities for the sake of analysis. To illustrate this point, we first turn to the minimization of a function F {\displaystyle F} of n {\displaystyle n} real variables over a convex set C {\displaystyle C} ; the minimizer x {\displaystyle x} is characterized by the condition
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 dynamics and inequality constraints.
One particularly important class of equilibrium problems which has aided in the rise of projected dynamical systems has been that of variational inequalities. The formalization of projected dynamical systems began in the 1990s in Section 5.3 of the paper of Dupuis and Ishii.
Since then the study of complementarity problems and variational inequalities has expanded enormously. Areas of mathematics and science that contributed to the development of complementarity theory include: optimization, equilibrium problems, variational inequality theory, fixed point theory, topological degree theory and nonlinear analysis.
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. Many well-known problem types are special cases of, or may be reduced to MCP.
Hilbert's twentieth problem; History of variational principles in physics; ... Variational inequality; Variational vector field; W. Weierstrass–Erdmann condition