# Variational analysis

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Variational_analysis
> Markdown URL: https://mediated.wiki/source/Variational_analysis.md
> Source: https://en.wikipedia.org/wiki/Variational_analysis
> Source revision: 1343770568
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

In [mathematics](/source/mathematics), '''variational analysis''' is the combination and extension of methods from [convex optimization](/source/convex_optimization) and the classical [calculus of variations](/source/calculus_of_variations) to a more general theory.{{sfn|Rockafellar|Wets|2009}} This includes the more general problems of [optimization theory](/source/optimization_theory), including topics in [set-valued analysis](/source/set-valued_analysis), e.g. [generalized derivative](/source/generalized_derivative)s.

In the [Mathematics Subject Classification](/source/Mathematics_Subject_Classification) scheme (MSC2010), the field of "Set-valued and variational analysis"  is coded by "49J53".<ref>{{cite web| url=http://www.ams.org/mathscinet/msc/msc2010.html?t=49Jxx&btn=Current| title=49J53 Set-valued and variational analysis|date=5 July 2010}}</ref>

== 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](/source/R._Tyrrell_Rockafellar) and [Roger J-B Wets](/source/Roger_J-B_Wets).{{sfn|Rockafellar|Wets|2009}}{{fv|date=April 2024}}

== Existence of minima ==
A classical result is that a [lower semicontinuous](/source/lower_semicontinuous) function on a [compact set](/source/compact_set) attains its minimum. Results from variational analysis such as [Ekeland's variational principle](/source/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.<ref>{{Cite journal |last=Borwein |first=J. M. |last2=Preiss |first2=D. |date=1987 |title=A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions |url=https://www.ams.org/tran/1987-303-02/S0002-9947-1987-0902782-7/ |journal=Transactions of the American Mathematical Society |language=en |volume=303 |issue=2 |pages=517–527 |doi=10.1090/S0002-9947-1987-0902782-7 |issn=0002-9947|hdl=1959.13/940776 |hdl-access=free |url-access=subscription }}</ref>

== Generalized derivatives ==
The classical [Fermat's theorem](/source/Fermat's_theorem_(stationary_points)) 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](/source/derivative) must be zero at that point. For problems where a [smooth function](/source/smooth_function) 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](/source/Lagrange_multiplier)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](/source/convex_function)s by generalizing the notion of derivative to that of [subderivative](/source/subderivative). Further generalization of the notion of the derivative such as the [Clarke generalized gradient](/source/Clarke_generalized_derivative) allow the results to be extended to nonsmooth [locally Lipschitz](/source/locally_Lipschitz) functions.<ref>
Frank H. Clarke, ''Optimization and Nonsmooth Analysis'', SIAM, 1990.
</ref>

== See also ==

* {{annotated link|Convex analysis}}
* {{annotated link|Functional analysis}}
* {{annotated link|Oriented projective geometry}}
* [Optimization](/source/Mathematical_optimization)

== Citations ==
{{reflist}}

== References ==

* {{Rockafellar Wets Variational Analysis 2009 Springer}} <!-- {{sfn|Rockafellar|Wets|2009|p=}} --> https://doi.org/10.1007/978-3-642-02431-3
* Ekeland, Ivar; Te{{Acute}}mam, 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 ==
*{{Commonscat-inline}}

{{Convex analysis and variational analysis}}

Category:Variational analysis

---
Adapted from the Wikipedia article [Variational analysis](https://en.wikipedia.org/wiki/Variational_analysis) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Variational_analysis?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.