Journal of Global Optimization

, Volume 50, Issue 1, pp 3–22 | Cite as

Codifferential method for minimizing nonsmooth DC functions

Article

Abstract

In this paper, a new algorithm to locally minimize nonsmooth functions represented as a difference of two convex functions (DC functions) is proposed. The algorithm is based on the concept of codifferential. It is assumed that DC decomposition of the objective function is known a priori. We develop an algorithm to compute descent directions using a few elements from codifferential. The convergence of the minimization algorithm is studied and its comparison with different versions of the bundle methods using results of numerical experiments is given.

Keywords

Nonsmooth optimization Nonconvex optimization DC functions Sub-differential Codifferential 

JEL Classification

C61 

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References

  1. 1.
    Bagirov A.M.: A method for minimizing of quasidifferentiable functions. Optim. Methods Softw. 17(1), 31–60 (2002)CrossRefGoogle Scholar
  2. 2.
    Bagirov A.M.: Max- min separability. Optim. Methods Softw. 20(2–3), 271–290 (2005)Google Scholar
  3. 3.
    Bagirov A.M., Karasozen B., Sezer M.: Discrete gradient method: derivative-free method for nonsmooth optimization. J. Optim. Theor. Appl. 137, 317–334 (2008)CrossRefGoogle Scholar
  4. 4.
    Bagirov A.M., Ganjehlou A.N.: A quasisecant method for minimizing nonsmooth functions. Optim. Methods Softw. 25(1), 3–18 (2010)CrossRefGoogle Scholar
  5. 5.
    Bagirov, A.M., Ganjehlou, A.N., Ugon, J., Tor, A.H.: Truncated codifferential method for nonsmooth convex optimization. Pac. J. Optim. (in press)Google Scholar
  6. 6.
    Bagirov A.M., Yearwood J.: A new nonsmooth optimization algorithm for minimum sum-of-squares clustering problems. Eur. J. Oper. Res. 170(2), 578–596 (2006)CrossRefGoogle Scholar
  7. 7.
    Burke J.V., Lewis A.S., Overton M.L.: A robust gradient sampling algorithm for nonsmooth, nonconvex optimization. SIAM J. Optim. 15, 751–779 (2005)CrossRefGoogle Scholar
  8. 8.
    Demyanov V.F., Bagirov A.M., Rubinov A.M.: A method of truncated codifferential with application to some problems of cluster analysis. J. Glob. Optim. 23(1), 63–80 (2002)CrossRefGoogle Scholar
  9. 9.
    Demyanov, V.F., Rubinov, A.M.: Constructive nonsmooth analysis Peter Lang. Frankfurt am Main (1995)Google Scholar
  10. 10.
    Frangioni A.: Solving semidefinite quadratic problems within nonsmooth optimization algorithms. Comput. Oper. Res. 23, 1099–1118 (1996)CrossRefGoogle Scholar
  11. 11.
    Fuduli A., Gaudioso M., Giallombardo G.: Minimizing nonconvex nonsmooth functions via cutting planes and proximity control. SIAM J. Optim. 14, 743–756 (2004)CrossRefGoogle Scholar
  12. 12.
    Fuduli A., Gaudioso M., Giallombardo G.: A DC piecewise affine model and a bundling technique in nonconvex nonsmooth minimization. Optim. Methods Softw. 19, 89–102 (2004)CrossRefGoogle Scholar
  13. 13.
    Gaudioso M., Gorgone E., Monaco M.F.: Piecewise linear approximations in nonconvex nonsmooth optimization. Numerische Mathematik 113, 73–88 (2009)Google Scholar
  14. 14.
    Gaudioso M., Gorgone E.: Gradient set splitting in nonconvex nonsmooth numerical optimization. Optim. Methods Softw. 25(1), 59–74 (2010)CrossRefGoogle Scholar
  15. 15.
    Haarala N., Miettinen K., Makela M.M.: New limited memory bundle method for large-scale nonsmooth optimization. Optim. Methods Softw. 19(6), 673–692 (2004)CrossRefGoogle Scholar
  16. 16.
    Hiriart-Urruty J.B., Lemarechal C.: Convex Analysis and Minimization Algorithms, Vols. 1 and 2. Springer, Heidelberg (1993)Google Scholar
  17. 17.
    Kiwiel K.C.: Methods of Descent for Nondifferentiable Optimization Lecture Notes in Mathematics, Vol. 1133. Springer, Berlin (1985)Google Scholar
  18. 18.
    Kiwiel K.C.: A dual method for certain positive semidefinite quadratic programming problems. SIAM J. Sci. Stat. Comput. 10, 175–186 (1989)CrossRefGoogle Scholar
  19. 19.
    Lukśan L., Vlćek J.: A bundle Newton method for nonsmooth unconstrained minimization. Math. Program. 83, 373–391 (1998)Google Scholar
  20. 20.
    Lukśan L., Vlćek J.: Globally convergent variable metric method for convex nonsmooth unconstrained minimization. J. Optim. Theor. Appl. 102(3), 593–613 (1999)CrossRefGoogle Scholar
  21. 21.
    Lukśan L., Vlćek J.: Algorithm 811: NDA: Algorithms for nondifferentiable optimization. ACM Trans. Math. Softw. 27(2), 193–213 (2001)CrossRefGoogle Scholar
  22. 22.
    Makela M.M., Neittaanmaki P.: Nonsmooth Optimization. World Scientific, Singapore (1992)Google Scholar
  23. 23.
    Makela M.M.: Survey of bundle methods for nonsmooth optimization. Optim. Methods Softw. 17(1), 1–29 (2002)CrossRefGoogle Scholar
  24. 24.
    Nurminski E.A.: Convergence of the suitable affine subspace method for finding the least distance to a simplex. Comput. Math. Math. Phys. 45(11), 1915–1922 (2005)Google Scholar
  25. 25.
    Nurminski E.A.: Projection onto polyhedra in outer representation. Comput. Math. Math. Phys. 48(3), 367–375 (2008)CrossRefGoogle Scholar
  26. 26.
    Wolfe P.H.: Finding the nearest point in a polytope. Math. Program. 11(2), 128–149 (1976)CrossRefGoogle Scholar
  27. 27.
    Zaffaroni A.: Continuous approximations, codifferentiable functions and minimization methods. In: Demyanov, V.F., Rubinov, A.M. (eds) Nonconvex Optimization and its Applications, Quasidifferentiability and Related Topics, pp. 361–391. Kluwer Academic Publishers, Dordrecht (2000)Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  1. 1.Centre for Informatics and Applied Optimization, School of Information Technology and Mathematical SciencesUniversity of BallaratBallarat, VictoriaAustralia

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