Optimization problems

January 24th, 2006 ~ Posted in: Mathematics

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.

Mathematics ~

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