doi: 10.3934/jimo.2020157

Optimality results for a specific fractional problem

1. 

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

* Corresponding author: Khadija Hamdaoui

Received  July 2019 Revised  July 2020 Published  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. MordukhovichN. 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. ZhengZ. 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

show all references

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. MordukhovichN. 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. ZhengZ. 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

[1]

Ying Gao, Xinmin Yang, Kok Lay Teo. Optimality conditions for approximate solutions of vector optimization problems. Journal of Industrial & Management Optimization, 2011, 7 (2) : 483-496. doi: 10.3934/jimo.2011.7.483

[2]

J. Alberto Conejero, Marko Kostić, Pedro J. Miana, Marina Murillo-Arcila. Distributionally chaotic families of operators on Fréchet spaces. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1915-1939. doi: 10.3934/cpaa.2016022

[3]

J. Leonel Rocha, Sandra M. Aleixo. An extension of Gompertzian growth dynamics: Weibull and Fréchet models. Mathematical Biosciences & Engineering, 2013, 10 (2) : 379-398. doi: 10.3934/mbe.2013.10.379

[4]

Mansoureh Alavi Hejazi, Soghra Nobakhtian. Optimality conditions for multiobjective fractional programming, via convexificators. Journal of Industrial & Management Optimization, 2020, 16 (2) : 623-631. doi: 10.3934/jimo.2018170

[5]

Luong V. Nguyen. A note on optimality conditions for optimal exit time problems. Mathematical Control & Related Fields, 2015, 5 (2) : 291-303. doi: 10.3934/mcrf.2015.5.291

[6]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017

[7]

Jiawei Chen, Shengjie Li, Jen-Chih Yao. Vector-valued separation functions and constrained vector optimization problems: optimality and saddle points. Journal of Industrial & Management Optimization, 2020, 16 (2) : 707-724. doi: 10.3934/jimo.2018174

[8]

Xiang-Kai Sun, Xian-Jun Long, Hong-Yong Fu, Xiao-Bing Li. Some characterizations of robust optimal solutions for uncertain fractional optimization and applications. Journal of Industrial & Management Optimization, 2017, 13 (2) : 803-824. doi: 10.3934/jimo.2016047

[9]

Éder Rítis Aragão Costa. An extension of the concept of exponential dichotomy in Fréchet spaces which is stable under perturbation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 845-868. doi: 10.3934/cpaa.2019041

[10]

Nuno R. O. Bastos, Rui A. C. Ferreira, Delfim F. M. Torres. Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 417-437. doi: 10.3934/dcds.2011.29.417

[11]

Ricardo Almeida. Optimality conditions for fractional variational problems with free terminal time. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 1-19. doi: 10.3934/dcdss.2018001

[12]

Xian-Jun Long, Nan-Jing Huang, Zhi-Bin Liu. Optimality conditions, duality and saddle points for nondifferentiable multiobjective fractional programs. Journal of Industrial & Management Optimization, 2008, 4 (2) : 287-298. doi: 10.3934/jimo.2008.4.287

[13]

Xian-Jun Long, Jing Quan. Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 361-370. doi: 10.3934/naco.2011.1.361

[14]

Xiao-Bing Li, Qi-Lin Wang, Zhi Lin. Optimality conditions and duality for minimax fractional programming problems with data uncertainty. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1133-1151. doi: 10.3934/jimo.2018089

[15]

Francis Clarke. A general theorem on necessary conditions in optimal control. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 485-503. doi: 10.3934/dcds.2011.29.485

[16]

Lucas Bonifacius, Ira Neitzel. Second order optimality conditions for optimal control of quasilinear parabolic equations. Mathematical Control & Related Fields, 2018, 8 (1) : 1-34. doi: 10.3934/mcrf.2018001

[17]

Sofia O. Lopes, Fernando A. C. C. Fontes, Maria do Rosário de Pinho. On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 559-575. doi: 10.3934/dcds.2011.29.559

[18]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033

[19]

Enkhbat Rentsen, J. Zhou, K. L. Teo. A global optimization approach to fractional optimal control. Journal of Industrial & Management Optimization, 2016, 12 (1) : 73-82. doi: 10.3934/jimo.2016.12.73

[20]

Jianxiong Ye, An Li. Necessary optimality conditions for nonautonomous optimal control problems and its applications to bilevel optimal control. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1399-1419. doi: 10.3934/jimo.2018101

2019 Impact Factor: 1.366

Article outline

[Back to Top]