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Fundamental regions in convex programming

In linear programming, the fundamental regions are polyhedra, since those are the intersection of half-spaces defined by linear inequalities. In semidefinite programming, the fundamental regions are ...
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What is the relation between different generalizations of linear programming?

Linear programming subsumed by each of Semidefinite programming (SDP) Convex programming (CXP) SOS programming (SSP) Is there any relation between each pair in the three? Are all three equivalent in ...