First Order Conditions for Semidefinite Representations of Convex Sets Defined by Rational or Singular Polynomials
| dc.creator | Nie, Jiawang | |
| dc.date | 2008-06-28 | |
| dc.date.accessioned | 2026-07-07T09:47:22Z | |
| dc.date.available | 2026-07-07T09:47:22Z | |
| dc.description | 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={x\in 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. | |
| dc.description | 25 pages | |
| dc.identifier | https://arxiv.org/abs/0806.4721 | |
| dc.identifier | http://arxiv.org/abs/0806.4721 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/163853 | |
| dc.subject | Optimization and Control | |
| dc.subject | Algebraic Geometry | |
| dc.title | First Order Conditions for Semidefinite Representations of Convex Sets Defined by Rational or Singular Polynomials | |
| dc.type | text |