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## Saddle-point type optimality criteria, duality and a new approach for solving nonsmooth fractional continuous-time programming problems

 Department of Numerical Mathematics and Optimization, Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Belgrade, Serbia

Received  October 2021 Revised  January 2022 Early access February 2022

Fund Project: This research was supported by the Science Fund of the Republic of Serbia, Grant No. 7744592, Integrability and Extremal Problems in Mechanics, Geometry and Combinatorics-MEGIC

In this paper, fractional continuous-time programming problems with inequality phase constraints are considered. Optimality conditions and duality results under a certain regularity condition are derived. All functions are assumed to be nondifferentiable. These results improve and generalize a number of existing results in the area of fractional continuous-time programming. We provide a practical example to illustrate our results.

Citation: Aleksandar Jović. Saddle-point type optimality criteria, duality and a new approach for solving nonsmooth fractional continuous-time programming problems. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022025
##### References:
 [1] A. V. Arutunov, S. E. Zhukovskiy and B. Marinkovic, Theorems of the alternative for systems of convex inequalities, Set-Valued and Variational Analysis, 27 (2019), 51-70.  doi: 10.1007/s11228-017-0406-y. [2] R. Bellman, Bottleneck problems and dynamic programming, Proc. Natl. Acad. Sci., 39 (1953), 947-951.  doi: 10.1073/pnas.39.9.947. [3] A. J. V. Brandao, M. A. Rojas-Medar and G. N. Silva, Nonsmooth continuous-time optimization problems: Necessary conditions, Comput. Math. Appl., 41 (2001), 1477-1486.  doi: 10.1016/S0898-1221(01)00112-2. [4] B. D. Craven, Mathematical Programming and Control Theory, Chapman Hall, London, 1978. [5] B. D. Craven, Fractional Programming, Heldermann Verlag, Berlin, 1988. [6] G. Gol'stein, Theory of Convex Programming, Trans. Math. Mono. Amer. Math. Soc., Providence, RI., 1972. [7] S. Nobakhtian and M. Pouryayevali, Optimality conditions and duality for nonsmooth fractional continuous-time problems, Journal of Optimization Theory and Applications, 152 (2012), 245-255.  doi: 10.1007/s10957-010-9693-2. [8] S. Schaible, Bibliography in fractional programming, Zeitschrift fur Operations Research, 26 (1982), 211-241.  doi: 10.1007/bf01917115. [9] A. M. Stancu, Mathematical Programming with Type-I Functions, Matrix Rom., Bucharest, 2013. [10] A. M. Stancu and I. M. Stancu-Minasian, Sufficiency criteria in continuous-time nonlinear programming under generalized ($\alpha$, $\rho$)-($\eta$, $\theta$) - type Ⅰ invexity, Rev. Roumaine Math. Pures Appl., 56 (2011), 169-179. [11] A. M. Stancu and I. M. Stancu-Minasian, Carathéodory-John-type sufficiency criteria in continuous-time nonlinear programming under generalized ($\alpha$, $\rho$)-($\eta$, $\theta$) - type Ⅰ invexity, Math. Reports, 16 (2012), 345-354. [12] I. M. Stancu-Minasian, Fractional Programming. Theory, Methods and Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. doi: 10.1007/978-94-009-0035-6. [13] I. M. Stancu-Minasian, An eighth bibliography of fractional programming, Optimization, 66 (2017), 439-470.  doi: 10.1080/02331934.2016.1276179. [14] I. M. Stancu-Minasian, A ninth bibliography of fractional programming. With supplemental material "A classification due to the bibliography on fractional programming" by I. M. Stancu-Minasian, Optimization, 68 (2019), 2125-2169.  doi: 10.1080/02331934.2019.1632250. [15] I. M. Stancu-Minasian and S. Tigan, Continuous time linear-fractional programming. The minimum-risk approach, RAIRO-Oper.Res., 34 (2000), 397-409.  doi: 10.1051/ro:2000121. [16] C. Wen and H. Wu, Using the Dinkelbach-type algorithm to solve the continuous-time linear fractional Programming Problems, Journal of Global Optimization, 49 (2011), 237-263.  doi: 10.1007/s10898-010-9542-8. [17] C. Wen and H. Wu, Approximate solutions and duality theorems for continuous-time linear fractional programming problems, Numerical Functional Analysis and Optimization, 33 (2012), 80-129.  doi: 10.1080/01630563.2011.629312. [18] C. Wen and H. Wu, Using the parametric approach to solve the continuous-time linear fractional max-min problems, Journal of Global Optimization, 54 (2012), 129-153.  doi: 10.1007/s10898-011-9751-9. [19] G. J. Zalmai, A continuous time generalization of Gordan's transposition theorem, J. Math. Anal. Appl., 110 (1985), 130-140.  doi: 10.1016/0022-247X(85)90339-7. [20] G. J. Zalmai, The Fritz John and Kuhn-Tucker optimality conditions in continuous-time programming, J. Math. Anal. Appl., 110 (1985), 503-518.  doi: 10.1016/0022-247X(85)90312-9. [21] G. J. Zalmai, Duality for a class of continuous-time homogeneous fractional programming problems, Z. Oper. Res., Ser. A-B 30 (1986), 43-48.  doi: 10.1007/BF01918630. [22] G. J. Zalmai, Optimality conditions and duality models for a class of nonsmooth continuous-time fractional programming problems, J. Stat. Manag. Sy. St., 1 (1998), 141-173.  doi: 10.1080/09720510.1998.10700983.

