Mathematical Programming

, Volume 160, Issue 1–2, pp 149–191 | Cite as

Sparse sums of squares on finite abelian groups and improved semidefinite lifts

  • Hamza Fawzi
  • James Saunderson
  • Pablo A. Parrilo
Full Length Paper Series A


Let G be a finite abelian group. This paper is concerned with nonnegative functions on G that are sparse with respect to the Fourier basis. We establish combinatorial conditions on subsets \({\mathcal {S}}\) and \({\mathcal {T}}\) of Fourier basis elements under which nonnegative functions with Fourier support \({\mathcal {S}}\) are sums of squares of functions with Fourier support \({\mathcal {T}}\). Our combinatorial condition involves constructing a chordal cover of a graph related to G and \({\mathcal {S}}\) (the Cayley graph \({{\mathrm{Cay}}}(\widehat{G},{\mathcal {S}})\)) with maximal cliques related to \({\mathcal {T}}\). Our result relies on two main ingredients: the decomposition of sparse positive semidefinite matrices with a chordal sparsity pattern, as well as a simple but key observation exploiting the structure of the Fourier basis elements of G (the characters of G). We apply our general result to two examples. First, in the case where \(G = {\mathbb {Z}}_2^n\), by constructing a particular chordal cover of the half-cube graph, we prove that any nonnegative quadratic form in n binary variables is a sum of squares of functions of degree at most \(\left\lceil n/2 \right\rceil \), establishing a conjecture of Laurent. Second, we consider nonnegative functions of degree d on \({\mathbb {Z}}_N\) (when d divides N). By constructing a particular chordal cover of the dth power of the N-cycle, we prove that any such function is a sum of squares of functions with at most \(3d\log (N/d)\) nonzero Fourier coefficients. Dually this shows that a certain cyclic polytope in \({\mathbb {R}}^{2d}\) with N vertices can be expressed as a projection of a section of the cone of positive semidefinite matrices of size \(3d\log (N/d)\). Putting \(N=d^2\) gives a family of polytopes in \({\mathbb {R}}^{2d}\) with linear programming extension complexity \(\varOmega (d^2)\) and semidefinite programming extension complexity \(O(d\log (d))\). To the best of our knowledge, this is the first explicit family of polytopes \((P_d)\) in increasing dimensions where \({{\mathrm{xc_{\text {PSD}}}}}(P_d) = o({{\mathrm{xc_{\text {LP}}}}}(P_d))\), where \({{\mathrm{xc_{\text {PSD}}}}}\) and \({{\mathrm{xc_{\text {LP}}}}}\) are respectively the SDP and LP extension complexity.

Mathematics Subject Classification

90C22 52B12 52B55 

Supplementary material


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016

Authors and Affiliations

  • Hamza Fawzi
    • 1
  • James Saunderson
    • 2
  • Pablo A. Parrilo
    • 1
  1. 1.Laboratory for Information and Decision Systems, Department of Electrical Engineering and Computer ScienceMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Department of Electrical EngineeringUniversity of WashingtonSeattleUSA

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