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- W2949982767 abstract "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 S_D(f) = {x in D: f(x) >= 0}. Here D={xin R^n: g_1(x) >= 0, ..., g_m(x) >= 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 mc{D}, we prove that S_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. When f(x) is a polynomial or rational function having singularities on the boundary of S_D(f), a perspective transformation is introduced to find some explicit semidefinite representations for S_D(f) under certain conditions. In the particular case n=2, if the Laurent expansion of f(x) around one singular point has only two consecutive homogeneous parts, we show that S_D(f) always admits an explicitly constructible semidefinite representation." @default.
- W2949982767 created "2019-06-27" @default.
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- W2949982767 date "2008-06-28" @default.
- W2949982767 modified "2023-09-23" @default.
- W2949982767 title "First Order Conditions for Semidefinite Representations of Convex Sets Defined by Rational or Singular Polynomials" @default.
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