Semidefinite representation of convex sets

Abstract

Let $${S =\{x\in \mathbb{R}^n: g_1(x)\geq 0, \ldots, g_m(x)\geq 0\}}$$be a semialgebraic set defined by multivariate polynomials g i (x). Assume S is convex, compact and has nonempty interior. Let $${S_i =\{x\in \mathbb{R}^n:\, g_i(x)\geq 0\}}$$, and ∂ S (resp. ∂ S i ) be the boundary of S (resp. S i ). This paper, as does the subject of semidefinite programming (SDP), concerns linear matrix inequalities (LMIs). The set S is said to have an LMI representation if it equals the set of solutions to some LMI and it is known that some convex S may not be LMI representable (Helton and Vinnikov in Commun Pure Appl Math 60(5):654–674, 2007). A question arising from Nesterov and Nemirovski (SIAM studies in applied mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1994), see Helton and Vinnikov in Commun Pure Appl Math 60(5):654–674, 2007 and Nemirovski in Plenary lecture, International Congress of Mathematicians (ICM), Madrid, Spain, 2006, is: given a subset S of $${\mathbb{R}^n}$$, does there exist an LMI representable set Ŝ in some higher dimensional space $${ \mathbb{R}^{n+N}}$$whose projection down onto $${\mathbb{R}^n}$$equals S. Such S is called semidefinite representable or SDP representable. This paper addresses the SDP representability problem. The following are the main contributions of this paper: (i) assume g i (x) are all concave on S. If the positive definite Lagrange Hessian condition holds, i.e., the Hessian of the Lagrange function for optimization problem of minimizing any nonzero linear function l T x on S is positive definite at the minimizer, then S is SDP representable. (ii) If each g i (x) is either sos-concave ( − ∇2 g i (x) = W(x)T W(x) for some possibly nonsquare matrix polynomial W(x)) or strictly quasi-concave on S, then S is SDP representable. (iii) If each S i is either sos-convex or poscurv-convex (S i is compact convex, whose boundary has positive curvature and is nonsingular, i.e., ∇g i (x) ≠ 0 on ∂ S i ∩ S), then S is SDP representable. This also holds for S i for which ∂ S i ∩ S extends smoothly to the boundary of a poscurv-convex set containing S. (iv) We give the complexity of Schmudgen and Putinar’s matrix Positivstellensatz, which are critical to the proofs of (i)–(iii).