Sum-of-Squares Hierarchies for Binary Polynomial Optimization (original) (raw)
2021, Integer Programming and Combinatorial Optimization
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 fmin 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 worst-case error fmin − 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. Keywords: Binary polynomial optimization • Lasserre hierarchy • Sum-of-squares polynomials • Fourier analysis • Krawtchouk polynomials • Polynomial kernels • Semidefinite programming This optimization problem is NP-hard in general, already for d = 2. Indeed, as is well-known, one can model an instance of max-cut on the complete graph K n with edge weights w = (w ij) as a problem of the form (1) by setting: f (x) = − 1≤i<j≤n w ij (x i − x j) 2 ,
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