2 research outputs found
Sum-of-squares hierarchies for binary polynomial optimization
We consider the sum-of-squares hierarchy of approximations for the problem of
minimizing a polynomial over the boolean hypercube
. This hierarchy provides for each integer a lower bound on the minimum of , given by
the largest scalar for which the polynomial is a
sum-of-squares on with degree at most . We analyze the
quality of these bounds by estimating the worst-case error
in terms of the least roots of the Krawtchouk polynomials. As a consequence,
for fixed , we can show that this worst-case error in the
regime is of the order as tends
to . Our proof combines classical Fourier analysis on
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 and another
hierarchy of upper bounds , for which we are also able to establish
the same error analysis. Our analysis extends to the minimization of a
polynomial over the -ary cube .Comment: 23 pages, 1 figure. Fixed a typo in Theorem 1 and Theorem
Sum-of-squares hierarchies for binary polynomial optimization
We consider the sum-of-squares hierarchy of approximations for the problem of minimizing a polynomial f over the boolean hypercube Bn={0,1}n. This hierarchy provides for each integer rβN a lower bound f(r) on the minimum fmin of f, given by the largest scalar Ξ» for which the polynomial fβΞ» is a sum-of-squares on Bn with degree at most 2r. We analyze the quality of these bounds by estimating the worst-case error fminβf(r) in terms of the least roots of the Krawtchouk polynomials. As a consequence, for fixed tβ[0,1/2], we can show that this worst-case error in the regime rβtβ
n is of the order 1/2βt(1βt)βββββββ as n tends to β. Our proof combines classical Fourier analysis on Bn 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