Sum-of-squares hierarchies for binary polynomial optimization
Résumé
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.