First Order Conditions for Semidefinite Representations of Convex Sets Defined by Rational or Singular Polynomials

TL;DR

This paper establishes conditions under which convex sets defined by rational or singular polynomials can be represented as semidefinite programs, expanding the understanding of SDP representability for complex polynomial-defined sets.

Contribution

It provides explicit semidefinite representation conditions for convex sets defined by rational or singular polynomials, using Positivstellensatz certificates and perspective transformations.

Findings

Semidefinite representations are possible under preordering and q-module concavity conditions.

Explicit SDP representations are constructed for sets with boundary singularities using perspective transformations.

In two dimensions, sets with certain Laurent expansion properties always admit explicit SDP representations.

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={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.