Skip to main content
Book cover

International Conference on Learning and Intelligent Optimization

LION 2017: Learning and Intelligent Optimization pp 166–179Cite as

Decomposition Descent Method for Limit Optimization Problems

  • 933 Accesses

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 10556)

Abstract

We consider a general limit optimization problem whose goal function need not be smooth in general and only approximation sequences are known instead of exact values of this function. We suggest to apply a two-level approach where approximate solutions of a sequence of mixed variational inequality problems are inserted in the iterative scheme of a selective decomposition descent method. Its convergence is attained under coercivity type conditions.

Keywords

  • Optimization problems
  • Limit problems
  • Non-smooth functions
  • Mixed variational inequality
  • Decomposition descent method
  • Coercivity conditions

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-69404-7_12
  • Chapter length: 14 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   59.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-69404-7
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   79.99
Price excludes VAT (USA)
Learn about institutional subscriptions

References

  1. Burges, C.J.C.: A tutorial on support vector machines for pattern recognition. Data Mining Know. Disc. 2, 121–167 (1998)

    CrossRef  Google Scholar 

  2. Cevher, V., Becker, S., Schmidt, M.: Convex optimization for big data. Signal Process. Magaz. 31, 32–43 (2014)

    CrossRef  Google Scholar 

  3. Facchinei, F., Scutari, G., Sagratella, S.: Parallel selective algorithms for nonconvex big data optimization. IEEE Trans. Sig. Process. 63, 1874–1889 (2015)

    CrossRef  MathSciNet  Google Scholar 

  4. Tseng, P., Yun, S.: A coordinate gradient descent method for nonsmooth separable minimization. Math. Progr. 117, 387–423 (2010)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. Richtárik, P., Takáč, M.: Parallel coordinate descent methods for big data optimization. Math. Program. 156, 433–484 (2016)

    CrossRef  MATH  MathSciNet  Google Scholar 

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

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. Alart, P., Lemaire, B.: Penalization in non-classical convex programming via variational convergence. Math. Program. 51, 307–331 (1991)

    CrossRef  MATH  Google Scholar 

  8. Cominetti, R.: Coupling the proximal point algorithm with approximation methods. J. Optim. Theor. Appl. 95, 581–600 (1997)

    CrossRef  MATH  MathSciNet  Google Scholar 

  9. Salmon, G., Nguyen, V.H., Strodiot, J.J.: Coupling the auxiliary problem principle and epiconvergence theory for solving general variational inequalities. J. Optim. Theor. Appl. 104, 629–657 (2000)

    CrossRef  MATH  Google Scholar 

  10. Konnov, I.V.: An inexact penalty method for non stationary generalized variational inequalities. Set-Valued Variat. Anal. 23, 239–248 (2015)

    CrossRef  MATH  MathSciNet  Google Scholar 

  11. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  12. Ermoliev, Y.M., Norkin, V.I., Wets, R.J.B.: The minimization of semicontinuous functions: mollifier subgradient. SIAM J. Contr. Optim. 33, 149–167 (1995)

    CrossRef  MATH  MathSciNet  Google Scholar 

  13. Czarnecki, M.-O., Rifford, L.: Approximation and regularization of lipschitz functions: convergence of the gradients. Trans. Amer. Math. Soc. 358, 4467–4520 (2006)

    CrossRef  MATH  MathSciNet  Google Scholar 

  14. Gwinner, J.: On the penalty method for constrained variational inequalities. In: Hiriart-Urruty, J.-B., Oettli, W., Stoer, J. (eds.) Optimization: Theory and Algorithms, pp. 197–211. Marcel Dekker, New York (1981)

    Google Scholar 

  15. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. The Math. Stud. 63, 127–149 (1994)

    MATH  MathSciNet  Google Scholar 

  16. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic Publishers, Dordrecht (1996)

    CrossRef  MATH  Google Scholar 

  17. Tibshirani, R.: Regression shrinkage and selection via the lasso. J. Royal Stat. Soc. Ser. B. 58, 267–288 (1996)

    MATH  MathSciNet  Google Scholar 

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

    CrossRef  MATH  Google Scholar 

Download references

Acknowledgement

The results of 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 author was also supported by Russian Foundation for Basic Research, project No. 16-01-00109 and by grant No. 297689 from Academy of Finland.

Author information

Authors and Affiliations

  1. Institute of Computational Mathematics and Information Technologies, Kazan Federal University, Kremlevskaya St. 18, 420008, Kazan, Russia

    Igor Konnov

Authors
  1. Igor Konnov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Igor Konnov .

Editor information

Editors and Affiliations

  1. University of Trento, Trento, Italy

    Roberto Battiti

  2. University of Calabria, Rende, Italy

    Dmitri E. Kvasov

  3. University of Calabria, Rende, Italy

    Yaroslav D. Sergeyev

Rights and permissions

Reprints and Permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Konnov, I. (2017). Decomposition Descent Method for Limit Optimization Problems. In: Battiti, R., Kvasov, D., Sergeyev, Y. (eds) Learning and Intelligent Optimization. LION 2017. Lecture Notes in Computer Science(), vol 10556. Springer, Cham. https://doi.org/10.1007/978-3-319-69404-7_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-69404-7_12

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-69403-0

  • Online ISBN: 978-3-319-69404-7

  • eBook Packages: Computer ScienceComputer Science (R0)