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Optimization problems

Two easy problems. The first one illustrates a general idea that is useful (it can be used to help in analytically deriving the FTA, for instance): show that an even polynomial achieves its minimum on $\R$ . The second, while interesting, doesn’t have any immediately interesting applications I know of:

Define the convex (a.k.a. Fenchel) conjugate of a function $f: \R^n \rightarrow \R \cup \{\infty\}$ to be the function $f^\star : \R^n \rightarrow \R \cup \{\infty\}$ defined by

$\displaystyle f^{\star}\left(x^\star\right) = \sup\left\{\left\langle x^\star,x\right\rangle - f\left(x\right) : x \in \mathbb{R}^n \right\} = - \inf\left\{f\left(x\right) - \left\langle x^\star,x\right\rangle : x \in \mathbb{R}^n \right\}$

show that if $f_p = \frac{|x|^p}{p}$ and $\frac{1}{p} + \frac{1}{q} = 1$ with p>1 but not necessarily an integer, then $f^\star = f_q$ .

Possibly relevant posts:

Equivalence of the $\gamma_2$ norm and the $\|\cdot\|_{\infty \rightarrow 1}^\star$ norm (1/27/2010)
An observation on the norming functionals of the $(\infty,1)^\star$ norm ball (2/24/2010)
Trace dual of the $\|\cdot\|_{\infty\rightarrow 1}$ norm (1/19/2010)

Tags: convex, optimization, polynomial

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Jan 24th, 2006 | Posted in Mathematics

Tags: convex, optimization, polynomial

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JuanPablo

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Jan 26th, 2006 at 09:29 | #1

a.k.a. Legendre transform (in physics gives the equivalence between Hamiltonian and Lagrangian mechanics)
Alex

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Jan 26th, 2006 at 14:05 | #2

Thanks for the info. It’s pretty strange, that such a seemingly abstract concept would show up in physics. But then again, happens all the time