“Fenchel–Moreau theorem”的意思、由来-开放百科全书

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

.^[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 of theorem

Let

be a Hausdorff locally convex space, for any extended real valued function

it follows that

if and only if one of the following is true

References

1. ^¹²{{cite book |last1=Borwein |first1=Jonathan |authorlink1=Jonathan Borwein|last2=Lewis |first2=Adrian |title=Convex Analysis and Nonlinear Optimization: Theory and Examples| edition=2 |year=2006 |publisher=Springer |isbn=9780387295701|pages=76–77}}
2. ^{{cite book |last=Zălinescu |first=Constantin |title=Convex analysis in general vector spaces |publisher=World Scientific Publishing Co., Inc. |isbn=981-238-067-1 |mr=1921556 |issue=J |year=2002 |location=River Edge, NJ |pages=75–79}}
3. ^{{cite journal | author = Hang-Chin Lai | author2 = Lai-Jui Lin | date=May 1988 | title = The Fenchel-Moreau Theorem for Set Functions | journal = Proceedings of the American Mathematical Society | publisher = American Mathematical Society | volume = 103 | issue = 1 | pages = 85–90 | doi = 10.2307/2047532 | url = | format = | accessdate = }}
4. ^{{cite journal|title=A generalization of the Fenchel–Moreau theorem|author=Shozo Koshi|author2=Naoto Komuro|journal=Proc. Japan Acad. Ser. A Math. Sci.|volume=59|issue=5|year=1983|pages=178–181}}