ar X iv : 0 70 5 . 40 68 v 2 [ m at h . O C ] 1 3 Ju n 20 07 SEMIDEFINITE REPRESENTATION OF CONVEX SETS

Let S = {x ∈ R n : g 1 (x) ≥ 0, · · · , gm(x) ≥ 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 ∈ R n : g i (x) ≥ 0} and ∂S i = {x ∈ R n : g i (x) = 0} be its boundary. 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 [6]. A question arising from [13], see [6, 14], is: given S ∈ R n , does there exist an LMI representable setˆS in some higher dimensional space R n+N whose projection down onto 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 (PDLH) 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 convex, compact and has smooth boundary with positive curvature), 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 Schmüdgen and Putinar's matrix Positivstellensatz, which are critical to the proofs of (i)-(iii).