Skip to main content

Part of the book series: Applied Optimization ((APOP,volume 24))

Abstract

In this paper, the unconstrained minimization problem of locally Lipschitz functions is considered. A numerical method for solving such problems based on continuous approximations to the Clarke sub differential is proposed and studied. An algorithm for the construction of the continuous approximations is described. Results of numerical experiments are presented.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. M. Bagirov. A method of approximating a subdifferential Zh. Vichisl. Mat. Mat. Fiz., V. 32, No.4, pp. 652-658, 1992. English transl. in Comput. Maths Math. Phys. V. 32. No. 4, pp. 561 - 566. 1992.

    Google Scholar 

  2. A. M. Bagirov. Continuous approximation to a subdifferential of a function of a maximum. Kibernetika i sistemniy analiz. 4, pp. 180-184, 1993. English transl. in Cybernet, and System Anal. 4, pp. 626 - 630. 1994.

    Google Scholar 

  3. A. M. Bagirov and A. A. Gasanov. A method of approximating a quasidifferential Zh. Vichisl. Mat. Mat. Fiz., V. 35, No.4, pp. 511-519, 1995. English transl. in Comput. Maths Math. Phys., V. 35, No. 4, pp. 403 - 409, 1995.

    Google Scholar 

  4. F. H. Clarke. Optimization and Nonsmooth Analysis. John Wiley, New York, 1983.

    MATH  Google Scholar 

  5. A. A. Goldstein. Optimization of Lipschitz continuous functions. Math. Program., 13, pp. 14 -22, 1977.

    Google Scholar 

  6. P. Hansen and B. Jaumard. Lipshitz optimization. In: Horst, R. and Pardalos, P.M., eds. Handbook of Global Optimization. Kluwer Academic Publishers, Dordrecht Boston London, pp. 407 - 493, 1995.

    Google Scholar 

  7. C. Lemarechal. Nondifferentiable optimization, subgradient and e- subgradient methods. Lecture Notes in Optimization and Operations Research, 17, Springer verlag, New York, 1976.

    Google Scholar 

  8. C. Lemarechal. Numerical experiments in nonsmooth optimization. In: Progress in nondifferentiable optimization. Ed. E.A. Nurminski. CP - 82 - 5. International Institute for Applied System Analysis: Laxenburg, Austria, pp. 61 - 84, 1982.

    Google Scholar 

  9. R. Mifflin. Semismooth and semiconvex functions in constrained optimization. SIAM Journal on Control and Optimization, 15, pp. 957 - 972, 1977.

    Google Scholar 

  10. J. D. Pinter. Global Optimization in Action. Kluwer Academic Publishers, Dordrecht Boston London, 1996.

    MATH  Google Scholar 

  11. E. Polak, D. Q. Mayne and Y. Y. Wardi. On the extension of constrained optimization algorithms from differentiable to nondifferentiable problems, SIAM Journal on Control and Optimization, 21, pp. 179 - 203, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  12. E. Polak and D. Q. Mayne. Algorithm models for nondifferentiable optimization. SIAM Journal on Control and Optimization, 23, pp. 477 - 491, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  13. P. H. Wolfe. Finding the nearest point in a polytope. Math. Program. V. 11, No. 2, pp. 128 - 149, 1976.

    Google Scholar 

  14. H. Xu, A. M. Rubinov and B. M. Glover. Continuous approximations to generalized Jacobians with application to nonsmooth least-squares minimization. Research Paper, 17/96, University of Ballarat, Australia, 1996.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Kluwer Academic Publishers, Boston

About this chapter

Cite this chapter

Bagirov, A.M., Gadjiev, N.K. (1998). A Monotonous Method for Unconstrained Lipschitz Optimization. In: De Leone, R., Murli, A., Pardalos, P.M., Toraldo, G. (eds) High Performance Algorithms and Software in Nonlinear Optimization. Applied Optimization, vol 24. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3279-4_2

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-3279-4_2

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-3281-7

  • Online ISBN: 978-1-4613-3279-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics