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

Abstract. A set S ⊆ R is called to be Semidefinite (SDP) 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 construct SDP representations. The contributions of this paper are: (i) For bounded SDP representable sets W1, · · · ,Wm, we give an explicit construction of an SDP representation for conv(∪ k=1Wk). 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 ∂S is positively curved, and necessary condition: ∂S 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 ∂S where they vanish. The gaps between them are ∂S having positive versus nonnegative curvature and smooth versus nonsmooth points. 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 ∂cT , the maximum subset of ∂T contained in ∂conv(T ). We prove sufficient conditions for SDP representability: ∂cT is positively curved, and necessary conditions: ∂cT has nonnegative curvature at smooth points and on nondegenerate corners. The gaps between sufficient and necessary conditions are similar to case (ii). The positive definite Lagrange Hessian (PDLH) condition is also discussed.