Semidefinite representation of convex sets

Let S = {x ∈ R : g1(x) ≥ 0, · · · , gm(x) ≥ 0} be a semialgebraic set defined by multivariate polynomials gi(x). Assume S is convex, compact and has nonempty interior. Let Si = {x ∈ R : gi(x) ≥ 0}, and ∂S (resp. ∂Si) be the boundary of S (resp. Si). 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 [9]. A question arising from [15], see [9,16], is: given a subset S of R, does there exist an LMI representable set Ŝ in some higher dimensional space R whose projection down onto R 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 gi(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 x on S is positive definite at the minimizer, then S is SDP representable. (ii) If each gi(x) is either sos-concave (−∇gi(x) = W (x) W (x) for some possibly nonsquare matrix polynomial W (x)) or strictly quasiconcave on S, then S is SDP representable. (iii) If each Si is either sos-convex or poscurv-convex (Si is compact convex, whose boundary has positive curvature and is nonsingular, i.e. ∇gi(x) 6= 0 on ∂Si ∩ S), then S is SDP representable. This also holds for Si for which ∂Si ∩ 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).