Skip to main content
SpringerLink
Log in

Steklov convexification and a trajectory method for global optimization of multivariate quartic polynomials

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

Abstract

The Steklov function \(\mu _f(\cdot ,t)\) is defined to average a continuous function f at each point of its domain by using a window of size given by \(t>0\). It has traditionally been used to approximate f smoothly with small values of t. In this paper, we first find a concise and useful expression for \(\mu _f\) for the case when f is a multivariate quartic polynomial. Then we show that, for large enough t, \(\mu _f(\cdot ,t)\) is convex; in other words, \(\mu _f(\cdot ,t)\) convexifies f. We provide an easy-to-compute formula for t with which \(\mu _f\) convexifies certain classes of polynomials. We present an algorithm which constructs, via an ODE involving \(\mu _f\), a trajectory x(t) emanating from the minimizer of the convexified f and ending at x(0), an estimate of the global minimizer of f. For a family of quartic polynomials, we provide an estimate for the size of a ball that contains all its global minimizers. Finally, we illustrate the working of our method by means of numerous computational examples.

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

Similar content being viewed by others

References

  1. Arıkan, O., Burachik, R.S., Kaya, C.Y.: “Backward differential flow” may not converge to a global minimizer of polynomials. J. Optim. Theory Appl. 167, 401–408 (2015)

    Article  MathSciNet  Google Scholar 

  2. Arıkan, O., Burachik, R.S., Kaya, C.Y.: Steklov regularization and trajectory methods for univariate global optimization. J. Glob. Optim. 76(1), 91–120 (2020)

    Article  MathSciNet  Google Scholar 

  3. Attouch, H., Chbani, Z., Peypouquet, J., Redont, P.: Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity. Math. Program. 168(1–2), 123–175 (2018)

    Article  MathSciNet  Google Scholar 

  4. Borelli, R.L., Coleman, C.S.: Differential Equations: A Modeling Perspective. Wiley, Hoboken (2004)

    Google Scholar 

  5. Boţ, R.I., Csetnek, E.R.: Convergence rates for forward–backward dynamical systems associated with strongly monotone inclusions. J. Math. Anal. Appl. 457(2), 1135–1152 (2018)

    Article  MathSciNet  Google Scholar 

  6. Cai, J.-F., Liu, H., Wang, Y.: Fast rank-one alternating minimization algorithm for phase retrieval. J. Sci. Comput. 79(1), 128–147 (2019)

    Article  MathSciNet  Google Scholar 

  7. Cardoso, J.F.: Blind signal separation: statistical principles. Proc. IEEE 90, 2009–2026 (1998)

    Article  Google Scholar 

  8. Chen, X.: Smoothing methods for nonsmooth, nonconvex minimization. Math. Program. Ser. B 134, 71–99 (2012)

    Article  MathSciNet  Google Scholar 

  9. Ermoliev, Y.M., Norkin, V.I., Wets, R.J.-B.: The minimization of semicontinuous functions: mollifier subgradients. SIAM J. Control Optim. 32, 149–167 (1995)

    Article  MathSciNet  Google Scholar 

  10. Feng, C., Lagoa, C. M., Sznaier, M.: Hybrid system identification via sparse polynomial optimization. In: American Control Conference (ACC) (2010)

  11. Garmanjani, R., Vicente, L.N.: Smoothing and worst-case complexity for direct-search methods in nonsmooth optimization. IMA J. Num. Anal. 33, 1008–1028 (2013)

    Article  MathSciNet  Google Scholar 

  12. Goldberg, D.E.: Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, Reading (1989)

    MATH  Google Scholar 

  13. Gupal, A.M.: On a method for the minimization of almost-differentiable functions. Cybern. Syst. Anal. 13, 115–117 (1977)

    Article  MathSciNet  Google Scholar 

  14. Jeyakumar, V., Kim, S., Lee, G.M., Li, G.: Semidefinite programming relaxation methods for global optimization problems with sparse polynomials and unbounded semialgebraic feasible sets. J. Glob. Optim. 65, 175–190 (2016)

    Article  MathSciNet  Google Scholar 

  15. Jeyakumar, V., Lasserre, J.B., Li, G.: On polynomial optimization over non-compact semi-algebraic sets. J. Optim. Theory Appl. 163(3), 707–718 (2014)

    Article  MathSciNet  Google Scholar 

  16. Kim, S., Kojima, M., Waki, H., Yamashita, M.: Algorithm920: SFSDP: a sparse version of full semidefinite programming relaxation for sensor network localization problems. ACM Trans. Math. Softw. 38, 1–19 (2012)

