Sum-of-Squares Hierarchies for Binary Polynomial Optimization

Abstract

We consider the sum-of-squares hierarchy of approximations for the problem of minimizing a polynomial f over the boolean hypercube B n = {0, 1} n . This hierarchy provides for each integer r ∈ N a lower bound f (r ) on the minimum f min of f , given by the largest scalar λ for which the polynomial f − λ is a sum-of-squares on B n with degree at most 2r . We analyze the quality of these bounds by estimating the worstcase error f min − f (r ) in terms of the least roots of the Krawtchouk polynomials. As a consequence, for fixed t ∈ [0, 1/2], we can show that this worst-case error in the regime r ≈ t • n is of the order 1/2 − √ t(1 − t) as n tends to ∞. Our proof combines classical Fourier analysis on B n with the polynomial kernel technique and existing results on the extremal roots of Krawtchouk polynomials. This link to roots of orthogonal polynomials relies on a connection between the hierarchy of lower bounds f (r ) and another hierarchy of upper bounds f (r ) , for which we are also able to establish the same error analysis. Our analysis extends to the minimization of a polynomial over the q-ary cube (Z/qZ) n . Furthermore, our results apply to the setting of matrix-valued polynomials.

An effective version of Schmüdgen’s Positivstellensatz for the hypercube

“…For the case of the hypercube 2 a degree bound in O(1∕ ) for Schmüdgen-type certificates is obtained in [10], thus showing that one can take c ≤ 1 in the above mentioned result of [7]. This result holds in fact for a weaker hierarchy of bounds obtained by restricting in (5) to decompositions of the polynomial f − involving factors J that are nonnegative scalars (instead of sums of squares), also known as Handelman-type decompositions (thus replacing the preordering Q(B n ) r by its subset H r of polynomials having a Handelman-type decomposition). The analysis in [10] relies on employing the Bernstein operator r , which has the property of mapping a polynomial nonnegative over the hypercube to a polynomial in the set…”

“…We have so far focused our discussion on positive results concerning sum-ofsquares representations. That is, results that give upper bounds on the error of Lasserre's bound (5); or equivalently on the required degree of Schmüdgen-type positivity certificates. In order to put these results in context, it would be interesting to have complementary negative results, thus giving lower bounds on the convergence rate of the Lasserre hierarchy.…”

“…Writing Q((1 − x 2 ) 3 ) r for the corresponding (truncated) preordering, Stengle shows that only when r = Ω(1∕ √ ) . In other words, he shows for f (x) = 1 − x 2 that the Lasserre-type bound f (r) obtained by replacing Q(1 − x 2 ) r in (5) by Q((1 − x 2 ) 3 ) r satisfies:…”

“…and for optimization over the binary cube {−1, 1} n in [5]. There, the authors use kernels that are invariant under the symmetry of S n−1 and {−1, 1} n , respectively.…”

“…(20)). This question is motivated by the fact that for the analysis of the analogous hierarchies for the unit sphere in [4] and for the boolean hypercube in [5] the existence of such a constant (depending only on d) was in fact shown.…”