2 research outputs found

    Sum-of-squares hierarchies for binary polynomial optimization

    Get PDF
    We consider the sum-of-squares hierarchy of approximations for the problem of minimizing a polynomial ff over the boolean hypercube Bn={0,1}n\mathbb{B}^{n}=\{0,1\}^n. This hierarchy provides for each integer r∈Nr \in \mathbb{N} a lower bound f(r)f_{(r)} on the minimum fmin⁑f_{\min} of ff, given by the largest scalar Ξ»\lambda for which the polynomial fβˆ’Ξ»f - \lambda is a sum-of-squares on Bn\mathbb{B}^{n} with degree at most 2r2r. We analyze the quality of these bounds by estimating the worst-case error fminβ‘βˆ’f(r)f_{\min} - f_{(r)} in terms of the least roots of the Krawtchouk polynomials. As a consequence, for fixed t∈[0,1/2]t \in [0, 1/2], we can show that this worst-case error in the regime rβ‰ˆtβ‹…nr \approx t \cdot n is of the order 1/2βˆ’t(1βˆ’t)1/2 - \sqrt{t(1-t)} as nn tends to ∞\infty. Our proof combines classical Fourier analysis on Bn\mathbb{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)f_{(r)} and another hierarchy of upper bounds f(r)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 qq-ary cube (Z/qZ)n(\mathbb{Z}/q\mathbb{Z})^{n}.Comment: 23 pages, 1 figure. Fixed a typo in Theorem 1 and Theorem

    Sum-of-squares hierarchies for binary polynomial optimization

    Get PDF
    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 r∈N a lower bound f(r) on the minimum fmin of f, given by the largest scalar Ξ» for which the polynomial fβˆ’Ξ» 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 fminβˆ’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 rβ‰ˆtβ‹…n is of the order 1/2βˆ’t(1βˆ’t)βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’βˆš as n tends to ∞. 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
    corecore