Exact Semidefinite Programming Relaxations with Truncated Moment Matrix for Binary Polynomial Optimization Problems

Sakaue, Shinsaku; Kim, Sun‐Young; Ito, Naoki

doi:10.1137/16m105544x

Cited by 13 publications

(19 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Hence Corollary 1 tells us that the relative error of both hierarchies tends to 0 as r /n → 1/2. We thus 'asymptotically' recover the exactness result (3) of [36].…”

Section: Asymptotic Analysis For Both Hierarchiessupporting

confidence: 72%

“…The bounds f (r ) have finite convergence: f (r ) = f min for r ≥ n [18,21]. In fact, it has been shown in [36] that the bound f (r ) is exact already for 2r ≥ n + d − 1. That is,…”

Section: The Sum-of-squares Hierarchy On the Boolean Cubementioning

confidence: 99%

“…In addition, it is shown in [36] that the bound f (r ) is exact for 2r ≥ n + d − 2 when the polynomial f has only monomials of even degree. This extends an earlier result of [12] shown for quadratic forms (d = 2), which applies in particular to the case of max-cut.…”

Section: The Sum-of-squares Hierarchy On the Boolean Cubementioning

confidence: 99%

“…The case d = 2 (which includes max-cut) was treated in [12], positively answering a question posed in [22]. Extending the strategy of [12], the general case was settled in [36]. These exactness results are best possible when d is even and n is odd [15].…”

Section: Related Workmentioning

confidence: 99%

See 3 more Smart Citations

Sum-of-Squares Hierarchies for Binary Polynomial Optimization

Slot

Laurent

2021

Lecture Notes in Computer Science

View full text Add to dashboard Cite

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 worstcase 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.

show abstract

“…Hence Corollary 1 tells us that the relative error of both hierarchies tends to 0 as r /n → 1/2. We thus 'asymptotically' recover the exactness result (3) of [36].…”

Section: Asymptotic Analysis For Both Hierarchiessupporting

confidence: 72%

“…The bounds f (r ) have finite convergence: f (r ) = f min for r ≥ n [18,21]. In fact, it has been shown in [36] that the bound f (r ) is exact already for 2r ≥ n + d − 1. That is,…”

Section: The Sum-of-squares Hierarchy On the Boolean Cubementioning

confidence: 99%

Section: The Sum-of-squares Hierarchy On the Boolean Cubementioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

See 2 more Smart Citations

Sum-of-Squares Hierarchies for Binary Polynomial Optimization

Slot

Laurent

2021

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…In the literature, such an upper bound is only known for two extreme cases: Z n 2 and Z N . We refer interested readers to [39] and [17] for more details. In fact, even in these two cases, upper bounds of the FSOS sparsity can be extremely different, as we will see in the following example: Suppose that N = 2 n and n = 2 k for k ≥ 1.…”

mentioning

confidence: 99%

Computing sparse Fourier sum of squares on finite abelian groups in quasi-linear time

Yang¹,

Yao²,

Zhi³

2022

Preprint

View full text Add to dashboard Cite

The problem of verifying the nonnegativity of a real valued function on a finite set is a long-standing challenging problem, which has received extensive attention from both mathematicians and computer scientists. Given a finite set X together with a function F : X → R, if we equip X a group structure G via a bijection ϕ : G → X, then effectively verifying the nonnegativity of F on X is equivalent to computing a sparse Fourier sum of squares (FSOS) certificate of f = F • ϕ on G. In this paper, we show that by performing the fast (inverse) Fourier transform and finding a local minimal Fourier support, we are able to compute a sparse FSOS certificate of f on G with complexity O(|G| log |G| + |G|t 4 + poly(t)), which is quasi-linear in the order of G and polynomial in the FSOS sparsity t of f . We demonstrate the efficiency of the proposed algorithm by numerical experiments on various abelian groups of order up to 10 6 . It is also noticeable that different choices of group structures on X would result in different FSOS sparsities of F . Along this line, we investigate upper bounds for FSOS sparsities with respect to different choices of group structures, which generalize and refine existing results in the literature. More precisely, (i) we give an upper bound for FSOS sparsities of nonnegative functions on the product and the quotient of two finite abelian groups respectively; (ii) we prove the equivalence between finding the group structure for the Fourier-sparsest representation of F and solving an integer linear programming problem.

show abstract