Talk:Convex analysis

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 Field:  Analysis

Is there a difference between convex analysis and convex geometry? Perhaps the two should be merged. ??? --Kompik 10:51, 5 October 2007 (UTC)


I think this page should probably be expanded from its current 1 sentence description of convex analysis. Perhaps definition of a convex function/set should be given, as well as some basic properties and applications should be given (and mentions of main articles, for instance brief discussion of convex minimization). I would propose a structure like:

  1. Intro/Definitions (Convex set/function)
  2. Separating/Supporting hyperplane
  3. Conjugate/Biconjugate
  4. Convex Optimization

This is just off the top of my head. Does any one else have thoughts on the topic? Zfeinst (talk) 18:17, 23 February 2012 (UTC)

Your suggestion is good. An extension might also be informed by the spirit of "the bible" of convex analysis, Rockafellar's book, which presents convex analysis as an (oriented) extensions of (real) linear analysis:
  • affine manifolds generalized to half spaces and other convex sets,
  • the set containing the derivative generalized to the subdifferential set,
  • linear operators (and perhaps bilinear operators and bimodules?) generalized and positive linear operators to oriented monotone processes of convex or concave type (unpopular with the hoi polloi but justly praised by the leaders---Robinson, Aubin, several authors from the Romanian school, Borwein) to bifunctions (but this was where the spirit was willing but the mind was weak, for me).
There have been attempts, e.g. by Hörmander and other Swedes, etc., to develop complex convex analysis.
Good editor Isheden can warn you that I am a fundamentalist about insisting on convex "minimization" unless stationary-points/saddlepoints or convex maximization be also considered.
Cheers,  Kiefer.Wolfowitz 19:39, 23 February 2012 (UTC)
Yes, I think using Rockafellar heavily is wise. And some use of Zalinescu + Aliprantis/Border would be good for more general spaces beyond the finite dimensional real spaces. And for optimization concepts I think Boyd would be a good general reference. Of course this isn't to say this page should cover everything, but merely should give some background and point to other pages for the main pages, which some might disagree about what should be included - for instance I would probably include more about convex conjugates than others might want if left to my own devices. I'll perhaps try to add in some of the very basics soon (for instance just defining convex sets), and the rest can be added subsequently.
As for convex minimization vs. maximization, I saw that discussion on the the convex optimization talk page. My 2 cents are that when I think of convex optimization I think of minimization, but I can't think of any reason to not consider maximization problems. Zfeinst (talk) 20:03, 23 February 2012 (UTC)