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

摘要

A set is called semidefinite representable or semidefinite programming (SDP) representable if it equals the projection of a higher dimensional set which is defined by some Linear Matrix Inequality (LMI). This paper discusses the semidefinite representability conditions for convex sets of the form ${S_{mathcal {D}}(f) ={xin mathcal {D} : f(x) geq 0 }}$ . Here, ${mathcal {D}={xin mathbb {R}^n : g_1(x) geq 0, ldots, g_m(x) geq 0 }}$ is a convex domain defined by some “nice” concave polynomials g i (x) (they satisfy certain concavity certificates), and f(x) is a polynomial or rational function. When f(x) is concave over ${mathcal {D}}$ , we prove that ${S_{mathcal {D}}(f) }$ 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: $$f(u) + nabla f(u)^T(x-u) -f(x) geq 0, quad forall , x, u in mathcal {D}.$$ When f(x) is a polynomial or rational function having singularities on the boundary of ${S_{mathcal {D}}(f)}$ , a perspective transformation is introduced to find some explicit semidefinite representations for ${S_{mathcal {D}}(f)}$ under certain conditions. In the special case n = 2, if the Laurent expansion of f(x) around one singular point has only two consecutive homogeneous parts, we show that ${S_{mathcal {D}}(f)}$ always admits an explicitly constructible semidefinite representation.