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Theoretical Computer Science

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  • $\begingroup$ Relative to what? Also exact optimization is not generally in P, as far as I know. $\endgroup$ Commented Apr 4, 2014 at 6:13
  • $\begingroup$ Convex optimization is NP-hard. For example, it is coNP-complete to determine a given matrix is copositive, and the set of all copositive matrices (of given size) forms a closed convex cone. $\endgroup$ Commented Apr 4, 2014 at 6:30
  • $\begingroup$ @Yoshio Hmm, this course seemed to give the impression that is was as solved as linear programming. Maybe is it NP-hard to determine whether or something is convex (like the example you gave), yet within P for actually solving the problem once you prove that? $\endgroup$ Commented Apr 4, 2014 at 18:00
  • $\begingroup$ @Sasho No, exact optimization (in general) is NP Hard, and even slightly non-convex problems become quickly intractable. (assuming P!=NP) $\endgroup$ Commented Apr 4, 2014 at 18:02
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    $\begingroup$ The mildest assumptions that work AFAIK are a separation oracle for the convex set, or a membership oracle and a point strictly inside the set (sufficiently far away from all boundaries). And even under these assumptions, exact optimization is possible only for LP problems. Even for SDPs you need to be very careful. Maybe you need to make your way through the course before you give an answer :) $\endgroup$ Commented Apr 4, 2014 at 18:44