On the effective Putinar’s Positivstellensatz and moment approximation

Abstract

The Positivstellensätze of Putinar and Schmüdgen show that any polynomial f positive on a compact semialgebraic set can be represented using sums of squares. Recently, there has been large interest in proving effective versions of these results, namely to show bounds on the required degree of the sums of squares in such representations. These effective Positivstellensätze have direct implications for the convergence rate of the celebrated moment-SOS hierarchy in polynomial optimization. In this paper, we restrict to the fundamental case of the hypercube B n = [−1, 1] n . We show an upper degree bound for Putinar-type representations on B n of the order O(fmax/f min ), where fmax, f min are the maximum and minimum of f on B n , respectively. Previously, specialized results of this kind were available only for Schmüdgentype representations and not for Putinar-type ones. Complementing this upper degree bound, we show a lower degree bound in Ω( 8 fmax/f min ). This is the first lower bound for Putinar-type representations on a semialgebraic set with nonempty interior described by a standard set of inequalities.

Algebraic certificates for the truncated moment problem

“…A second one is to refine the quantifier-elimination bound given in Corollary 3. Unlike the viewpoint of the so-called effective Putinar Positivstellensatz introduced in [1], for which degree bounds depend on the polynomial itself, it is clear from our analysis that for the complexity analysis of the TMP one needs to give uniform degree bounds that only depend on the input of the TMP. One way of getting such uniform bounds is to consider manifestly positive polynomials such as those in 1 + Q(g ) for compact K = S(g ).…”

“…In this work we make use of quantifier elimination for determining bounds on the complexity of computing certificates for unrepresentability by using quantitative results from [2]. These also rely on recent advances in the complexity analysis of Putinar Positivstellensatz [1].…”

“…The problem of bounding the degree D of the summands in a Putinar certificate f = k i=0 σ i g i for a polynomial f ∈ P(K) d , is called the effective Putinar Positivstellensatz, see [24,1]. The work [1] gives a bound for D as a function of d, n, of the polynomial f and of geometrical parameters of K, see [1, Th. 1.7], cf.…”

“…We first recall the following bound on the degree of a Putinar representation of a polynomial in Int(P(K) d ), for an archimedean quadratic module Q(g ), given in [1]. Let Under the following assumptions:…”

“…Given n, d ∈ N and a sequence of real numbers y = (y α ) α∈N n d indexed by N n d = {α ∈ N n : n i=1 α i ≤ d}, the truncated moment problem (below TMP) is the question of deciding 1 CNRS, LAAS, Université de Toulouse, France. 2 Faculty of Electrical Engineering, Czech Technical University in Prague, Czechia.…”