A New Definition for Generalized First Derivative of Nonsmooth Functions

ABSTRACT

In this paper, we define a functional optimization problem corresponding to smooth functions which its optimal solution is first derivative of these functions in a domain. These functional optimization problems are applied for non-smooth functions which by solving these problems we obtain a kind of generalized first derivatives. For this purpose, a linear programming problem corresponding functional optimization problem is obtained which their optimal solutions give the approximate generalized first derivative. We show the efficiency of our approach by obtaining derivative and generalized derivative of some smooth and nonsmooth functions respectively in some illustrative examples.

In this paper, we define a functional optimization problem corresponding to smooth functions which its optimal solution is first derivative of these functions in a domain. These functional optimization problems are applied for non-smooth functions which by solving these problems we obtain a kind of generalized first derivatives. For this purpose, a linear programming problem corresponding functional optimization problem is obtained which their optimal solutions give the approximate generalized first derivative. We show the efficiency of our approach by obtaining derivative and generalized derivative of some smooth and nonsmooth functions respectively in some illustrative examples.

KEYWORDS

Generalized Derivative, Smooth and Nonsmooth Functions, Fourier analysis, Linear Programming, Functional Optimization

Generalized Derivative, Smooth and Nonsmooth Functions, Fourier analysis, Linear Programming, Functional Optimization

Cite this paper

nullA. Kamyad, M. Skandari and H. Erfanian, "A New Definition for Generalized First Derivative of Nonsmooth Functions,"*Applied Mathematics*, Vol. 2 No. 10, 2011, pp. 1252-1257. doi: 10.4236/am.2011.210174.

nullA. Kamyad, M. Skandari and H. Erfanian, "A New Definition for Generalized First Derivative of Nonsmooth Functions,"

References

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[6] B. S. Mordukhovich, “Gene-ralized Differential Calculus for Nonsmooth and Set-Valued Mappings,” Journal of Mathematical Analysis and Applications, Vol. 183, No. 1, 1994, pp. 250-288. doi:10.1006/jmaa.1994.1144

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[8] B. Mordukho-vich, J. S. Treiman and Q. J. Zhu, “An Extended Extremal Principle with Applications to Multiobjective Optimi-zation,” SIAM Journal on Optimization, Vol. 14, 2003, pp. 359-379. doi:10.1137/S1052623402414701

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[13] A. D. Ioffe, “Metric Regularity and Subdifferential Calculus,” Russian Mathematical Surveys, Vol. 55, No. 3, 2000, pp. 501-558. doi:10.1070/RM2000v055n03ABEH000292

[14] M. S. Gowda and G. Ravindran, “Algebraic Univalence Theo-rems for Nonsmooth Functions,” Journal of Mathematical Analysis and Applications, Vol. 252, No. 2, 2000, pp. 917-935. doi:10.1006/jmaa.2000.7171

[15] R. T. Rock-afellar and R. J. Wets, “Variational Analysis,” Springer, New York, 1997.

[16] P. Michel and J.-P. Penot, “Calcul Sous-Diff′erentiel Pour des Fonctions Lipschitziennes et Non-Lipschitziennes,” CR Academic Science Paris, Ser. I Math. Vol. 298, 1985, pp. 269-272.

[17] J. S. Treiman, “Lagrange Multipliers for Nonconvex Generalized Gra-dients with Equality, Inequality and Set Constraints,” SIAM Journal on Optimization, Vol. 37, 1999, pp. 1313-1329. doi:10.1137/S0363012996306595

[18] E. Stade, “Fourier Analysis, USA,” Wiley, New York, 2005.

[19] M. S. Bazaraa, J. J. Javis and H. D. Sheralli, “Linear Programming,” Wiley & Sons, New York, 1990.

[20] M. S. Bazaraa, H. D. Sheralli and C. M. Shetty, “Nonlinear Programming: Theory and Applica-tion,” Wiley & Sons, New York, 2006.

[1] F. H. Clarke, “Optimization and Non-Smooth Analysis,” Wiley, New York, 1983.

