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July  2019, 15(3): 1133-1151. doi: 10.3934/jimo.2018089

Optimality conditions and duality for minimax fractional programming problems with data uncertainty

College of Sciences, Chongqing Jiaotong University, Chongqing, 400074, China

1Corresponding author

Received  February 2017 Revised  February 2018 Published  July 2018

In this paper, we consider minimax nondifferentiable fractional programming problems with data uncertainty in both the objective and constraints. Via robust optimization, we establish the necessary and sufficient optimality conditions for an uncertain minimax convex-concave fractional programming problem under the robust subdifferentiable constraint qualification. Making use of these optimality conditions, we further obtain strong duality results between the robust counterpart of this programming problem and the optimistic counterpart of its conventional Wolf type and Mond-Weir type dual problems. We also show that the optimistic counterpart of the Wolf type dual of an uncertain minimax linear fractional programming problem with scenario uncertainty (or interval uncertainty) in objective function and constraints is a simple linear programming, and show that the robust strong duality results in sense of Wolf type always hold for this linear minimax fractional programming problem.

Citation: 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
References:
[1]

A. Beck and A. Ben-Tal, Duality in robust optimization: Primal worst equals dual best, Oper. Res. Lett., 37 (2009), 1-6. doi: 10.1016/j.orl.2008.09.010. Google Scholar

[2]

A. Ben-Tal, L. E. Ghaoui and A. Nemirovski, Robust Optimization, Princeton Series in Applied Mathemathics, 2009. doi: 10.1515/9781400831050. Google Scholar

[3]

J. M. Borwein and J. D. Vanderwerff, Convex Functions: Constructions, Characterizations and Counterexamples, Cambridge University Press, 2010. doi: 10.1017/CBO9781139087322. Google Scholar

[4]

R. I. Bot, S. M. Grad and G. Wanka, Duality in Vector Optimization, Springer-Verlag Berlin Heidelberg, 2009. doi: 10.1007/978-3-642-02886-1. Google Scholar

[5]

R. I. BotI. B. Hodrea and G. Wanka, Farkas-type results for fractional programming problems, Nonlinear Anal., 67 (2007), 1690-1703. doi: 10.1016/j.na.2006.07.041. Google Scholar

[6]

R. I. Bot, Conjugate Duality in Convex Optimization, Springer-Verlag Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-04900-2. Google Scholar

[7]

R. I. BotS. M. Grad and G. Wanka, New regularity conditions for strong and total Fenchel-Lagrange duality in infinite dimensional spaces, Nonlinear Anal., 69 (2008), 323-336. doi: 10.1016/j.na.2007.05.021. Google Scholar

[8]

F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley and Sons, Inc., New York, 1983. Google Scholar

[9]

W. Dinkelbach, On nonlinear fractional programming, Manage. Sci., 13 (1967), 492-498. doi: 10.1287/mnsc.13.7.492. Google Scholar

[10]

J. B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms I, Springer, Berlin, 1993. doi: 10.1007/978-3-662-02796-7. Google Scholar

[11]

V. JeyakumarG. Li and S. Srisatkunarajah, Strong duality for robust minmax fractional programming problem, Eur. J. Oper. Res., 228 (2013), 331-336. doi: 10.1016/j.ejor.2013.02.015. Google Scholar

[12]

V. Jeyakumar and G. Li, Strong duality in robust convex programming: Complete characterizations, SIAM J. Optim., 20 (2010), 3384-3407. doi: 10.1137/100791841. Google Scholar

[13]

V. JeyakumarG. Li and J. H. Wang, Some robust convex programs without a duality gap, J. Convex Anal., 20 (2013), 377-394. Google Scholar

[14]

V. Jeyakumar and G. Li, Robust duality for fractional programming under data uncertainty, J. Optim. Theor. Appl., 151 (2011), 292-303. doi: 10.1007/s10957-011-9896-1. Google Scholar

[15]

V. Jeyakumar, Constraint qualifications characterizing lagrangian duality in convex optimization, J. Optim. Theo. Appl., 136 (2008), 31-41. doi: 10.1007/s10957-007-9294-x. Google Scholar

[16]

V. Jeyakumar and G. Li, Characterizing robust set containments and solutions of uncertain linear programs without qualifications, Oper. Res. Lett., 38 (2010), 188-194. doi: 10.1016/j.orl.2009.12.004. Google Scholar

[17]

O. L. Mangasarian, Set containment characterization, J. Global Optim., 24 (2002), 473-480. doi: 10.1023/A:1021207718605. Google Scholar

[18]

R. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970. Google Scholar

[19]

S. Schaible, Parameter-free convex equivalent and dual programs of fractional programming problems, Z. Oper. Res., 18 (1974), 187-196. Google Scholar

[20]

S. Schaible, Fractional programming: A recent survey, J. Stat. Manag. Syst., 5 (2002), 63-86. doi: 10.1080/09720510.2002.10701051. Google Scholar

[21]

X. K. Sun and Y. Cai, On robust duality for fractional programming with uncertainty data, Positivity, 18 (2014), 9-28. doi: 10.1007/s11117-013-0227-7. Google Scholar

[22]

X. K. SunY. Cai and J. Zeng, Farkas-type results fro constraint fractional programming with DC functions, Optim. Lett., 8 (2014), 2299-2313. doi: 10.1007/s11590-014-0737-7. Google Scholar

[23]