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##### References:
 [1] A. V. Arutunov, S. E. Zhukovskiy and B. Marinkovic, Theorems of the alternative for systems of convex inequalities, Set-Valued and Variational Analysis, 27 (2019), 51-70.  doi: 10.1007/s11228-017-0406-y. [2] R. Bellman, Bottleneck problems and dynamic programming, Proc. Natl. Acad. Sci., 39 (1953), 947-951.  doi: 10.1073/pnas.39.9.947. [3] A. J. V. Brandao, M. A. Rojas-Medar and G. N. Silva, Nonsmooth continuous-time optimization problems: Necessary conditions, Comput. Math. Appl., 41 (2001), 1477-1486.  doi: 10.1016/S0898-1221(01)00112-2. [4] B. D. Craven, Mathematical Programming and Control Theory, Chapman Hall, London, 1978. [5] B. D. Craven, Fractional Programming, Heldermann Verlag, Berlin, 1988. [6] G. Gol'stein, Theory of Convex Programming, Trans. Math. Mono. Amer. Math. Soc., Providence, RI., 1972. [7] S. Nobakhtian and M. Pouryayevali, Optimality conditions and duality for nonsmooth fractional continuous-time problems, Journal of Optimization Theory and Applications, 152 (2012), 245-255.  doi: 10.1007/s10957-010-9693-2. [8] S. Schaible, Bibliography in fractional programming, Zeitschrift fur Operations Research, 26 (1982), 211-241.  doi: 10.1007/bf01917115. [9] A. M. Stancu, Mathematical Programming with Type-I Functions, Matrix Rom., Bucharest, 2013. [10] A. M. Stancu and I. M. Stancu-Minasian, Sufficiency criteria in continuous-time nonlinear programming under generalized ($\alpha$, $\rho$)-($\eta$, $\theta$) - type Ⅰ invexity, Rev. Roumaine Math. Pures Appl., 56 (2011), 169-179. [11] A. M. Stancu and I. M. Stancu-Minasian, Carathéodory-John-type sufficiency criteria in continuous-time nonlinear programming under generalized ($\alpha$, $\rho$)-($\eta$, $\theta$) - type Ⅰ invexity, Math. Reports, 16 (2012), 345-354. [12] I. M. Stancu-Minasian, Fractional Programming. Theory, Methods and Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. doi: 10.1007/978-94-009-0035-6. [13] I. M. Stancu-Minasian, An eighth bibliography of fractional programming, Optimization, 66 (2017), 439-470.  doi: 10.1080/02331934.2016.1276179. [14] I. M. Stancu-Minasian, A ninth bibliography of fractional programming. With supplemental material "A classification due to the bibliography on fractional programming" by I. M. Stancu-Minasian, Optimization, 68 (2019), 2125-2169.  doi: 10.1080/02331934.2019.1632250. [15] I. M. Stancu-Minasian and S. Tigan, Continuous time linear-fractional programming. The minimum-risk approach, RAIRO-Oper.Res., 34 (2000), 397-409.  doi: 10.1051/ro:2000121. [16] C. Wen and H. Wu, Using the Dinkelbach-type algorithm to solve the continuous-time linear fractional Programming Problems, Journal of Global Optimization, 49 (2011), 237-263.  doi: 10.1007/s10898-010-9542-8. [17] C. Wen and H. Wu, Approximate solutions and duality theorems for continuous-time linear fractional programming problems, Numerical Functional Analysis and Optimization, 33 (2012), 80-129.  doi: 10.1080/01630563.2011.629312. [18] C. Wen and H. Wu, Using the parametric approach to solve the continuous-time linear fractional max-min problems, Journal of Global Optimization, 54 (2012), 129-153.  doi: 10.1007/s10898-011-9751-9. [19] G. J. Zalmai, A continuous time generalization of Gordan's transposition theorem, J. Math. Anal. Appl., 110 (1985), 130-140.  doi: 10.1016/0022-247X(85)90339-7. [20] G. J. Zalmai, The Fritz John and Kuhn-Tucker optimality conditions in continuous-time programming, J. Math. Anal. Appl., 110 (1985), 503-518.  doi: 10.1016/0022-247X(85)90312-9. [21] G. J. Zalmai, Duality for a class of continuous-time homogeneous fractional programming problems, Z. Oper. Res., Ser. A-B 30 (1986), 43-48.  doi: 10.1007/BF01918630. [22] G. J. Zalmai, Optimality conditions and duality models for a class of nonsmooth continuous-time fractional programming problems, J. Stat. Manag. Sy. St., 1 (1998), 141-173.  doi: 10.1080/09720510.1998.10700983.
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