Proof of the direction of directional derivative is convex

Asked Jan 24 '15 at 05:41

Active Jan 24 '15 at 06:05

Viewed 771 times

How to prove the following is cvx in $v$ (the direction of directional derivative):

$h(v) = $inf $_{a \geq 0} \frac {f(x+av)-f(x)}{a}$ with $f$ is cvx and $x \in$dom $f$.

It looks like a pointwise infimum? Any hint of proving this? I have no idea where I can start.

edited Jan 24 '15 at 06:05

asked Jan 24 '15 at 05:41

sleeve chen

2

Does cvx mean convex? Why so cryptic? I presume you mean $a\ge 0$? Show that $h$ is positive homogeneous and subadditive. I had forgotten I had answered part of this a long time ago: http://en.wikipedia.org/wiki/Convex_function#Properties. – copper.hat Jan 24 '15 at 06:00
Excuse me, I meant the following link: http://math.stackexchange.com/a/304608/27978. – copper.hat Jan 24 '15 at 17:02

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