Fenchel-Moreau and Primal/Dual Optimization Problems: Difference between revisions

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Let''X'' be a normed vector space, and let ''X*'' denote its topological dual. Given an extended real-valued function <math> f: X \to \mathbb{R} \cup \{+\infty\} </math>, its ''convex conjugate'' <math>f^*:X^* \to \mathbb{R} \cup \{+\infty\}</math> is defined by  
Let''X'' be a normed vector space, and let ''X*'' denote its topological dual. Given an extended real-valued function <math> f: X \to \mathbb{R} \cup \{+\infty\} </math>, its ''convex conjugate'' <math>f^*:X^* \to \mathbb{R} \cup \{+\infty\}</math> is defined by  


:<math> test </math>
:<math> f^*(u):=\sup_{x \in X} \{ \langle u,x \rangle - f(x) \} \quad \forall u \in X^*. </math>


An immediate consequence of this definition is ''Young's Inequality'',
:<math> f^*(u) +f(x) \geq \langle u,x \rangle \quad \forall x \in X, u \in X^* . </math>
Furthermore, it follows directly from the definition that, for ''any'' function ''f'', its conjugate function ''f*'' is convex and lower semicontinuous.
In a similar way, for any function ''f'', its the ''biconjugate'' function <math> f^{**}:X \to \mathbb{R} \cup \{+\infty\} <\math> is defined by
:<math> f^{**}(x):=\sup_{u \in X^*} \{ \langle u,x \rangle - f^*(u) \} \quad \forall u \in X^*. </math>
As above, for any function ''f'', its biconjugate function ''f^*'' is convex and lower semicontinuous. Furthermore, by Young's inequality, we always have
:<math> f^{**}(x) \leq f(x) \quad \forall x \in X. <\math>
In this way, it is clear that, in order for equality to hold, it is necessary for ''f'' to be convex and lower semicontinuous. The heart of Fenchel-Moreau Theorem is that this is also sufficient.
  <ref name="Brezis" />
  <ref name="Brezis" />



Revision as of 22:33, 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 , its convex conjugate is defined by

An immediate consequence of this definition is Young's Inequality,

Furthermore, it follows directly from the definition that, for any function f, its conjugate function f* is convex and lower semicontinuous.

In a similar way, for any function f, its the biconjugate function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{**}:X \to \mathbb{R} \cup \{+\infty\} <\math> is defined by :<math> f^{**}(x):=\sup_{u \in X^*} \{ \langle u,x \rangle - f^*(u) \} \quad \forall u \in X^*. }

As above, for any function f, its biconjugate function f^* is convex and lower semicontinuous. Furthermore, by Young's inequality, we always have

<math> f^{**}(x) \leq f(x) \quad \forall x \in X. <\math>

In this way, it is clear that, in order for equality to hold, it is necessary for f to be convex and lower semicontinuous. The heart of Fenchel-Moreau Theorem is that this is also sufficient.


[1]

References

  1. H. Brezis, Functional Analysis.