Fennel Theorem - Moreau

A function that is not lower semicontinuous . By the Fenchel-Moreau theorem, this function is not equal to its second conjugate.

The Fenchel – Moreau theorem is a necessary and sufficient condition for a real-valued function to be equal to its double convex conjugation . Moreover, for any function, it is true that $f^{**}\leqslant f$ ${\ displaystyle f ^ {**} \ leqslant f}$ ${\ displaystyle f ^ {**} \ leqslant f}$ ^[1] ^[2] .

The statement can be considered as a generalization ^[1] . It is used in duality theory to prove strong duality (via ).

The theorem was proved for a finite-dimensional case by Werner Fenchel in 1949 , and for an infinite-dimensional case, by Jean-Jacques Moreau in 1960 ^[3] .

Statement of the theorem

Let be $(X,\tau )$ ${\ displaystyle (X, \ tau)}$ $(X,\tau )$ is a Hausdorff locally convex space . For any function with values on the extended number line $f:X\to \mathbb {R} \cup \{\pm \infty \}$ ${\ displaystyle f: X \ to \ mathbb {R} \ cup \ {\ pm \ infty \}}$ $f:X\to \mathbb {R} \cup \{\pm \infty \}$ follows that $f=f^{**}$ ${\ displaystyle f = f ^ {**}}$ $f=f^{**}$ where $f^{*}$ ${\ displaystyle f ^ {*}}$ $f^{*}$ - convex conjugation to $f$ ${\ displaystyle f}$ $f$ , if and only if one of the following conditions is true:

$f$ ${\ displaystyle f}$ $f$ is lower semicontinuous and convex function ,
$f\equiv +\infty$ ${\ displaystyle f \ equiv + \ infty}$ $f\equiv +\infty$ , or
$f\equiv -\infty$ ${\ displaystyle f \ equiv - \ infty}$ $f\equiv -\infty$ ^[1] ^[4] ^[5] .

In a geometric formulation, the theorem states that the necessary and sufficient condition for a function epigraph to be the intersection of epigraphs of affine functions is the convexity and closeness of this function ^[3] .

Notes

↑ ¹ ² ³ Borwein, Lewis, 2006 , p. 76–77.
↑ Zălinescu, 2002 , p. 75–79.
↑ ¹ ² Tikhomirov V. Convexity geometry // Quantum. - 2003. - No. 4.
↑ Lai, Lin, 1988 , p. 85–90.
↑ Koshi, Komuro, 1983 , p. 178–181.

Literature

Ioffe A. D., Tikhomirov V. M. Duality of convex functions and extremal problems . - UMN. - 1968. - T. 23, No. 6 (144). - S. 51–116.
Strekalovsky A.S. Introduction to convex analysis . - Irkutsk State University, 2009.
Jonathan Borwein, Adrian Lewis. Convex Analysis and Nonlinear Optimization: Theory and Examples. - 2. - Springer, 2006. - ISBN 9780387295701 .
Constantin Zălinescu. Convex analysis in general vector spaces. - River Edge, NJ: World Scientific Publishing Co., Inc., 2002. - ISBN 981-238-067-1 .
Hang-Chin Lai, Lai-Jui Lin. The Fenchel-Moreau Theorem for Set Functions // Proceedings of the American Mathematical Society. - American Mathematical Society, 1988. - May (vol. 103). - DOI : 10.2307 / 2047532 .
Shozo Koshi, Naoto Komuro. A generalization of the Fenchel – Moreau theorem // Proc. Japan Acad. Ser. A math. Sci. . - 1983 .-- T. 59 , no. 5 .

[_8d235279be84b86d-1] ¹ ² ³ Borwein, Lewis, 2006 , p. 76–77.

[_248c306d9fb30428-2] Zălinescu, 2002 , p. 75–79.

[тихомиров-квант-3] ¹ ² Tikhomirov V. Convexity geometry // Quantum. - 2003. - No. 4.

[_d341a9dc1e71f9df-4] Lai, Lin, 1988 , p. 85–90.

[_84499d67eb2a958c-5] Koshi, Komuro, 1983 , p. 178–181.

Fennel Theorem - Moreau

Statement of the theorem

Notes

Literature

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