Sufficient and Necessary Conditions for Semidefinite Representability of Convex Hulls and Sets

• Authors: J. William Helton;Jiawang Nie
• Venue: SIAM Journal on Optimization
• Year: 2009

A set $S\subseteq\mathbb{R}^n$ is called semidefinite programming (SDP) representable or semidefinite representable if $S$ equals the projection of a set in higher dimensional space which is describable by some linear matrix inequality (LMI). Clearly, if $S$ is SDP representable, then $S$ must be convex and semialgebraic (it is describable by conjunctions and disjunctions of polynomial equalities or inequalities). This paper proves sufficient conditions and necessary conditions for SDP representability of convex sets and convex hulls by proposing a new approach to constructing SDP representations. The contributions of this paper are: (i) For bounded SDP representable sets $W_1,\dots,W_m$, we give an explicit construction of an SDP representation for $\mathrm{conv}(\cup_{k=1}^mW_k)$. This provides a technique for building global SDP representations from the local ones. (ii) For the SDP representability of a compact convex semialgebraic set $S$, we prove sufficient: the boundary $\partial S$ is nonsingular and positively curved, while necessary is: $\partial S$ has nonnegative curvature at each nonsingular point. In terms of defining polynomials for $S$, nonsingular boundary amounts to them having nonvanishing gradient at each point on $\partial S$ and the curvature condition can be expressed as their strict versus nonstrict quasi-concavity at those points on $\partial S$ where they vanish. The gaps between them are $\partial S$ having or not having singular points either of the gradient or of the curvature's positivity. A sufficient condition bypassing the gaps is when some defining polynomials of $S$ satisfy an algebraic condition called sos-concavity. (iii) For the SDP representability of the convex hull of a compact nonconvex semialgebraic set $T$, we find that the critical object is $\partial_c T$, the maximum subset of $\partial T$ contained in $\partial\mathrm{conv}(T)$. We prove sufficient for SDP representability: $\partial_c T$ is nonsingular and positively curved, and necessary is: $\partial_c T$ has nonnegative curvature at nonsingular points. The gaps between our sufficient and necessary conditions are similar to case (ii). The positive definite Lagrange Hessian (PDLH) condition, which meshes well with constructions, is also discussed.