TITLE: More general than semidefinite program?
QUESTION [12 upvotes]: I was TAing my convex optimization class and explaining that Linear Programs are a special case of Second Order Cone Programs, which are themselves special cases of Semidefinite Programs.  My question is, is there any well-established class of optimization problems that is more general than semidefinite programs?  Conic optimization problems would be an example of this, but I'm hoping for something a little more algebraic, if it exists.

REPLY [5 votes]: If you like, you might look at cones of sums of squares of polynomials (cones of PSD matrices are the same thing as cones of sums of squares of linear polynomials). This is the starting point of the modern technique of solving optimisation problems on semi-algebraic sets, due to J.Lasserre and others.
More generally, you might look at cones of nonnegative polynomials, and this opens up the whole Hilbert 17th problem business.