Fenchel–Moreau theorem
In convex analysis, the Fenchel–Moreau theorem (named after Werner Fenchel and Jean Jacques Moreau) or Fenchel biconjugation theorem (or just biconjugation theorem) is a theorem which gives necessary and sufficient conditions for a function to be equal to its biconjugate. This is in contrast to the general property that for any function [math]\displaystyle{ f^{**} \leq f }[/math].[1][2] This can be seen as a generalization of the bipolar theorem.[1] It is used in duality theory to prove strong duality (via the perturbation function).
Statement
Let [math]\displaystyle{ (X,\tau) }[/math] be a Hausdorff locally convex space, for any extended real valued function [math]\displaystyle{ f: X \to \mathbb{R} \cup \{\pm \infty\} }[/math] it follows that [math]\displaystyle{ f = f^{**} }[/math] if and only if one of the following is true
- [math]\displaystyle{ f }[/math] is a proper, lower semi-continuous, and convex function,
- [math]\displaystyle{ f \equiv +\infty }[/math], or
- [math]\displaystyle{ f \equiv -\infty }[/math].[1][3][4]
References
- ↑ 1.0 1.1 1.2 Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. pp. 76–77. ISBN 9780387295701.
- ↑ Zălinescu, Constantin (2002). Convex analysis in general vector spaces. River Edge, NJ: World Scientific Publishing Co., Inc.. pp. 75–79. ISBN 981-238-067-1.
- ↑ Hang-Chin Lai; Lai-Jui Lin (May 1988). "The Fenchel-Moreau Theorem for Set Functions". Proceedings of the American Mathematical Society (American Mathematical Society) 103 (1): 85–90. doi:10.2307/2047532.
- ↑ Shozo Koshi; Naoto Komuro (1983). "A generalization of the Fenchel–Moreau theorem". Proc. Japan Acad. Ser. A Math. Sci. 59 (5): 178–181.
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