Reference request for Newton's method on a convex/concave function

Question

The proof featured at http://planetmath.org/newtonsmethodworksforconvexrealfunctions shows that, roughly speaking, Newton's method is unconditionally convergent for convex/concave functions.

I am looking for a textbook or paper reference of this fact. I have found part of it in Michael Spivak's Calculus: enter image description here

You could look at the textbook Convex Optimization, by Boyd and Vandenberghe which is free online. — littleO, Feb 20 '15 at 02:15
I do not think there will be much there, as Boyd is interested in maximizing $f$, and hence he will try to find roots of $f^\prime$ (using quasi-Newton methods). However, $f^\prime$ is not necessarily convex/concave whenever $f$ is. — parsiad, Mar 11 '15 at 00:00
Ah, good point, I didn't understand your question when I posted that comment. — littleO, Mar 11 '15 at 00:07

parsiad · Accepted Answer · 2015-03-11T16:26:49.100

7

I have not found any references from books, but the cached version of a PlanetMath page (currently down) captures all of the relevant cases (attached). I am posting it to answer the question for other readers.

edited Mar 11 '15 at 16:26

answered Mar 10 '15 at 23:58

parsiad

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Reference request for Newton's method on a convex/concave function

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