Fenchel-Moreau and Primal/Dual Optimization Problems

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 $f:X\to \mathbb {R} \cup \{+\infty \}$, its convex conjugate $f^{*}:X^{*}\to \mathbb {R} \cup \{+\infty \}$ is defined by

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