Semidefinite representation of convex sets - Helton, J. William; Nie, Jiawang (Mathematical programming, 2010)
Let be a semialgebraic set defined by multivariate polynomials g i ( x ). Assume S is convex, compact and has nonempty interior. Let , 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 , does there exist an LMI representable set Ŝ in some higher dimensional space whose projection down onto 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 Schmüdgen and Putinar’s matrix Positivstellensatz, which are critical to the proofs of (i)–(iii).