Skip to main content
SpringerLink
Log in

DC decomposition of nonconvex polynomials with algebraic techniques

  • Full Length Paper
  • Series B
  • Published:
Mathematical Programming Submit manuscript

Abstract

We consider the problem of decomposing a multivariate polynomial as the difference of two convex polynomials. We introduce algebraic techniques which reduce this task to linear, second order cone, and semidefinite programming. This allows us to optimize over subsets of valid difference of convex decompositions (dcds) and find ones that speed up the convex–concave procedure. We prove, however, that optimizing over the entire set of dcds is NP-hard.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. For a strongly NP-hard problem, even a pseudo-polynomial time algorithm cannot exist unless P=NP [16].

  2. If we do not add the condition on the input that \(f\ne g\), the problem would again be NP-hard (in fact, this is even easier to prove). However, we believe that in any interesting instance of this question, one would have \(f\ne g\).

  3. Note that \(\text{ Tr } H_h({\bar{x}})\) (resp. \(\lambda _{\max } H_h({\bar{x}})\)) gives the average (resp. maximum) of \(y^TH_h({\bar{x}})y\) over \(\{y~|~||y||=1\}\).

  4. A variant of this proposition in the quadratic case appears in [9, Proposition 12].

  5. The notion of sos-convexity has already appeared in the study of semidefinite representability of convex sets [19] and in applications such as shaped-constrained regression in statistics [31].

  6. In general, constructing polynomials that are convex but not sos-convex seems to be a nontrivial task [4]. A complete characterization of the dimensions and degrees for which convexity and sos-convexity are equivalent is given in [5].

References

  1. Ahmadi, A.A., Majumdar, A.: DSOS and SDSOS optimization: LP and SOCP-based alternatives to sum of squares optimization. In: Proceedings of the 48th Annual Conference on Information Sciences and Systems. Princeton University (2014)

  2. Ahmadi, A.A., Majumdar, A.: DSOS and SDSOS optimization: more tractable alternatives to sum of squares and semidefinite optimization. In preparation (2017)

  3. Ahmadi, A.A., Olshevsky, A., Parrilo, P.A., Tsitsiklis, J.N.: NP-hardness of deciding convexity of quartic polynomials and related problems. Math. Progr. 137(1–2), 453–476 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ahmadi, A.A., Parrilo, P.A.: A convex polynomial that is not sos-convex. Math. Progr. 135(1–2), 275–292 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ahmadi, A.A., Parrilo, P.A.: A complete characterization of the gap between convexity and sos-convexity. SIAM J. Optim. 23(2), 811–833 (2013). Also arXiv:1111.4587

  6. Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Progr. 95(1), 3–51 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Alvarado, A., Scutari, G., Pang, J.: A new decomposition method for multiuser dc-programming and its applications. Signal Process. IEEE Trans. 62(11), 2984–2998 (2014)

    Article  MathSciNet  Google Scholar 

  8. Argyriou, A., Hauser, R., Micchelli, C.A., Pontil, M.: A DC-programming algorithm for kernel selection. In: Proceedings of the 23rd International Conference on Machine Learning, pp. 41–48. ACM (2006)

  9. Bomze, I., Locatelli, M.: Undominated dc decompositions of quadratic functions and applications to branch-and-bound approaches. Comput. Optim. Appl. 28(2), 227–245 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chapelle, O., Do, C.B., Teo, C.H., Le, Q.V., Smola, A.J.: Tighter bounds for structured estimation. In: Advances in Neural Information Processing Systems, pp. 281–288 (2009)

  11. de Klerk, E., Laurent, M.: On the Lasserre hierarchy of semidefinite programming relaxations of convex polynomial optimization problems (2010). http://www.optimization-online.org/DB-FILE/2010/11/2800.pdf

  12. Dür, M.: A parametric characterization of local optimality. Math. Methods Oper. Res. 57(1), 101–109 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Floudas, C., Pardalos, P.: Optimization in Computational Chemistry and Molecular Biology: Local and Global Approaches, vol. 40. Springer, New York (2013)

    Google Scholar 

  14. Folland, G.: How to integrate a polynomial over a sphere. Am. Math. Month. 108(5), 446–448 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fung, G., Mangasarian, O.: Semi-supervised support vector machines for unlabeled data classification. Optim. Methods Softw. 15(1), 29–44 (2001)

    Article  MATH  Google Scholar 

  16. Garey, M.R., Johnson, D.S.: Computers and Intractability. W. H. Freeman and Co., San Francisco (1979)

    MATH  Google Scholar 

  17. Gulpinar, N., Hoai An, L.T., Moeini, M.: Robust investment strategies with discrete asset choice constraints using DC programming. Optimization 59(1), 45–62 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hartman, P.: On functions representable as a difference of convex functions. Pac. J. Math 9(3), 707–713 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  19. Helton, J.W., Nie, J.: Semidefinite representation of convex sets. Math. Progr. 122(1), 21–64 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hilbert, D.: Über die Darstellung Definiter Formen als Summe von Formenquadraten. Math. Ann. 32, 342–350 (1888)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hillestad, R., Jacobsen, S.: Reverse convex programming. Appl. Math. Optim. 6(1), 63–78 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hiriart-Urruty, J.B.: Generalized differentiability, duality and optimization for problems dealing with differences of convex functions. In: Convexity and Duality in Optimization, pp. 37–70. Springer (1985)

