Sum-of-squares hierarchies for binary polynomial optimization
Slot,Lucas Laurent,Monique
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.
Description
Date
2021-05
Journal Title
Journal ISSN
Volume Title
Publisher
Springer Verlag
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Keywords
Binary polynomial optimization, Lasserre hierarchy, sum of squares polynomial, polynomial kernel, Semidefinite programming, Fourier Analysis, Krawtchouk polynomials
Citation
Slot, L & Laurent, M 2021, Sum-of-squares hierarchies for binary polynomial optimization. in M Singh & D P Williamson (eds), Integer Programming and Combinatorial Optimization - Proceedings of the 22nd International Conference, IPCO 2021. Lecture Notes in Computer Science, vol. 12707, Springer Verlag, Cham, pp. 43-57, International Conference on Integer Programming and Combinatorial Optimization, Atlanta, Georgia, United States, 19/05/21. https://doi.org/10.1007/978-3-030-73879-2_4
License
info:eu-repo/semantics/closedAccess