Abstract: | 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).
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