Variational inequality
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Variational inequality is a mathematical theory intended for the study of equilibrium problems. Guido Stampacchia and Philip Hartman put forth the theory in 1966 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 Rn. Given a subset K of Rn and a mapping F : K → Rn, the finite-dimensional variational inequality problem associated with K is
where <·,·> is the standard inner product on Rn.
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.[citation needed]
[edit] Examples
Consider the problem of finding the minimal value of a continuous differentiable function f over a closed interval I = [a,b]. Let x be the point in I where the minimum occurs. Three cases can occur:
- 1) if a < x < b then f ′(x)=0;
- 2) if x = a then f ′(x) ≥ 0;
- 3) if x = b then f ′(x) ≤ 0.
These conditions can be summarized as the problem of
