# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020157
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## Optimality results for a specific fractional problem

 1 LAMA, FSDM, Sidi Mohamed Ben Abdellah University, Fez, Morocco

Received  July 2019 Revised  July 2020 Early access November 2020

In this paper, one minimizes a fractional function over a compact set. Using an exact separation theorem, one gives necessary optimality conditions for strict optimal solutions in terms of Fréchet subdifferentials. All data are assumed locally Lipschitz.

Citation: Nazih Abderrazzak Gadhi, Khadija Hamdaoui. Optimality results for a specific fractional problem. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020157
##### References:
 [1] A. Y. Kruger, On Fréchet subdifferentials, Journal of Mathematical Sciences, 116 (2003), 3325-3358.  doi: 10.1023/A:1023673105317.  Google Scholar [2] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, Grundlehren Series (Fundamental Principles of Mathematical Sciences) 330, Springer, Berlin, 2006.  Google Scholar [3] B. S. Mordukhovich and Y. Shao, Nonsmooth sequential analysis in Asplund spaces, Transactions of the American Mathematical Society, 348 (1996), 1235-1280.  doi: 10.1090/S0002-9947-96-01543-7.  Google Scholar [4] B. S. Mordukhovich, N. M. Nam and N. D. Yen, Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming, Optimization, 55 (2006), 685-708.  doi: 10.1080/02331930600816395.  Google Scholar [5] R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, Springer-Verlag, Berlin, 1993.  Google Scholar [6] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processe, Wiley, New York, 1962.  Google Scholar [7] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar [8] X. Y. Zheng, Z. Yang and J. Zou, Exact separation theorem for closed sets in Asplund spaces, Optimization, 66 (2017), 1065-1077.  doi: 10.1080/02331934.2017.1316503.  Google Scholar

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##### References:
 [1] A. Y. Kruger, On Fréchet subdifferentials, Journal of Mathematical Sciences, 116 (2003), 3325-3358.  doi: 10.1023/A:1023673105317.  Google Scholar [2] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, Grundlehren Series (Fundamental Principles of Mathematical Sciences) 330, Springer, Berlin, 2006.  Google Scholar [3] B. S. Mordukhovich and Y. Shao, Nonsmooth sequential analysis in Asplund spaces, Transactions of the American Mathematical Society, 348 (1996), 1235-1280.  doi: 10.1090/S0002-9947-96-01543-7.  Google Scholar [4] B. S. Mordukhovich, N. M. Nam and N. D. Yen, Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming, Optimization, 55 (2006), 685-708.  doi: 10.1080/02331930600816395.  Google Scholar [5] R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, Springer-Verlag, Berlin, 1993.  Google Scholar [6] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processe, Wiley, New York, 1962.  Google Scholar [7] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar [8] X. Y. Zheng, Z. Yang and J. Zou, Exact separation theorem for closed sets in Asplund spaces, Optimization, 66 (2017), 1065-1077.  doi: 10.1080/02331934.2017.1316503.  Google Scholar
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