First order conditions for semidefinite representations of convex sets defined by rational or singular polynomials

A set is called semidefinite representable or semidefinite programming (SDP) representable if it can be represented as the projection of a higher dimensional set which is represented by some Linear Matrix Inequality (LMI). This paper discuss the semidefinite representability conditions for convex sets of the form SD(f) = {x ∈ D : f(x) ≥ 0}. Here D = {x ∈ R n : g1(x) ≥ 0, · · · , gm(x) ≥ 0} is a convex domain defined by some “nice” concave polynomials gi(x) (they satisfy certain concavity certificates), and f(x) is a polynomial or rational function. When f(x) is concave over D, we prove that SD(f) has some explicit semidefinite representations under certain conditions called preordering concavity or q-module concavity, which are based on the Positivstellensatz certificates for the first order concavity criteria: