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Sum-of-squares hierarchies for binary polynomial optimization

Slot,Lucas
; 
Laurent,Monique
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 worst-case 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.
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Publisher Copyright: © 2021, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.
Date
2023-02
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Tilburg University Research Output Collection
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Binary polynomial optimization, Lasserre hierarchy, Sum-of-squares polynomials, Fourier analysis, Krawtchouk polynomials, Polynomial kernels, Semidefinite programming, Polynomial matrices, RELAXATIONS, CUT, REPRESENTATIONS, COMPLEXITY, MATRICES
Citation
Slot, L & Laurent, M 2023, 'Sum-of-squares hierarchies for binary polynomial optimization', Mathematical Programming, vol. 197, no. 2, pp. 621-660. https://doi.org/10.1007/s10107-021-01745-9
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https://hdl.handle.net/20.500.14602/76589
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