• Previous Article
    Development of opened-network data envelopment analysis models under uncertainty
  • JIMO Home
  • This Issue
  • Next Article
    Multi-objective optimization model for planning metro-based underground logistics system network: Nanjing case study
doi: 10.3934/jimo.2022025
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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. ArutunovS. 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. BrandaoM. 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.

show all references

References:
[1]

A. V. ArutunovS. 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. BrandaoM. 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.

[1]

Joon Kwon, Panayotis Mertikopoulos. A continuous-time approach to online optimization. Journal of Dynamics and Games, 2017, 4 (2) : 125-148. doi: 10.3934/jdg.2017008

[2]

Hanqing Jin, Xun Yu Zhou. Continuous-time portfolio selection under ambiguity. Mathematical Control and Related Fields, 2015, 5 (3) : 475-488. doi: 10.3934/mcrf.2015.5.475

[3]

Fritz Colonius, Guilherme Mazanti. Decay rates for stabilization of linear continuous-time systems with random switching. Mathematical Control and Related Fields, 2019, 9 (1) : 39-58. doi: 10.3934/mcrf.2019002

[4]

Shui-Nee Chow, Xiaojing Ye, Hongyuan Zha, Haomin Zhou. Influence prediction for continuous-time information propagation on networks. Networks and Heterogeneous Media, 2018, 13 (4) : 567-583. doi: 10.3934/nhm.2018026

[5]

J. C. Dallon, Lynnae C. Despain, Emily J. Evans, Christopher P. Grant. A continuous-time stochastic model of cell motion in the presence of a chemoattractant. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4839-4852. doi: 10.3934/dcdsb.2020129

[6]

Hui Meng, Fei Lung Yuen, Tak Kuen Siu, Hailiang Yang. Optimal portfolio in a continuous-time self-exciting threshold model. Journal of Industrial and Management Optimization, 2013, 9 (2) : 487-504. doi: 10.3934/jimo.2013.9.487

[7]

Wenpin Tang, Xun Yu Zhou. Tail probability estimates of continuous-time simulated annealing processes. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022015

[8]

Andy Hammerlindl, Bernd Krauskopf, Gemma Mason, Hinke M. Osinga. Determining the global manifold structure of a continuous-time heterodimensional cycle. Journal of Computational Dynamics, 2022, 9 (3) : 393-419. doi: 10.3934/jcd.2022008

[9]

Ran Dong, Xuerong Mao. Asymptotic stabilization of continuous-time periodic stochastic systems by feedback control based on periodic discrete-time observations. Mathematical Control and Related Fields, 2020, 10 (4) : 715-734. doi: 10.3934/mcrf.2020017

[10]

Huijuan Li, Junxia Wang. Input-to-state stability of continuous-time systems via finite-time Lyapunov functions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 841-857. doi: 10.3934/dcdsb.2019192

[11]

Lakhdar Aggoun, Lakdere Benkherouf. A Markov modulated continuous-time capture-recapture population estimation model. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 1057-1075. doi: 10.3934/dcdsb.2005.5.1057

[12]

Ping Chen, Haixiang Yao. Continuous-time mean-variance portfolio selection with no-shorting constraints and regime-switching. Journal of Industrial and Management Optimization, 2020, 16 (2) : 531-551. doi: 10.3934/jimo.2018166

[13]

Zhigang Zeng, Tingwen Huang. New passivity analysis of continuous-time recurrent neural networks with multiple discrete delays. Journal of Industrial and Management Optimization, 2011, 7 (2) : 283-289. doi: 10.3934/jimo.2011.7.283

[14]

Huai-Nian Zhu, Cheng-Ke Zhang, Zhuo Jin. Continuous-time mean-variance asset-liability management with stochastic interest rates and inflation risks. Journal of Industrial and Management Optimization, 2020, 16 (2) : 813-834. doi: 10.3934/jimo.2018180

[15]

Willem Mélange, Herwig Bruneel, Bart Steyaert, Dieter Claeys, Joris Walraevens. A continuous-time queueing model with class clustering and global FCFS service discipline. Journal of Industrial and Management Optimization, 2014, 10 (1) : 193-206. doi: 10.3934/jimo.2014.10.193

[16]

Haixiang Yao, Zhongfei Li, Xun Li, Yan Zeng. Optimal Sharpe ratio in continuous-time markets with and without a risk-free asset. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1273-1290. doi: 10.3934/jimo.2016072

[17]

Qian Zhang, Huaicheng Yan, Jun Cheng, Xisheng Zhan, Kaibo Shi. Fault detection filtering for continuous-time singular systems under a dynamic event-triggered mechanism. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022023

[18]

Zheng Dou, Shaoyong Lai. Optimal contracts and asset prices in a continuous-time delegated portfolio management problem. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022083

[19]

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

[20]

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

2021 Impact Factor: 1.411

Metrics

  • PDF downloads (187)
  • HTML views (119)
  • Cited by (0)

Other articles
by authors

[Back to Top]