Sufficient and Necessary Conditions for Semidefinite Representability of Convex Hulls and Sets
| dc.creator | Helton, J. William | |
| dc.creator | Nie, Jiawang | |
| dc.date | 2007-09-25 | |
| dc.date | 2008-12-07 | |
| dc.date.accessioned | 2026-07-07T12:09:32Z | |
| dc.date.available | 2026-07-07T12:09:32Z | |
| dc.description | A set $S\subseteq \re^n$ is called to be {\it Semidefinite (SDP)} representable if $S$ equals the projection of a set in higher dimensional space which is describable by some Linear Matrix Inequality (LMI). The contributions of this paper are: (i) For bounded SDP representable sets $W_1,...,W_m$, we give an explicit construction of an SDP representation for $\cv{\cup_{k=1}^mW_k}$. 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 condition: the boundary $\bdS$ is positively curved, and necessary condition: $\bdS$ has nonnegative curvature at smooth points and on nondegenerate corners. This amounts to the strict versus nonstrict quasi-concavity of defining polynomials on those points on $\bdS$ where they vanish. The gaps between them are $\bdS$ having positive versus nonnegative curvature and smooth versus nonsmooth points. A sufficient condition bypassing the gaps is when some defining polynomials of $S$ are sos-concave. (iii) For the SDP representability of the convex hull of a compact nonconvex semialgebraic set $T$, we find that the critical object is $\pt_cT$, the maximum subset of $\pt T$ contained in $\pt \cv{T}$. We prove sufficient conditions for SDP representability: $\pt_cT$ is positively curved, and necessary conditions: $\pt_cT$ has nonnegative curvature at smooth points and on nondegenerate corners. The gaps between them are similar to case (ii). The positive definite Lagrange Hessian (PDLH) condition is also discussed. | |
| dc.description | 28 pages, revised version made on December 7, 2008 | |
| dc.identifier | https://arxiv.org/abs/0709.4017 | |
| dc.identifier | http://arxiv.org/abs/0709.4017 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/209661 | |
| dc.subject | Optimization and Control | |
| dc.subject | Algebraic Geometry | |
| dc.title | Sufficient and Necessary Conditions for Semidefinite Representability of Convex Hulls and Sets | |
| dc.type | text |