Skip to main content
SpringerLink
Log in

Splitting Method with Adaptive Step-Size

  • Conference paper
  • First Online:
Mathematical Optimization Theory and Operations Research (MOTOR 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11548))

Abstract

We suggest the modified splitting method for mixed variational inequalities and prove its convergence under rather mild assumptions. This method maintains the basic convergence properties but does not require any iterative step-size search procedure. It involves a simple adaptive step-size choice, which takes into account the problem behavior along the iterative sequence. The key element of this approach is a given majorant step-size sequence converging to zero. The next decreased value of step-size is taken only when the current iterate does not give a sufficient descent of the objective function. This descent value is estimated with the help of an Armijo-type condition, similar to the rule used in the inexact step-size linesearch. If the current iterate gives a sufficient descent, we can even take an increasing step-size value at the next iterate. Preliminary results of computational experiments confirm the efficiency of the proposed modification in comparison with the ordinary splitting method using the inexact step-size linesearch procedure.

The results of the first author in this work were obtained within the state assignment of the Ministry of Science and Education of Russia, project No. 1.460.2016/1.4. In this work, the first author was also supported by the RFBR grant, project No. 19-01-00431.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two monotone operators. SIAM. J. Num. Anal. 16(6), 964–979 (1979)

    Article  Google Scholar 

  2. Gabay, D.: Application of the method of multipliers to variational inequalities. In: Fortin, M., Glowinski, R. (eds.) Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, pp. 299–331. North-Holland, Amsterdam (1983)

    Chapter  Google Scholar 

  3. Fukushima, M., Mine, H.: A generalized proximal point algorithm for certain non-convex minimization problems. Int. J. Syst. Sci. 12, 989–1000 (1981)

    Article  Google Scholar 

  4. Patriksson, M.: Cost approximations: a unified framework of descent algorithms for nonlinear programs. SIAM J. Optim. 8(2), 561–582 (1998)

    Article  MathSciNet  Google Scholar 

  5. Patriksson, M.: Nonlinear Programming and Variational Inequality Problems: A Unified Approach. Kluwer, Dordrecht (1999)

    Book  Google Scholar 

  6. Konnov, I.V., Kum, S.: Descent methods for mixed variational inequalities in a Hilbert space. Nonlinear Anal. Theory Methods Appl. 47(1), 561–572 (2001)

    Article  MathSciNet  Google Scholar 

  7. Konnov, I.V.: Iterative solution methods for mixed equilibrium problems and variational inequalities with non-smooth functions. In: Haugen, I.N., Nilsen, A.S. (eds.) Game Theory: Strategies, Equilibria, and Theorems, pp. 117–160. NOVA, Hauppauge (2008)

    Google Scholar 

  8. Konnov, I.V.: Descent methods for mixed variational inequalities with non-smooth mappings. In: Reich, S., Zaslavski, A.J. (eds.) Optimization Theory and Related Topics. Contemporary Mathematics, vol. 568, pp. 121–138. American Mathematical Society, Providence (2012)

    Chapter  Google Scholar 

  9. Konnov, I.V.: Sequential threshold control in descent splitting methods for decomposal optimization problems. Optim. Methods Softw. 30(6), 1238–1254 (2015)

    Article  MathSciNet  Google Scholar 

  10. Konnov, I.V., Salahuddin: Two-level iterative method for non-stationary mixed variational inequalities. Russ. Mathem. (Iz. VUZ) 61(10), 44–53 (2017)

    Article  MathSciNet  Google Scholar 

  11. Konnov, I.: Conditional gradient method without line-search. Russ. Math. 62(1), 82–85 (2018)

    Article  Google Scholar 

  12. Konnov, I.: A simple adaptive step-size choice for iterative optimization methods. Adv. Model. Optim. 20(2), 353–369 (2018)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Computational Mathematics and Information Technologies, Department of System Analysis and Information Technologies, Kazan Federal University, Kazan, Russia

    Igor Konnov

  2. Institute of Computational Mathematics and Information Technologies, Department of Data Mining and Operations Research, Kazan Federal University, Kazan, Russia

    Olga Pinyagina

Authors
  1. Igor Konnov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Olga Pinyagina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Olga Pinyagina .

Editor information

Editors and Affiliations

  1. Krasovsky Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    Michael Khachay

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Yury Kochetov

  3. University of Florida, Gainesville, FL, USA

    Panos Pardalos

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Konnov, I., Pinyagina, O. (2019). Splitting Method with Adaptive Step-Size. In: Khachay, M., Kochetov, Y., Pardalos, P. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2019. Lecture Notes in Computer Science(), vol 11548. Springer, Cham. https://doi.org/10.1007/978-3-030-22629-9_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-22629-9_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-22628-2

  • Online ISBN: 978-3-030-22629-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation