Skip to main content
SpringerLink
Log in

An Effective Nonsmooth Optimization Algorithm for Locally Lipschitz Functions

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

To construct an effective minimization algorithm for locally Lipschitz functions, we show how to compute a descent direction satisfying Armijo’s condition. We present a finitely terminating algorithm to construct an approximating set for the Goldstein subdifferential leading to the desired descent direction. Using this direction, we propose a minimization algorithm for locally Lipschitz functions and prove its convergence. Finally, we implement our algorithm with matrix laboratory (MATLAB) codes and report our testing results. The comparative numerical results attest to the efficiency of the proposed algorithm.

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. Shor, N.Z.: Minimization Methods for Non-differentiable Functions. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  2. Frangioni, A.: Generalized bundle methods. SIAM J. Optim. 13(1), 117–156 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Fuduli, A., Gaudioso, M., Giallombardo, G.: Minimizing nonconvex nonsmooth functions via cutting planes and proximity control. SIAM J. Optim. 14(3), 743–756 (2003)

    Article  MathSciNet  Google Scholar 

  4. Fuduli, A., Gaudioso, M., Giallombardo, G.: A DC piecewise affine model and a bundling technique in nonconvex nonsmooth minimization. Optim. Methods Softw. 19(1), 89–102 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gaudioso, M., Gorgone, E., Monaco, M.F.: Piecewise linear approximations in nonconvex nonsmooth optimization. Numer. Math. 113(1), 73–88 (2009)

    MathSciNet  MATH  Google Scholar 

  6. Gaudioso, M., Monaco, M.F.: A bundle type approach to the unconstrained minimization of convex nonsmooth functions. Math. Program. 23(2), 216–226 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms II: Advanced Theory and Bundle Methods. Springer, Berlin (1993)

    MATH  Google Scholar 

  8. Kiwiel, K.C.: Methods of Descent for Nondifferentiable Optimization. Lecture Notes in Mathematics, vol. 1133. Springer, Berlin (1985)

    MATH  Google Scholar 

  9. Makela, M.M., Neittaanmaki, P.: Nonsmooth Optimization. World Scientific, Singapore (1992)

    Google Scholar 

  10. Mifflin, R.: An algorithm for constrained optimization with semismooth functions. Math. Oper. Res. 2(2), 191–207 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  11. Wolfe, P.H.: A method of conjugate subgradients of minimizing nondifferentiable convex functions. Math. Program. Stud. 3, 145–173 (1975)

    Article  MathSciNet  Google Scholar 

  12. Mayne, D.Q., Polak, E.: Nondifferential optimization via adaptive smoothing. J. Optim. Theory Appl. 43, 19–30 (1984)

    Article  Google Scholar 

  13. Polak, E., Royset, J.O.: Algorithms for finite and semi-infinite min-max-min problems using adaptive smoothing techniques. J. Optim. Theory Appl. 119(3), 421–457 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hare, W., Sagastizábal, C.: Computing proximal points of nonconvex functions. Math. Program. 116(1–2), 221–258 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hare, W., Sagastizábal, C.: A VU-algorithm for convex minimization. Math. Program. 104(2–3), 583–608 (2005)

    MathSciNet  Google Scholar 

  16. Kaplan, A., Tichatschke, R.: Proximal point methods and nonconvex optimization. J. Glob. Optim. 13(4), 389–406 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bagirov, A.M.: Continuous subdifferential approximations and their applications. J. Math. Sci. 115, 2567–2609 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bagirov, A.M., Karasözen, B., Sezer, M.: Discrete gradient method: derivative-free method for nonsmooth optimization. J. Optim. Theory Appl. 137(2), 317–334 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Burke, J.V., Lewis, A.S., Overton, M.L.: A robust gradient sampling algorithm for nonsmooth, nonconvex optimization. SIAM J. Optim. 15, 571–779 (2005)

    Article  MathSciNet  Google Scholar 

  20. Kiwiel, K.C.: Convergence of the gradient sampling algorithm for nonsmooth nonconvex optimization. SIAM J. Optim. 18(2), 379–388 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Clarke, F.H.: Optimization and Nonsmooth Analysis. Society for Industrial and Applied Mathematics, Philadelphia (1987)

    Google Scholar 

  22. Goldstein, A.A.: Optimization of Lipschitz continuous functions. Math. Program. 13, 14–22 (1977)

    Article  MATH  Google Scholar 

  23. Bertsekas, D.P., Mitter, S.K.: A descent numerical method for optimization problems with nondifferentiable cost functionals. SIAM J. Optim. 11, 637–652 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer Series in Operations Research and Financial Engineering. Springer, New York (2006)

    MATH  Google Scholar 

  25. Bruke, J.V., Lewis, A.S., Overton, M.L.: Approximating subdifferential by random sampling of gradient. Math. Oper. Res. 27, 576–584 (2002)

    Google Scholar 

  26. Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 959–972 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  27. Daugavet, V.A.: Modification of the Wolfe method. Zh. Vychisl. Mat. Mat. Fiz. 21, 504–508 (1981)

    MathSciNet  MATH  Google Scholar 

  28. Mitchell, B.F., Demyanov, V.F., Malozemov, V.N.: Finding the point of a polyhedron closest to the origin. SIAM J. Control Optim. 12, 19–26 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wolfe, P.: Finding the nearest point in a polytope. Math. Program. 11, 128–149 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  30. Anstreicher, K.M., Lee, J.: A masked spectral bound for maximum-entropy sampling. In: mODa 7—Advances in Model-Oriented Design and Analysis. Contributions to Statistics, pp. 1–12. Physica, Heidelberg (2004)

    Chapter  Google Scholar 

  31. Trefethen, L.N.: Pseudospectra of linear operators. SIAM Rev. 39, 383–406 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  32. Bruke, J.V., Lewis, A.S., Overton, M.L.: Robust stability and a criss-cross algorithm for pseudospectra. SIAM J. Numer. Anal. 23, 359–375 (2003)

    Google Scholar 

Download references

Acknowledgements

The first author thanks the Research Council of Sharif University of Technology for supporting this work.

Author information

Authors and Affiliations

  1. Faculty of Mathematical Sciences, Sharif University of Technology, Tehran, Iran

    Nezam Mahdavi-Amiri

  2. Department of Mathematical Sciences, University of Mazandaran, Babolsar, Iran

    Rohollah Yousefpour

Authors
  1. Nezam Mahdavi-Amiri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rohollah Yousefpour
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nezam Mahdavi-Amiri.

Additional information

Communicated by Vladimir F. Dem’yanov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mahdavi-Amiri, N., Yousefpour, R. An Effective Nonsmooth Optimization Algorithm for Locally Lipschitz Functions. J Optim Theory Appl 155, 180–195 (2012). https://doi.org/10.1007/s10957-012-0024-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-012-0024-7

Keywords

Search

Navigation