Fenchel-Moreau and Primal/Dual Optimization Problems: Difference between revisions
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==Background on Conjugate Functions== | ==Background on Conjugate Functions== | ||
Let''X'' be a normed vector space, and let ''X*'' denote its topological dual. Given an extended real-valued function ''f: X→[0,+∞]'', its ''convex conjugate'' ''f*:X* → [0, +∞]'' is defined by | |||
:<math> test </math> | |||
<ref name="Brezis" /> | <ref name="Brezis" /> | ||
Revision as of 22:20, 7 April 2020
The Fenchel-Moreau Theorem is a fundamental result in convex analysis, characterizing the class of functions for which a function equals its biconjugate. A key consequence of this theorem is the equivalence of primal and dual optimization problems.
Background on Conjugate Functions
LetX be a normed vector space, and let X* denote its topological dual. Given an extended real-valued function f: X→[0,+∞], its convex conjugate f*:X* → [0, +∞] is defined by
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References
- ↑ H. Brezis, Functional Analysis.