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 | Let''X'' be a normed vector space, and let ''X*'' denote its topological dual. Given an extended real-valued function ''f'': ''X''→[0,+{{math|+ ∞}}], its ''convex conjugate'' ''f*'':''X*'' → [0, +{{math|+ ∞}}] is defined by | ||
:<math> test </math> | :<math> test </math> | ||
Revision as of 22:23, 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,+Template:Math], its convex conjugate f*:X* → [0, +Template:Math] is defined by
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References
- ↑ H. Brezis, Functional Analysis.