Skip to main content

Advertisement

Springer Nature Link
Log in

Cutting-plane method based on epigraph approximation with discarding the cutting planes

  • System Analysis and Operations Research
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

Propose a method for solving a mathematical programming problem from the class of cutting methods. In our method, on each step the epigraph of the objective function is embedded into a specifically constructed polyhedral set, and on this set an auxiliary linear function is minimized in order to construct the iteration point. Proposed method does not require that each approximation set is embedded in the previous ones. This feature lets us periodically discard additional constraints that form the approximation sets obtained during the solution process. Prove the method’s convergence and discuss possible implementations.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Bulatov, V.P., Metody pogruzheniya v zadachakh optimizatsii (Immersion Methods in Optimization Problems), Novosibirsk: Nauka, 1977.

    Google Scholar 

  2. Bulatov, V.P. and Khamisov, O.V., Cutting Methods in E n+1 for Global Optimization Problems on a Class of Functions, Zh. Vychisl. Mat. Mat. Fiz., 2007, vol. 47, no. 11, pp. 1830–1842.

    MathSciNet  Google Scholar 

  3. Zabotin, I.Ya., On Some Immersion-Severance Algorithms for Mathematical Programming Problems, Izv. Irkutsk. Gos. Univ., Ser. Mat., 2011, vol. 4, no. 2, pp. 91–101.

    MATH  Google Scholar 

  4. Zabotin, I.Ya. and Yarullin, R.S., On One Approach to Constructing Branch and Bound Algorithms with Discarding Separating Planes, Izv. Vyssh. Uchebn. Zaved., Mat., 2013, no. 3, pp. 74–79.

    MathSciNet  Google Scholar 

  5. Zangwill, W.I., Nonlinear Programming: A Unified Approach, Englewood Cliffs: Prentice Hall, 1969. Translated under the title Nelineinoe programmirovanie, Moscow: Sovetskoe Radio, 1973.

    Google Scholar 

  6. Kolokolov, A.A., Regular Partitions and Cuts in Integer Programming, Sib. Zh. Issled. Oper., 1994, vol. 1, no. 2, pp. 18–39.

    MATH  MathSciNet  Google Scholar 

  7. Nesterov, Yu.E., Vvedenie v vypukluyu optimizatsiyu (Introduction to Convex Optimization), Moscow: MTsNMO, 2010.

    Google Scholar 

  8. Nurminskii, E.A., The Separating Planes Method with Bounded Memory for Convex Nonsmooth Optimization Problems, Vychisl. Metod. Programm., 2006, vol. 7, pp. 133–137.

    Google Scholar 

  9. Polyak, B.T., Vvedenie v optimizatsiyu (Introduction to Optimization), Moscow: Nauka, 1983.

    MATH  Google Scholar 

  10. Kelley, J.E., The Cutting-Plane Method for Solving Convex Programs, SIAM J., 1960, vol. 8, no. 4, pp. 703–712.

    MathSciNet  Google Scholar 

  11. Lemarechal, C., Nemirovskii, A., and Nesterov, Yu., New Variants of Bundle Methods, Math. Program., 1995, vol. 69, pp. 111–148.

    Article  MATH  MathSciNet  Google Scholar 

  12. Vasil’ev, F.P., Chislennye metody resheniya ekstremal’nykh zadach (Numerical Methods for Extremal Problems), Moscow: Nauka, 1988.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kazan (Volga Region) Federal University, Kazan, Russia

    I. Ya. Zabotin & R. S. Yarullin

Authors
  1. I. Ya. Zabotin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. S. Yarullin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. Ya. Zabotin.

Additional information

Original Russian Text © I.Ya. Zabotin, R.S. Yarullin, 2015, published in Avtomatika i Telemekhanika, 2015, No. 11, pp. 76–88.

This paper was recommended for publication by B.T. Polyak, a member of the Editorial Board

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zabotin, I.Y., Yarullin, R.S. Cutting-plane method based on epigraph approximation with discarding the cutting planes. Autom Remote Control 76, 1966–1975 (2015). https://doi.org/10.1134/S0005117915110065

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117915110065

Keywords

Search

Navigation