  23. Hoai An, L.T., Le, H.M., Tao, P.D., et al.: A DC programming approach for feature selection in support vector machines learning. Adv. Data Anal. Classif. 2(3), 259–278 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hoai An, L.T., Tao, P.D.: Solving a class of linearly constrained indefinite quadratic problems by dc algorithms. J. Glob. Optim. 11(3), 253–285 (1997)

    Article  MATH  Google Scholar 

  25. Horst, R., Thoai, N.: DC programming: overview. J. Optim. Theory Appl. 103(1), 1–43 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lanckriet, G., Sriperumbudur, B.: On the convergence of the concave–convex procedure. In: Advances in Neural Information Processing Systems, pp. 1759–1767 (2009)

  27. Lasserre, J.B.: Convexity in semialgebraic geometry and polynomial optimization. SIAM J. Optim. 19(4), 1995–2014 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ling, C., Nie, J., Qi, L., Ye, Y.: Biquadratic optimization over unit spheres and semidefinite programming relaxations. SIAM J. Optim. 20(3), 1286–1310 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lipp, T., Boyd, S.: Variations and extensions of the convex–concave procedure (2014). http://web.stanford.edu/~boyd/papers/cvx_ccv.html

  30. Lou, Y., Osher, S., Xin, J.: Computational aspects of constrained \(l_1-l_2\) minimization for compressive sensing. In: Modelling, Computation and Optimization in Information Systems and Management Sciences, pp. 169–180. Springer (2015)

  31. Magnani, A., Lall, S., Boyd, S.: Tractable fitting with convex polynomials via sum of squares. In: Proceedings of the 44th IEEE Conference on Decision and Control (2005)

  32. Megretski, A.: SPOT: systems polynomial optimization tools (2013)

  33. MOSEK Reference Manual: Version 7. Latest version http://www.mosek.com/ (2013)

  34. Parrilo, P.A.: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. Ph.D. thesis, California Institute of Technology (2000)

  35. Parrilo, P.A.: Semidefinite programming relaxations for semialgebraic problems. Math. Progr. 96(2, Ser. B), 293–320 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. Parrilo, P.A., Sturmfels, B.: Minimizing polynomial functions. Algor. Quant. Real Algebr. Geom. DIMACS Ser. Discrete Math. Theor. Comput. Sci. 60, 83–99 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  37. Piot, B., Geist, M., Pietquin, O.: Difference of convex functions programming for reinforcement learning. In: Advances in Neural Information Processing Systems, pp. 2519–2527 (2014)

  38. Reznick, B.: Some concrete aspects of Hilbert’s 17th problem. In: Contemporary Mathematics, vol. 253, pp. 251–272. American Mathematical Society (2000)

  39. Salakhutdinov, R., Roweis, S., Ghahramani, Z.: On the convergence of bound optimization algorithms. In: Proceedings of the Nineteenth Conference on Uncertainty in Artificial Intelligence, pp. 509–516. Morgan Kaufmann Publishers Inc. (2002)

  40. Tao, P.D., Hoai An, L.T.: Convex analysis approach to dc programming: theory, algorithms and applications. Acta Mathematica Vietnamica 22(1), 289–355 (1997)

    MathSciNet  MATH  Google Scholar 

  41. Tao, P.D.: Duality in dc (difference of convex functions) optimization. Subgradient methods. In: Trends in Mathematical Optimization, pp. 277–293. Springer (1988)

  42. Toland, J.: On subdifferential calculus and duality in non-convex optimization. Mémoires de la Société Mathématique de France 60, 177–183 (1979)

    Article  MATH  Google Scholar 

  43. Tuy, H.: A general deterministic approach to global optimization via dc programming. N.-Holl. Math. Stud. 129, 273–303 (1986)

    Article  MATH  Google Scholar 

  44. Tuy, H.: Global minimization of a difference of two convex functions. In: Nonlinear Analysis and Optimization, pp. 150–182. Springer (1987)

  45. Tuy, H.: DC optimization: theory, methods and algorithms. In: Handbook of Global Optimization, pp. 149–216. Springer (1995)

  46. Tuy, H., Horst, R.: Convergence and restart in branch-and-bound algorithms for global optimization. Application to concave minimization and dc optimization problems. Math. Progr. 41(1–3), 161–183 (1988)

    Article  MATH  Google Scholar 

  47. Wang, S., Schwing, A., Urtasun, R.: Efficient inference of continuous markov random fields with polynomial potentials. In: Advances in Neural Information Processing Systems, pp. 936–944 (2014)

  48. Yuille, A., Rangarajan, A.: The concave–convex procedure (CCCP). Adv. Neural Inf. Process. Syst. 2, 1033–1040 (2002)

    Google Scholar 

Download references

Acknowledgements

We would like to thank Pablo Parrilo for insightful discussions and Mirjam Dür for pointing out reference [9].

Author information

Authors and Affiliations

  1. ORFE, Princeton University, Sherrerd Hall, Princeton, NJ, 08540, USA

    Amir Ali Ahmadi & Georgina Hall

Authors
  1. Amir Ali Ahmadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Georgina Hall
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Georgina Hall.

Additional information

The authors are partially supported by the Young Investigator Program Award of the AFOSR and the CAREER Award of the NSF.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ahmadi, A.A., Hall, G. DC decomposition of nonconvex polynomials with algebraic techniques. Math. Program. 169, 69–94 (2018). https://doi.org/10.1007/s10107-017-1144-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-017-1144-5

Keywords

Mathematics Subject Classification

Search

Navigation