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

• 成果类型: Article
• 署名作者: Nie, Jiawang
• 署名单位: University of California System; University of California San Diego
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-009-0339-9
• 发表日期: 2012
• 页码: 1-36

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-D(f) = {x is an element of D : f (x) >= 0}. Here, D = {x is an element of 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 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: f (u) + del f (u)(T) (x - u) - f (x) >= 0, for all x, u is an element of D. 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 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-D(f) always admits an explicitly constructible semidefinite representation.