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

• Published in: SIAM journal on optimization Vol. 20; no. 2; pp. 759 - 791
• Main Authors: Helton, J William, Nie, Jiawang
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2009
• Subjects: Algebra; Boundaries; Convex analysis; Curvature; Gaps; Hulls; Hulls (structures); Mathematical analysis; Mathematical problems; Mathematical programming; Optimization; Studies
• ISSN: 1052-6234; 1095-7189
• DOI: 10.1137/07070526x

In terms of defining polynomials for S, nonsingular boundary amounts to them having nonvanishing gradient at each point on ∂S and the curvature condition can be expressed as their strict versus nonstrict quasiconcavity at those points on ∂S where they vanish. The gaps between them are ∂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 sosconcavity. (iii) For the SDP representability of the convex hull of a compact nonconvex semialgebraic set T, we find that the critical object is ∂cT, the maximum subset of ∂T contained in ∂conv(T). We prove sufficient for SDP representability: ∂cT is nonsingular and positively curved, and necessary is: ∂cT 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. [PUBLICATION ABSTRACT]