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Published in:
2013 | OriginalPaper | Chapter
4. Convex analysis
Author : Francis Clarke
Published in: Functional Analysis, Calculus of Variations and Optimal Control
Publisher: Springer London
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Abstract
The reader has been told that convex functions are of particular interest. One reason for this (among others) lies in the fact that a certain generalized calculus can be developed for convex functions (and sets), one that is often referred to nowadays as convex analysis. The subject has widespread applications. The role of the derivative in convex analysis is played by the subdifferential, the collection of subgradients. An element ζ of X
∗ is a subgradient of f at x (in the sense of convex analysis) if it satisfies the inequality The operation of conjugacy is also central to convex analysis. Before examining convex calculus however, we add to our toolkit a useful result on permuting min and max, a result called the minimax theorem.
$$f(y) - f(x) \: \geqslant \:\langle \, \zeta ,\,y-x \,\rangle ~\; \forall \,y\in\, X.$$