Let S = {x ∈ R n | g 1 (x) 0, . . . , g m (x) 0} be a basic closed semialgebraic set defined by real polynomials g i . Putinar's Positivstellensatz says that, under a certain condition stronger than compactness of S, every real polynomial f positive on S possesses a representation f = m i=0 i g i where g 0 := 1 and each i is a sum of squares of polynomials. Such a representation is a certificate for the nonnegativity of f on S. We give a bound on the degrees of the terms i g i in this representation which depends on the description of S, the degree of f and a measure of how close f is to having a zero on S. As a consequence, we get information about the convergence rate of Lasserre's procedure for optimization of a polynomial subject to polynomial constraints.
We show that Connes' embedding conjecture on von Neumann algebras is
equivalent to the existence of certain algebraic certificates for a polynomial
in noncommuting variables to satisfy the following nonnegativity condition: The
trace is nonnegative whenever self-adjoint contraction matrices of the same
size are substituted for the variables. These algebraic certificates involve
sums of hermitian squares and commutators. We prove that they always exist for
a similar nonnegativity condition where elements of separable II_1-factors are
considered instead of matrices. Under the presence of Connes' conjecture, we
derive degree bounds for the certificates
A basic closed semialgebraic subset S of R n is defined by simultaneous polynomial inequalities g 1 ≥ 0,. .. , gm ≥ 0. We give a short introduction to Lasserre's method for minimizing a polynomial f on a compact set S of this kind. It consists of successively solving tighter and tighter convex relaxations of this problem which can be formulated as semidefinite programs. We give a new short proof for the convergence of the optimal values of these relaxations to the infimum f * of f on S which is constructive and elementary. In the case where f possesses a unique minimizer x * , we prove that every sequence of "nearly" optimal solutions of the successive relaxations gives rise to a sequence of points in R n converging to x * .
Schmu¨dgen's Positivstellensatz roughly states that a polynomial f positive on a compact basic closed semialgebraic subset S of R n can be written as a sum of polynomials which are non-negative on S for certain obvious reasons. However, in general, you have to allow the degree of the summands to exceed largely the degree of f : Phenomena of this type are one of the main problems in the recently popular approximation of non-convex polynomial optimization problems by semidefinite programs. Prestel (Springer Monographs in Mathematics, Springer, Berlin, 2001) proved that there exists a bound on the degree of the summands computable from the following three parameters: The exact description of S; the degree of f and a measure of how close f is to having a zero on S: Roughly speaking, we make explicit the dependence on the second and third parameter. In doing so, the third parameter enters the bound only polynomially. r
Abstract. We show that all the coefficients of the polynomialare nonnegative whenever m ≤ 13 is a nonnegative integer and A and B are positive semidefinite matrices of the same size. This has previously been known only for m ≤ 7. The validity of the statement for arbitrary m has recently been shown to be equivalent to the Bessis-Moussa-Villani conjecture from theoretical physics. In our proof, we establish a connection to sums of hermitian squares of polynomials in noncommuting variables and to semidefinite programming. As a by-product we obtain an example of a real polynomial in two noncommuting variables having nonnegative trace on all symmetric matrices of the same size, yet not being a sum of hermitian squares and commutators.
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