Skip to main content
SpringerLink
Log in

Saddle points of rational functions

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

This paper concerns saddle points of rational functions, under general constraints. Based on optimality conditions, we propose an algorithm for computing saddle points. It uses Lasserre’s hierarchy of semidefinite relaxation. The algorithm can get a saddle point if it exists, or it can detect its nonexistence if it does not. Numerical experiments show that the algorithm is efficient for computing saddle points of rational functions.

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

Similar content being viewed by others

References

  1. Bertsekas, D., Nedić, A., Ozdaglar, A.: Convex Analysis and Optimization. Athena Scientific, Belmont (2003)

    MATH  Google Scholar 

  2. Benzi, M., Liesen, G.H.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)

    Article  MathSciNet  Google Scholar 

  3. Chen, Y., Lan, G., Ouyang, Y.: Optimal primal-dual methods for a class of saddle point problems. SIAM J. Optim. 24(4), 1779–1814 (2014)

    Article  MathSciNet  Google Scholar 

  4. Cox, B., Juditsky, A., Nemirovski, A.: Decomposition techniques for bilinear saddle point problems and variational inequalities with affine monotone operators. J. Optim. Theory Appl. 172(2), 402–435 (2017)

    Article  MathSciNet  Google Scholar 

  5. Demmel, J., Nie, J., Powers, V.: Representations of positive polynomials on non-compact semialgebraic sets via KKT ideals. J. Pure Appl. Algebra 209(1), 189–200 (2007)

    Article  MathSciNet  Google Scholar 

  6. Guo, F., Wang, L., Zhou, G.: Minimizing rational functions by exact Jacobian SDP relaxation applicable to finite sinjularities. J. Glob. Optim. 58(2), 261–284 (2014)

    Article  Google Scholar 

  7. He, Y., Monteiro, R.: Accelerating block-decomposition first-order methods for solving composite saddle-point and two-player nash equilibrium problems. SIAM J. Optim. 25(4), 2182–2211 (2015)

    Article  MathSciNet  Google Scholar 

  8. Henrion, D., Lasserre, J., Loefberg, J.: GloptiPoly 3: moments, optimization and semidefinite programming. Optim. Methods Softw. 24(4–5), 761–779 (2009)

    Article  MathSciNet  Google Scholar 

  9. Jibetean, D., de Klerk, E.: Global optimization of rational functions: a semidefinite programming approach. Math. Program. Ser. A 106, 93–109 (2006)

    Article  MathSciNet  Google Scholar 

  10. Lasserre, J.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2001)

    Article  MathSciNet  Google Scholar 

  11. Lasserre, J.: Introduction to Polynomial and Semi-algebraic Optimization. Cambridge University Press, Cambridge (2015)

    Book  Google Scholar 

  12. Laurent, M.: Revisiting two theorems of Curto and Fialkow on moment matrices. Proc. AMS 133(10), 2965–2976 (2005)

    Article  MathSciNet  Google Scholar 

  13. Laurent, M.: Optimization over polynomials: selected topics. In: Proceedings of the International Congress of Mathematicians (2014)

  14. Maistroskii, D.: Gradient methods for finding saddle points. Matekon 13, 3–22 (1977)

    Google Scholar 

  15. Monteiro, R., Svaiter, B.: Complexity of variants of Tseng’s modified F-B splitting and Korpelevich’s methods for hemivariational inequalities with applications to saddle-point and convex optimization problems. SIAM J. Optim. 21(4), 1688–1720 (2011)

    Article  MathSciNet  Google Scholar 

  16. Nedić, A., Ozdaglar, A.: Subgradient methods for saddle-point problems. J. Optim. Theory Appl. 142(1), 205–228 (2009)

    Article  MathSciNet  Google Scholar 

  17. Nemirovski, A.: Prox-method with rate of convergence O(1/t) for variational inequalities with Lipschitz continuous monotone operators and smooth convex–concave saddle point problems. SIAM J. Optim. 15(1), 229–251 (2004)

    Article  MathSciNet  Google Scholar 

  18. Nemirovski, A., Judin, D.B.: Cezare convergence of gradient method approximation of saddle points for convex–concave functions. Dokl. Akad. Nauk SSSR 239, 1056–1059 (1978)

    MathSciNet  Google Scholar 

  19. Nie, J., Demmel, J., Gu, M.: Global minimization of rational functions and the nearest GCDs. J. Glob. Optim. 40(4), 697–718 (2008)

    Article  MathSciNet  Google Scholar 

  20. Nie, J., Ranestad, K.: Algebraic degree of polynomial optimization. SIAM J. Optim. 20(1), 485–502 (2009)

    Article  MathSciNet  Google Scholar 

  21. Nie, J., Yang, Z., Zhou, G.: The saddle point problem of polynomials. Preprint (2018). arXiv:1809.01218

  22. Nie, J.: Certifying convergence of Lasserre’s hierarchy via flat truncation. Math. Program. Ser. A 142(1–2), 485–510 (2013)

    Article  MathSciNet  Google Scholar 

  23. Nie, J.: Polynomial optimization with real varieties. SIAM J. Optim. 23(3), 1634–1646 (2013)

    Article  MathSciNet  Google Scholar 

  24. Nie, J.: Linear optimization with cones of moments and nonnegative polynomials. Math. Program. 153, 247–274 (2015)

    Article  MathSciNet  Google Scholar 

  25. Nie, J.: Optimality conditions and finite convergence of Lasserre’s hierarchy. Math. Program. Ser. A 146(1–2), 97–121 (2014)

    Article  MathSciNet  Google Scholar 

  26. Nie, J.: Tight relaxations for polynomial optimization and lagrange multiplier expressions. Math. Program. (2018). https://doi.org/10.1007/s10107-018-1276-2

    Article  MathSciNet  Google Scholar 

  27. Von Nuemann, J.: Zur Theorie der Gesellschaftspiele. Math. Ann. 100, 295–320 (1928)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42, 969–984 (1993)

    Article  MathSciNet  Google Scholar 

  30. Scheiderer, C.: Positivity and sums of squares: a guide to recent results. In: Putinar, M., Sullivant, S. (eds.) Emerging Applications of Algebraic Geometry. The IMA Volumes in Mathematics and Its Applications, vol. 149, pp. 271–324. Springer, New York (2009)

    Google Scholar 

  31. Schweighofer, M.: Optimization of polynomials on compact semialgebraic sets. SIAM J. Optim. 15(3), 805–825 (2005)

    Article  MathSciNet  Google Scholar 

  32. Sturm, J.: SeDuMi 1.02: a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11 & 12, 625–653 (1999)

    Article  MathSciNet  Google Scholar 

  33. Zabotin, I.: A subgradient method for finding a saddle point of a convex–concave function. Issled. Prikl. Mat. 15, 6–12 (1988)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors wish to thank the editors and reviewers for their valuable comments that have improved the paper. Guangming Zhou was partially supported by the NSFC Grant 11671342.

Author information

Authors and Affiliations

  1. School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, 411105, Hunan, China

    Guangming Zhou

  2. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, Hunan, China

    Qin Wang & Wenjie Zhao

Authors
  1. Guangming Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wenjie Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guangming Zhou.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhou, G., Wang, Q. & Zhao, W. Saddle points of rational functions. Comput Optim Appl 75, 817–832 (2020). https://doi.org/10.1007/s10589-019-00141-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-019-00141-6

Keywords

Search

Navigation