Convex function: Difference between revisions

Revision as of 20:40, 3 September 2022

Revision as of 08:07, 9 September 2022

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or, more generally,

<math display=block>\langle \nabla f(x) - \nabla f(y), x-y \rangle \ge m \|x-y\|^2 </math>

where <math>\langle \cdot, \cdot\rangle</math> is any [[inner product]], and <math>\|\cdot\|</math> is anythe corresponding [[Norm (mathematics)|norm]]. Some authors, such as <ref name="ciarlet">{{cite book| title=Introduction to numerical linear algebra and optimisation|author=Philippe G. Ciarlet|publisher=Cambridge University Press |year=1989 |isbn=9780521339841}}</ref> refer to functions satisfying this inequality as [[Elliptic operator|elliptic]] functions.

An equivalent condition is the following:<ref name="nesterov">{{cite book|pages=[https://archive.org/details/introductorylect00nest/page/n79 63]–64|title=Introductory Lectures on Convex Optimization: A Basic Course|url=https://archive.org/details/introductorylect00nest|url-access=limited|author=Yurii Nesterov|publisher=Kluwer Academic Publishers|year=2004|isbn=9781402075537}}</ref>