science

Variational analysis

Variational analysis is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Variational analysis rather than just read about it. In short: In mathematics, variational analysis is the combination and extension of methods from convex optimization and the classical calculus of variations to a more general theory. This includes the more general problems of optimization theory, including topics in set-valued analysis, e.g. generalized derivatives.

Key takeaways

Variational analysis belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Variational analysis to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Variational analysis from memory before moving on to harder problems.

Reference excerpt

In mathematics, variational analysis is the combination and extension of methods from convex optimization and the classical calculus of variations 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 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 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 must be zero at that point. For problems where a 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 multipliers, 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 functions 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.

See also Convex analysis – Mathematics of convex functions and sets Functional analysis – Area of mathematics Oriented projective geometry Optimization

Citations

References Rockafellar, R. Tyrrell; Wets, Roger J.-B. (26 June 2009). Variational Analysis. Grundlehren der mathematischen Wissenschaften. Vol. 317. Berlin New York: Springer Science & Business Media. ISBN 9783642024313. OCLC 883392544. https://doi.org/10.1007/978-3-642-02431-3 Ekeland, Ivar; Témam, 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 Media related to Variational analysis at Wikimedia Commons

Worked examples

Example 1 — a first encounter with Variational analysis

Start with the simplest possible case. Write down what Variational analysis claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Variational analysis before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Variational analysis ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Variational analysis

In research: Variational analysis appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Variational analysis in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Variational analysis is common in secondary-school and first-year university syllabi. It links to neighbouring topics Variational analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Variational analysis outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Variational analysis in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Variational analysis means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Variational analysis out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Variational analysis in simple terms?

Why does Variational analysis matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Variational analysis?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Variational analysis.