On Generalized High Order Derivatives of Nonsmooth Functions

DOI: 10.4236/ajcm.2014.44028   PDF   HTML     2,862 Downloads   3,436 Views   Citations


In this paper, we proposed a Extension Definition to derive, simultaneously, the first, second and high order generalized derivatives for non-smooth functions, in which the involved functions are Riemann integrable but not necessarily locally Lipschitz or continuous. Indeed, we define a functional optimization problem corresponding to smooth functions where its optimal solutions are the first and second derivatives of these functions in a domain. Then by applying these functional optimization problems for non-smooth functions and using this method we obtain generalized first derivative (GFD) and generalized second derivative (GSD). Here, the optimization problem is approximated with a linear programming problem that by solving of which, we can obtain these derivatives, as simple as possible. We extend this approach for obtaining generalized high order derivatives (GHODs) of non-smooth functions, simultaneously. Finally, for efficiency of our approach some numerical examples have been presented.

Share and Cite:

Zeid, S. and Kamyad, A. (2014) On Generalized High Order Derivatives of Nonsmooth Functions. American Journal of Computational Mathematics, 4, 317-328. doi: 10.4236/ajcm.2014.44028.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Clarke, F.H. (1983) Optimization and Non-Smooth Analysis. Wiley, New York.
[2] Ioffe, A.D. (1993) A Lagrange Multiplier Rule with Small Convex-Valued Subdifferentials for Nonsmooth Problems of Mathematical Programming Involving Equality and Nonfunctional Constraints. Mathematical Programming, 588, 137-145.
[3] Mordukhovich, B.S. (1994) Generalized Differential Calculus for Nonsmooth and Set-Valued Mappings. Journal of Mathematical Analysis and Applications, 183, 250-288.
[4] Rockafellar, T. (1994) Nonsmooth Optimization, Mathematical Programming: State of the Art 1994. University of Michigan Press, Ann Arbor, 248-258.
[5] Mahdavi-Amiri, N. and Yousefpour, R. (2011) An Effective Optimization Algorithm for Locally Nonconvex Lipschitz Functions Based on Mollifier Subgradients. Bulletin of the Iranian Mathematical Society, 37, 171-198.
[6] Kamyad, A.V., Noori Skandari, M.H. and Erfanian, H.R. (2011) A New Definition for Generalized First Derivative of Nonsmooth Functions. Applied Mathematics, 2, 1252-1257.
[7] Rockafellar, R.T. and Wets, R.J. (1997) Variational Analysis. Springer, New York.
[8] Clarke, F.H., Ledyaev, Yu.S., Stern, R.J. and Wolenski, P.R. (1998) Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, Vol. 178. Springer-Verlag, New York.
[9] Mordukhovich, B. (1988) Approximation Methods in Problems of Optimization and Control. Nauka, Moscow.
[10] Mordukhovich, B. (1993) Complete Characterizations of Openness, Metric Regularity, and Lipschitzian Properties of Multifunctions. Transactions of the American Mathematical Society, 340, 1-35.
[11] Mordukhovich, B. (2006) Variational Analysis and Generalized Differentiation, Vol. 1-2. Springer, New York.
[12] Mordukhovich, B., Treiman, J.S. and Zhu, Q.J. (2003) An Extended Extremal Principle with Applications to MultiObjective Optimization. SIAM Journal on Optimization, 14, 359-379.
[13] Erfanian, H., Noori Skandari, M.H. and Kamyad, A.V. (2012) A Numerical Approach for Nonsmooth Ordinary Differential Equations. Journal of Vibration and Control.
[14] Bazaraa, M.S., Javis, J.J. and Sheralli, H.D. (1990) Linear Programming. Wiley and Sons, New York.
[15] Bazaraa, M.S., Sheralli, H.D. and Shetty, C.M. (2006) Nonlinear Programming: Theory and Application. Wiley and Sons, New York. http://dx.doi.org/10.1002/0471787779
[16] Erfanian, H., Noori Skandari, M.H. and Kamyad, A.V. (2013) A New Approach for the Generalized First Derivative and Extension It to the Generalized Second Derivative of Nonsmooth Functions, I. Journal of Intelligent Systems and Applications, 4, 100-107. http://dx.doi.org/10.5815/ijisa.2013.04.10
[17] Stade, E. (2005) Fourier Analysis. Wiley, New York. http://dx.doi.org/10.1002/9781118165508

comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.