    Article  Google Scholar 

  17. Kok, S., Sandrock, C.: Locating and characterizing the stationary points of the extended Rosenbrock function. Evol. Comput. 17(3), 437–453 (2009)

    Article  Google Scholar 

  18. Ling, C., He, H., Qi, L.: Improved approximation results on standard quartic polynomial optimization. Optim. Lett. 11(8), 1767–1782 (2017)

    Article  MathSciNet  Google Scholar 

  19. Luo, Z.-Q., Zhang, S.: A semidefinite relaxation scheme for multivariate quartic polynomial optimization with quadratic constraints. SIAM J. Optim. 20(4), 1716–1736 (2010)

    Article  MathSciNet  Google Scholar 

  20. Maricic, B., Luo, Z.-Q., Davidson, T.N.: Blind constant modulus equalization via convex optimization. IEEE Trans. Signal Process. 51, 805–818 (2003)

    Article  MathSciNet  Google Scholar 

  21. Nie, J.W.: Sum of squares method for sensor network localization. Comput. Optim. Appl. 43(2), 151–179 (2009)

    Article  MathSciNet  Google Scholar 

  22. Prudnikov, Igor M.: Subdifferentials of the first and second orders for Lipschitz functions. J. Optim. Theory Appl. 171(3), 906–930 (2016)

    Article  MathSciNet  Google Scholar 

  23. Qi, L.: Extrema of a real polynomial. J. Glob. Optim. 30(4), 405–433 (2004)

    Article  MathSciNet  Google Scholar 

  24. Qi, L., Teo, K.L.: Multivariate polynomial minimization and its application in signal processing. J. Glob. Optim. 26, 419–433 (2003)

    Article  MathSciNet  Google Scholar 

  25. Qi, L., Wan, Z., Yang, Y.-F.: Global minimization of normal quartic polynomials based on global descent directions. SIAM J. Optim. 15(1), 275–302 (2004)

    Article  MathSciNet  Google Scholar 

  26. Qing, A.: Dynamic differential evolution strategy and applications in electromagnetic inverse scattering problems. IEEE Trans. Geosci. Remote Sens. 44(1), 116–125 (2006)

    Article  Google Scholar 

  27. Sergeyev, Y.D., Kvasov, D.E.: Deterministic Global Optimization: an Introduction to the Diagonal Approach. Springer, Berlin (2017)

    Book  Google Scholar 

  28. Snyman, J.A., Kok, S.: A reassessment of the Snyman–Fatti dynamic search trajectory method for unconstrained global optimization. J. Glob. Optim. 43, 67–82 (2009)

    Article  MathSciNet  Google Scholar 

  29. Steklov, V.A.: Sur les expressions asymptotiques de certaines fonctions définies par les equations differentielles du second ordre et leurs applications au probleme du developement dúne fonction arbitraire en series procedant suivant les diverses fonctions. Commun. de la Soc. Math. de Kharkow, Sér. 2(10), 97–199 (1907). (In French)

    Google Scholar 

  30. Thng, I., Cantoni, A., Leung, Y.H.: Analytical solutions to the optimization of a quadratic cost function subject to linear and quadratic equality constraints. Appl. Math. Optim. 34, 161–382 (1996)

    Article  MathSciNet  Google Scholar 

  31. Wu, Z., Tian, J., Quan, J., Ugon, J.: Optimality conditions and optimization methods for quartic polynomial optimization. Appl. Math. Comput. 232, 968–982 (2014)

    MathSciNet  MATH  Google Scholar 

  32. Zhu, J., Zhao, S., Liu, G.: Solution to global minimization of polynomials by backward differential flow. J. Optim. Theory Appl. 161, 828–836 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors offer their warm thanks to the anonymous reviewers for their careful reading, and the comments and suggestions they made, which in turn have improved the manuscript. They are also grateful to the editors for efficiently handling the manuscript.

Author information

Authors and Affiliations

  1. Mathematics, UniSA STEM, University of South Australia, Mawson Lakes, SA, 5095, Australia

    Regina S. Burachik & C. Yalçın Kaya

Authors
  1. Regina S. Burachik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Yalçın Kaya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Regina S. Burachik.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Dedicated to our dear friend Marco António López Cerdá on his 70th birthday.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Burachik, R.S., Kaya, C.Y. Steklov convexification and a trajectory method for global optimization of multivariate quartic polynomials. Math. Program. 189, 187–216 (2021). https://doi.org/10.1007/s10107-020-01536-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-020-01536-8

Keywords

Mathematics Subject Classification

Search

Navigation