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Approximating Clarke’s subgradients of semismooth functions by divided differences

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Abstract

We show that the algorithm presented in an earlier paper by Studniarski (Numer. Math., 55:685–693, 1989) can be applied, after only a small modification, to approximate numerically Clarke’s subgradients of semismooth functions of two variables. Results of computational testing of this modified algorithm are also reported.

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References

  1. Bagirov, A.M.: A method for minimizing convex functions based on continuous approximations to the subdifferential. Optim. Methods Softw. 9, 1–17 (1998)

    MATH  MathSciNet  Google Scholar 

  2. Chaney, R.W.: Second-order necessary conditions in constrained semismooth optimization. SIAM J. Control Optim. 25, 1072–1081 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen, X., Nashed, Z., Qi, L.: Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM J. Numer. Anal. 38, 1200–1216 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  5. Hintermüller, M., Hinze, M.: A SQP-semismooth Newton-type algorithm applied to control of the instationary Navier–Stokes system subject to control constraints. SIAM J. Optim. 16, 1177–1200 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Hintermüller, M., Kovtunenko, V., Kunish, K.: Semismooth Newton methods for a class of unilaterally constrained variational problems. Adv. Math. Sci. Appl. 14, 513–535 (2004)

    MATH  MathSciNet  Google Scholar 

  7. Hintermüller, M., Ulbrich, M.: A mesh-independence result for semismooth Newton methods. Math. Program. 101, 151–184 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Facchinei, F., Pang, J.-S.: Finite-dimensional Variational Inequalities and Complementarity Problems, vol 2. Springer, Berlin Heidelberg New York (2003)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  11. Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  12. Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)

    Article  MathSciNet  Google Scholar 

  13. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton, New Jersey (1970)

    MATH  Google Scholar 

  14. Shor, N.Z.: Minimization Methods for Nondifferentiable Functions. Springer, Berlin Heidelberg New York (1985)

    Google Scholar 

  15. Studniarski, M.: An algorithm for calculating one subgradient of a convex function of two variables. Numer. Math. 55, 685–693 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  16. Śmietański, M.J.: An approximate Newton method for non-smooth equations with finite max functions. Numer. Algorithms 41, 219–238 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ulbrich, M.: Constrained optimal control of Navier–Stokes flow by semismooth Newton methods. Syst. Control. Lett. 48, 297–311 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Ulbrich, M.: Semismooth Newton methods for operator equations in function spaces. SIAM J. Optim. 13, 805–842 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics, University of Łódź, ul. S. Banacha 22, 90-238, Łódź, Poland

    Marcin Studniarski

  2. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt

    El Desouky Rahmo

Authors
  1. Marcin Studniarski
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  2. El Desouky Rahmo
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Correspondence to Marcin Studniarski.

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Studniarski, M., Rahmo, E.D. Approximating Clarke’s subgradients of semismooth functions by divided differences. Numer Algor 43, 385–392 (2006). https://doi.org/10.1007/s11075-007-9069-3

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  • DOI: https://doi.org/10.1007/s11075-007-9069-3

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