SOS Is Not Obviously Automatizable, Even Approximately

Author Ryan O'Donnell

Thumbnail PDF


  • Filesize: 470 kB
  • 10 pages

Document Identifiers

Author Details

Ryan O'Donnell

Cite AsGet BibTex

Ryan O'Donnell. SOS Is Not Obviously Automatizable, Even Approximately. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 59:1-59:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


Suppose we want to minimize a polynomial p(x) = p(x_1,...,x_n), subject to some polynomial constraints q_1(x),...,q_m(x) >_ 0, using the Sum-of-Squares (SOS) SDP hierarachy. Assume we are in the "explicitly bounded" ("Archimedean") case where the constraints include x_i^2 <_ 1 for all 1 <_ i <_ n. It is often stated that the degree-d version of the SOS hierarchy can be solved, to high accuracy, in time n^O(d). Indeed, I myself have stated this in several previous works. The point of this note is to state (or remind the reader) that this is not obviously true. The difficulty comes not from the "r" in the Ellipsoid Algorithm, but from the "R"; a priori, we only know an exponential upper bound on the number of bits needed to write down the SOS solution. An explicit example is given of a degree-2 SOS program illustrating the difficulty.
  • Sum-of-Squares
  • semidefinite programming


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Farid Alizadeh. Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM Journal on Optimization, 5(1):13-51, 1995. Google Scholar
  2. Sarah Allen, Ryan O'Donnell, and David Witmer. How to refute a random CSP. In Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science, 2015. Google Scholar
  3. Boaz Barak and David Steurer. Sum-of-squares proofs and the quest toward optimal algorithms. In Proceedings of the 2014 International Congress of Mathematicians. International Mathematical Union, 2014. Google Scholar
  4. Charles Delorme and Svatopluk Poljak. Laplacian eigenvalues and the maximum cut problem. Mathematical Programming, 62(1-3):557-574, 1993. Google Scholar
  5. Jack Edmonds. Systems of distinct representatives and linear algebra. Journal of Research of the National Bureau of Standards, 71B:241-245, 1967. Google Scholar
  6. Jeff Erickson. What is the actual time complexity of Gaussian elimination?, 2010.
  7. Dima Grigoriev. Linear lower bound on degrees of Positivstellensatz calculus proofs for the parity. Technical Report IHES/M 68, Insitut des Hautes Études Scientifiques, 1999. Google Scholar
  8. Dima Grigoriev. Complexity of Positivstellensatz proofs for the knapsack. Computational Complexity, 10(2):139-154, 2001. Google Scholar
  9. Dima Grigoriev and Nicolai Vorobjov. Complexity of Null- and Positivstellensatz proofs. Annals of Pure and Applied Logic, 113(1):153-160, 2001. Google Scholar
  10. Martin Grötschel, László Lovász, and Alexander Schrijver. The Ellipsoid Method and its consequences in combinatorial optimization. Combinatorica, 1(2):169-197, 1981. Google Scholar
  11. Martin Grötschel, László Lovász, and Alexander Schrijver. Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, 1988. Google Scholar
  12. Amir Hashemi and Daniel Lazard. Sharper complexity bounds for zero-dimensional gröbner bases and polynomial system solving. International Journal of Algebra and Computation, 21(05):703-713, 2011. Google Scholar
  13. Cédric Josz and Didier Henrion. Strong duality in Lasserre’s hierarchy for polynomial optimization. Optimization Letters, 10(1):3-10, 2016. Google Scholar
  14. Manuel Kauers, Ryan O'Donnell, Li-Yang Tan, and Yuan Zhou. Hypercontractive inequalities via SOS, and the Frankl-Rödl graph. Discrete Analysis, 4, 2016. Google Scholar
  15. Leonid Khachiyan. Polynomial algorithms in linear programming. USSR Computational Mathematics and Mathematical Physics, 20(1):53-72, 1980. Google Scholar
  16. Jean Lasserre. Optimisation globale et théorie des moments. Comptes Rendus de l'Académie des Sciences, 331(11):929-934, 2000. Google Scholar
  17. Jean Lasserre. Global optimization with polynomials and the problem of moments. SIAM Journal on Optimization, 11(3):796-817, 2001. Google Scholar
  18. Monique Laurent. Sums of squares, moment matrices and optimization over polynomials. Emerging Applications of Algebraic Geometry, 149:157-270, 2009. Google Scholar
  19. László Lovász. Semidefinite programs and combinatorial optimization. In Recent advances in algorithms and combinatorics, pages 137-194. Springer New York, 2003. URL:
  20. Yurii Nesterov. Squared functional systems and optimization problems, chapter 17, pages 405-440. Kluwer Academic Publishers, 2000. Google Scholar
  21. Ryan O'Donnell and Franklin Ta. Linear programming and semidefinite programming lecture 10 notes, 2011. URL:
  22. Ryan O'Donnell and Yuan Zhou. Approximability and proof complexity. In Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1537-1556, 2013. Google Scholar
  23. Pablo Parrilo. Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. PhD thesis, California Institute of Technology, 2000. Google Scholar
  24. Lorant Porkolab and Leonid Khachiyan. On the complexity of semidefinite programs. Journal of Global Optimization, 10(4):351-365, 1997. Google Scholar
  25. Motakuri Ramana. An exact duality theory for semidefinite programming and its complexity implications. Mathematical Programming, 77(1):129-162, 1997. Google Scholar
  26. Claus Scheiderer. Sums of squares of polynomials with rational coefficients. Journal of the European Mathematical Society, 18(7):1495-1513, 2016. Google Scholar
  27. Grant Schoenebeck. Linear level Lasserre lower bounds for certain k-CSPs. In Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, pages 593-602, 2008. Google Scholar
  28. Naum Shor. Class of global minimum bounds of polynomial functions. Cybernetics, 23(6):731-734, 1987. Google Scholar
  29. Kuduvally Swamy. On Sylvester’s criterion for positive-semidefinite matrices. IEEE Transactions on Automatic Control, 18(3):306-306, 1973. Google Scholar
  30. Sergey Tarasov and Mikhail Vyalyi. Semidefinite programming and arithmetic circuit evaluation. Discrete Applied Mathematics, 156(11):2070-2078, 2008. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail