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I am studying now convex sets and very interested in applications of them in Computer Science (maybe ACM problems) and other real life problems. Coud you please give some examples?

P.S. I am interested in examples that are strictly related to convex sets. And without this notion they would not be solved.

Turkhan Badalov
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    A local minimum is a global minimum in the case of a convex set. This is very useful for optimization problems. In the case of strictly convex sets, the minimum is moreover unique. – Peter Nov 16 '16 at 13:39
  • Linear programming and least squares problems alone give a wide variety of applied convex optimization problems – Ben Grossmann Nov 16 '16 at 13:41
  • @Omnomnomnom, but I am interested in examples that are strictly related to convex sets. And without this notion they would not be solved. – Turkhan Badalov Nov 16 '16 at 13:43
  • What do you mean by "strictly related to convex sets"? Isn't any convex optimization problem "related" (whatever that means) to certain convex sets (such as the constraint set, and the epigraph of the objective function)? – littleO Nov 16 '16 at 13:47
  • @littleO, of course, if you are given some problem where it is clearly given some convex set and asks you to minimize some function, that's okay. But is there any problem where its definition doesn't remind you complex-optimization, but actually it is solved using complex-optimization? – Turkhan Badalov Nov 16 '16 at 13:53
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    The book Convex Optimization by Boyd and Vandenberghe (free online) is filled with applications which do not appear to be convex at first sight, and yet with clever tricks or techniques can be formulated as convex problems. – littleO Nov 16 '16 at 13:58
  • @littleO, oh, great! Thank you very much! – Turkhan Badalov Nov 16 '16 at 14:00
  • Convex sets per se have little value. It is only when such set is represented by convex functions that you can apply optimization algorithms. – LinAlg Nov 16 '16 at 17:09

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Peter comment answer pretty well this question. It can be found misplaced at What are the main uses of Convex Functions? by Evgeny adding uniqueness. So, in my oppinion optimization is the key:

Every local extremum is also a global extremum and there exists only one such extremum.

Community
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Ernesto Iglesias
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    Quasiconvex functions and unimodal functions offer the same advantage. Also, you seem to talk about functions rather than sets. They are related via the epigraph, but you talk about the optimum of a function. – LinAlg Jan 30 '17 at 15:14
  • Yes you are right. Anyway I can't offer any answer outside basic optimization theory. It's not my field of expertice. I'm whaiting for someone else answer this question. – Ernesto Iglesias Jan 30 '17 at 20:31