Tight Sum-of-Squares lower bounds for binary polynomial optimization problems

We give two results concerning the power of the Sum-of-Squares(SoS)/Lasserre hierarchy. For binary polynomial optimization problems of degree $2d$ and an odd number of variables $n$, we prove that $\frac{n+2d-1}{2}$ levels of the SoS/Lasserre hierarchy are necessary to provide the exact optimal value. This matches the recent upper bound result by Sakaue, Takeda, Kim and Ito. Additionally, we study a conjecture by Laurent, who considered the linear representation of a set with no integral points. She showed that the Sherali-Adams hierarchy requires $n$ levels to detect the empty integer hull, and conjectured that the SoS/Lasserre rank for the same problem is $n-1$. We disprove this conjecture and derive lower and upper bounds for the rank

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 $\mathbb{B}^{n}=\{0,1\}^n$. This hierarchy provides for each integer $r \in \mathbb{N}$ a lower bound $f_{(r)}$ on the minimum $f_{\min}$ of $f$, given by the largest scalar $\lambda$ for which the polynomial $f - \lambda$ is a sum-of-squares on $\mathbb{B}^{n}$ 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 $t \in [0, 1/2]$, we can show that this worst-case error in the regime $r \approx t \cdot n$ is of the order $1/2 - \sqrt{t(1-t)}$ as $n$ tends to $\infty$. Our proof combines classical Fourier analysis on $\mathbb{B}^{n}$ 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 $f_{(r)}$ 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 $(\mathbb{Z}/q\mathbb{Z})^{n}$.Comment: 23 pages, 1 figure. Fixed a typo in Theorem 1 and Theorem

Sum-of-squares hierarchies for binary polynomial optimization

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 λ for which the polynomial f−λ 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 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 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