mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, variational analysis is the combination and extension of methods from

convex optimization Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems ...

and the classical

calculus of variations The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...

to a more general theory. This includes the more general problems of optimization theory, including topics in set-valued analysis, e.g. generalized derivatives. In the

Mathematics Subject Classification The Mathematics Subject Classification (MSC) is an alphanumerical classification scheme that has collaboratively been produced by staff of, and based on the coverage of, the two major mathematical reviewing databases, Mathematical Reviews and Zen ...

scheme (MSC2010), the field of "Set-valued and variational analysis" is coded by "49J53".

History

While this area of mathematics has a long history, the first use of the term "Variational analysis" in this sense was in an eponymous book by R. Tyrrell Rockafellar and Roger J-B Wets.

Existence of minima

A classical result is that a lower semicontinuous function on a

compact set In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...

attains its minimum. Results from variational analysis such as Ekeland's variational principle allow us to extend this result of lower semicontinuous functions on non-compact sets provided that the function has a lower bound and at the cost of adding a small perturbation to the function. A smooth variant is known as the Borwein-Press variational principle.

Generalized derivatives

The classical Fermat's theorem says that if a differentiable function attains its minimum at a point, and that point is an interior point of its domain, then its

derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

must be zero at that point. For problems where a

smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; t ...

must be minimized subject to constraints which can be expressed in the form of other smooth functions being equal to zero, the method of

Lagrange multiplier In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function (mathematics), function subject to constraint (mathematics), equation constraints (i.e., subject to the conditio ...

s, another classical result, gives necessary conditions in terms of the derivatives of the function. The ideas of these classical results can be extended to nondifferentiable

convex function In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of a function, graph of the function lies above or on the graph between the two points. Equivalently, a function is conve ...

s by generalizing the notion of derivative to that of subderivative. Further generalization of the notion of the derivative such as the Clarke generalized gradient allow the results to be extended to nonsmooth locally Lipschitz functions. Frank H. Clarke, ''Optimization and Nonsmooth Analysis'', SIAM, 1990.

Citations

References

* https://doi.org/10.1007/978-3-642-02431-3 * Ekeland, Ivar; Temam, Roger; Convex analysis and variational problems 1999 SIAM https://doi.org/10.1137/1.9781611971088 * Borwein, Jonathan M.; Zhu, Qiji J.; Techniques of Variational Analysis 2005 Springer https://doi.org/10.1007/0-387-28271-8

External links