[2] V. F. Demyanov and A. M. Rubinov, “Constructive Nonsmooth Analysis,” Verlag Peter Lang, New York, 1995.

[3] W. Schirotzek, “Nonsmooth Analysis,” Springer, New York, 2007. doi:10.1007/978-3-540-71333-3

[4] B. Mordukhovich, “Approximation Methods in Problems of Optimization and Control,” Nauka, Moscow, 1988.

[5] B. Mordukhovich, “Complete Characterizations of Openness, Metric Regularity, and Lipschitzian Properties of Multifunctions,” Transactions of the American Mathematical Society, Vol. 340, 1993, pp. 1-35. doi:10.2307/2154544

[6] B. S. Mordukhovich, “Gene-ralized Differential Calculus for Nonsmooth and Set-Valued Mappings,” Journal of Mathematical Analysis and Applications, Vol. 183, No. 1, 1994, pp. 250-288. doi:10.1006/jmaa.1994.1144

[7] B. Mordukhovich, “Variational Analysis and Generalized Differentiation,” Vol. 1-2, Springer, New York, 2006.

[8] B. Mordukho-vich, J. S. Treiman and Q. J. Zhu, “An Extended Extremal Principle with Applications to Multiobjective Optimi-zation,” SIAM Journal on Optimization, Vol. 14, 2003, pp. 359-379. doi:10.1137/S1052623402414701

[9] A. D. Ioffe, “Nonsmooth Analysis: Differential Calculus of Nondif-ferentiable Mapping,” Transactions of the American Ma-thematical Society, Vol. 266, 1981, pp. 1-56. doi:10.1090/S0002-9947-1981-0613784-7

[10] A. D. Ioffe, “Approximate Subdifferentials and Applications I: The Finite Dimensional Theory,” Transactions of the American Mathematical Society, Vol. 281, 1984, pp. 389-416.

[11] A. D. Ioffe, “On the Local Surjection Property,” Nonlinear Analysis, Vol. 11, 1987, pp. 565-592. doi:10.1016/0362-546X(87)90073-3

[12] A. D. Ioffe, “A Lagrange Multiplier Rule with Small Convex-Valued Subdifferentials Fornonsmooth Problems of Mathematical Programming Involving Equality and Nonfunctional Constraints,” Mathematical Programming, Vol. 588, 1993, pp. 137-145. doi:10.1007/BF01581262

[13] A. D. Ioffe, “Metric Regularity and Subdifferential Calculus,” Russian Mathematical Surveys, Vol. 55, No. 3, 2000, pp. 501-558. doi:10.1070/RM2000v055n03ABEH000292

[14] M. S. Gowda and G. Ravindran, “Algebraic Univalence Theo-rems for Nonsmooth Functions,” Journal of Mathematical Analysis and Applications, Vol. 252, No. 2, 2000, pp. 917-935. doi:10.1006/jmaa.2000.7171

[15] R. T. Rock-afellar and R. J. Wets, “Variational Analysis,” Springer, New York, 1997.

[16] P. Michel and J.-P. Penot, “Calcul Sous-Diff′erentiel Pour des Fonctions Lipschitziennes et Non-Lipschitziennes,” CR Academic Science Paris, Ser. I Math. Vol. 298, 1985, pp. 269-272.

[17] J. S. Treiman, “Lagrange Multipliers for Nonconvex Generalized Gra-dients with Equality, Inequality and Set Constraints,” SIAM Journal on Optimization, Vol. 37, 1999, pp. 1313-1329. doi:10.1137/S0363012996306595

[18] E. Stade, “Fourier Analysis, USA,” Wiley, New York, 2005.

[19] M. S. Bazaraa, J. J. Javis and H. D. Sheralli, “Linear Programming,” Wiley & Sons, New York, 1990.

[20] M. S. Bazaraa, H. D. Sheralli and C. M. Shetty, “Nonlinear Programming: Theory and Applica-tion,” Wiley & Sons, New York, 2006.