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 ...

, the term variational analysis usually denotes 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 probl ...

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 functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...

to a more general theory. This includes the more general problems of

optimization theory 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 ...

, including topics in

set-valued analysis A set-valued function (or correspondence) is a mathematical function that maps elements from one set, the domain of the function, to subsets of another set. Set-valued functions are used in a variety of mathematical fields, including optimizatio ...

, e.g.

generalized derivative In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, geometry, etc. Fréchet derivative The Fréchet ...

s. In the

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

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 Ralph Tyrrell Rockafellar (born February 10, 1935) is an American mathematician and one of the leading scholars in optimization theory and related fields of analysis and combinatorics. He is the author of four major books including the landmark ...

and

Roger J-B Wets Roger Jean-Baptiste Robert Wets (born February 1937) is a "pioneer" in stochastic programming and a leader in variational analysis who publishes as Roger J-B Wets. His research, expositions, graduate students, and his collaboration with R. Tyrr ...

Existence of Minima

A classical result is that a

lower semicontinuous In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, rou ...

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 by making precise the idea of a space having no "punctures" or "missing endpoints", i ...

attains its minimum. Results from variational analysis such as

Ekeland's variational principle In mathematical analysis, Ekeland's variational principle, discovered by Ivar Ekeland, is a theorem that asserts that there exist nearly optimal solutions to some optimization problems. Ekeland's principle can be used when the lower level set of a ...

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.

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 of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...

must be zero at that point. For problems where a

smooth function In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...

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 subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied ex ...

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 points on the graph of a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigra ...

s by generalizing the notion of derivative to that of

subderivative In mathematics, the subderivative, subgradient, and subdifferential generalize the derivative to convex functions which are not necessarily differentiable. Subderivatives arise in convex analysis, the study of convex functions, often in connection ...

. Further generalization of the notion of the derivative such as the

Clarke generalized gradient Clarke is a surname which means "clerk". The surname is of English and Irish origin and comes from the Latin . Variants include Clerk and Clark. Clarke is also uncommonly chosen as a given name. Irish surname origin Clarke is a popular surname in ...

allow the results to be extended to nonsmooth

locally Lipschitz In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exis ...

functions. Frank H. Clarke, ''Optimization and Nonsmooth Analysis'', SIAM, 1990.

Citations

References

External links