A New Approach for Generalized Partial Derivatives of Non-smooth Functions

Hamid Reza ERFANIAN, Mohammad Hadi Noori SKANDARI, Ali Vahidian KAMYAD


In this paper, we define the functional optimization problems corresponding to multi-variable smooth functions, so that their optimal solutions are partial derivatives of these functions. These functional optimization problems are applied for multi-variables nonsmooth functions, so that by solving these problems we obtain generalized partial derivatives. For this purpose, linear programming problems corresponding to the functional optimization problems are obtained, so that their optimal solutions give the approximate generalized partial derivatives. In some illustrative examples, we show the efficiency of our approach by obtaining the generalized partial derivative of some smooth and nonsmooth functions, respectively.



Generalized derivative, nonsmooth functions, optimization

Full Text:



FH Clarke. Optimization and Nonsmooth Analysis. John Wiley & Sons, New York, 1983.

FH Clarke, YS Ledyaev, YS Stern and PR Wolenski. Nonsmooth Analysis and Control Theory. Springer, New York, 1998.

BS Mordukhovich. Generalized differential calculus for nonsmooth and set-valued mappings. J. Math. Anal. Appl. 1994; 183, 250-88.

BS Mordukhovich. Variational Analysis and Generalized Differentiation. Springer, New York. 2006.

AV Kamyad, MH Noori Skandari and HR Erfanian. A new definition for generalized first derivative of nonsmooth functions. Appl. Math. 2011; 2, 1252-7.

HR Erfanian, MHN Skandari and AV Kamyad. A new approach for the generalized first derivative and extension it to the generalized second derivative of nonsmooth functions. Int. J. Intell. Syst. Appl 2013; 4, 100-7.

HR Erfanian, MH Noori Skandari and AV Kamyad. A numerical approach for nonsmooth ordinary differential equations. J. Vib. Contr. 2013; 19, 2124-36.

MS Bazaraa, JJ Javis and HD Sheralli. Linear Programming. Wiley & Sons, New York, 1990.

MS Bazaraa, HD Sheralli and CM Shetty. Nonlinear Programming: Theory and Application. Wiley & Sons, New York, 2006.


  • There are currently no refbacks.


Online ISSN: 2228-835X


Last updated: 17 May 2019