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


  1. 1.
    Agler, J., Helton, W., McCullough, S., Rodman, L.: Positive semidefinite matrices with a given sparsity pattern. Linear Algebra Appl. 107, 101–149 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barvinok, A.: A Course in Convexity, vol. 54. American Mathematical Society, Providence (2002)zbMATHGoogle Scholar
  3. 3.
    Bogomolov, Y., Fiorini, S., Maksimenko, A., Pashkovich, K.: Small extended formulations for cyclic polytopes. Discrete Comput Geom. 53(4), 809–816 (2015). doi: 10.1007/s00454-015-9682-1
  4. 4.
    Blekherman, G., Gouveia, J., Pfeiffer, J.: Sums of squares on the hypercube. arXiv preprint arXiv:1402.4199 (2014)
  5. 5.
    Blekherman, G., Parrilo, P.A., Thomas, R.R.: Semidefinite optimization and convex algebraic geometry. SIAM (2013)Google Scholar
  6. 6.
    Dumitrescu, B.: Positive Trigonometric Polynomials and Signal Processing Applications. Springer, Berlin (2007)zbMATHGoogle Scholar
  7. 7.
    Fawzi, H., Gouveia, J., Parrilo, P.A., Robinson, R.Z., Thomas, R.R.: Positive semidefinite rank. Math. Program. 153(1), 133–177 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fiorini, S., Kaibel, V., Pashkovich, K., Theis, D.O.: Combinatorial bounds on nonnegative rank and extended formulations. Discrete Math. 313(1), 67–83 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fawzi, H., Saunderson, J., Parrilo, P.A.: Equivariant semidefinite lifts of regular polygons. arXiv preprint arXiv:1409.4379 (2014)
  10. 10.
    Fawzi, H., Saunderson, J., Parrilo, P.A.: Equivariant semidefinite lifts and sum-of-squares hierarchies. SIAM J. Optim. 25(4), 2212–2243 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gale, D.: Neighborly and cyclic polytopes. In: Proceedings of the Seventh Symposium in Pure Mathematics of the American Mathematical Society, vol. 7, pp. 225–232 (1963)Google Scholar
  12. 12.
    Grone, R., Johnson, C.R., Sá, E.M., Wolkowicz, H.: Positive definite completions of partial hermitian matrices. Linear Algebra Appl. 58, 109–124 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gouveia, J., Parrilo, P.A., Thomas, R.R.: Lifts of convex sets and cone factorizations. Math. Oper. Res. 38(2), 248–264 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Griewank, A., Toint, P.L.: On the existence of convex decompositions of partially separable functions. Math. Program. 28(1), 25–49 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kaibel, V., Pashkovich, K.: Constructing extended formulations from reflection relations. In: Integer programming and combinatorial optimization, pp. 287–300. Springer, Berlin (2011)Google Scholar
  16. 16.
    Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Laurent, M.: Lower bound for the number of iterations in semidefinite hierarchies for the cut polytope. Math. Oper. Res. 28(4), 871–883 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Nesterov, Y.: Squared functional systems and optimization problems. In: High performance optimization, pp. 405–440. Springer, Berlin (2000)Google Scholar
  19. 19.
    Parrilo, P.A.: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, California Institute of Technology (2000)Google Scholar
  20. 20.
    Rudin, W.: Fourier Analysis on Groups. Wiley, New York (1990)CrossRefzbMATHGoogle Scholar
  21. 21.
    Terras, A.: Fourier Analysis on Finite Groups and Applications. Cambridge University Press, Cambridge (1999)CrossRefzbMATHGoogle Scholar
  22. 22.
    Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci. 43(3), 441–466 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ziegler, G.M.: Lectures on Polytopes, vol. 152. Springer, Berlin (1995)zbMATHGoogle Scholar

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

Personalised recommendations