X. K. SunZ. Y. Peng and X. L. Guo, Some characterizations of robust optimal solutions for uncertain convex optimization problems, Optim. Lett., 10 (2016), 1463-1478. doi: 10.1007/s11590-015-0946-8. Google Scholar

[24]

X. M. YangK. L. Teo and X. Q. Yang, Symmetric duality for a class of nonlinear fractional programming problems, J. Math. Anal. Appl., 271 (2002), 7-15. doi: 10.1016/S0022-247X(02)00042-2. Google Scholar

[25]

X. M. YangX. Q. Yang and K. L. Teo, Duality and saddle-point type optimality for generalized nonlinear fractional programming, J. Math. Anal. Appl., 289 (2004), 100-109. doi: 10.1016/j.jmaa.2003.08.029. Google Scholar

[26]

C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific, London, 2002. doi: 10.1142/9789812777096. Google Scholar

show all references

References:
[1]

A. Beck and A. Ben-Tal, Duality in robust optimization: Primal worst equals dual best, Oper. Res. Lett., 37 (2009), 1-6. doi: 10.1016/j.orl.2008.09.010. Google Scholar

[2]

A. Ben-Tal, L. E. Ghaoui and A. Nemirovski, Robust Optimization, Princeton Series in Applied Mathemathics, 2009. doi: 10.1515/9781400831050. Google Scholar

[3]

J. M. Borwein and J. D. Vanderwerff, Convex Functions: Constructions, Characterizations and Counterexamples, Cambridge University Press, 2010. doi: 10.1017/CBO9781139087322. Google Scholar

[4]

R. I. Bot, S. M. Grad and G. Wanka, Duality in Vector Optimization, Springer-Verlag Berlin Heidelberg, 2009. doi: 10.1007/978-3-642-02886-1. Google Scholar

[5]

R. I. BotI. B. Hodrea and G. Wanka, Farkas-type results for fractional programming problems, Nonlinear Anal., 67 (2007), 1690-1703. doi: 10.1016/j.na.2006.07.041. Google Scholar

[6]

R. I. Bot, Conjugate Duality in Convex Optimization, Springer-Verlag Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-04900-2. Google Scholar

[7]

R. I. BotS. M. Grad and G. Wanka, New regularity conditions for strong and total Fenchel-Lagrange duality in infinite dimensional spaces, Nonlinear Anal., 69 (2008), 323-336. doi: 10.1016/j.na.2007.05.021. Google Scholar

[8]

F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley and Sons, Inc., New York, 1983. Google Scholar

[9]

W. Dinkelbach, On nonlinear fractional programming, Manage. Sci., 13 (1967), 492-498. doi: 10.1287/mnsc.13.7.492. Google Scholar

[10]

J. B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms I, Springer, Berlin, 1993. doi: 10.1007/978-3-662-02796-7. Google Scholar

[11]

V. JeyakumarG. Li and S. Srisatkunarajah, Strong duality for robust minmax fractional programming problem, Eur. J. Oper. Res., 228 (2013), 331-336. doi: 10.1016/j.ejor.2013.02.015. Google Scholar

[12]

V. Jeyakumar and G. Li, Strong duality in robust convex programming: Complete characterizations, SIAM J. Optim., 20 (2010), 3384-3407. doi: 10.1137/100791841. Google Scholar

[13]

V. JeyakumarG. Li and J. H. Wang, Some robust convex programs without a duality gap, J. Convex Anal., 20 (2013), 377-394. Google Scholar

[14]

V. Jeyakumar and G. Li, Robust duality for fractional programming under data uncertainty, J. Optim. Theor. Appl., 151 (2011), 292-303. doi: 10.1007/s10957-011-9896-1. Google Scholar

[15]

V. Jeyakumar, Constraint qualifications characterizing lagrangian duality in convex optimization, J. Optim. Theo. Appl., 136 (2008), 31-41. doi: 10.1007/s10957-007-9294-x. Google Scholar

[16]

V. Jeyakumar and G. Li, Characterizing robust set containments and solutions of uncertain linear programs without qualifications, Oper. Res. Lett., 38 (2010), 188-194. doi: 10.1016/j.orl.2009.12.004. Google Scholar

[17]

O. L. Mangasarian, Set containment characterization, J. Global Optim., 24 (2002), 473-480. doi: 10.1023/A:1021207718605. Google Scholar

[18]

R. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970. Google Scholar

[19]

S. Schaible, Parameter-free convex equivalent and dual programs of fractional programming problems, Z. Oper. Res., 18 (1974), 187-196. Google Scholar

[20]

S. Schaible, Fractional programming: A recent survey, J. Stat. Manag. Syst., 5 (2002), 63-86. doi: 10.1080/09720510.2002.10701051. Google Scholar

[21]

X. K. Sun and Y. Cai, On robust duality for fractional programming with uncertainty data, Positivity, 18 (2014), 9-28. doi: 10.1007/s11117-013-0227-7. Google Scholar

[22]

X. K. SunY. Cai and J. Zeng, Farkas-type results fro constraint fractional programming with DC functions, Optim. Lett., 8 (2014), 2299-2313. doi: 10.1007/s11590-014-0737-7. Google Scholar

[23]

X. K. SunZ. Y. Peng and X. L. Guo, Some characterizations of robust optimal solutions for uncertain convex optimization problems, Optim. Lett., 10 (2016), 1463-1478. doi: 10.1007/s11590-015-0946-8. Google Scholar

[24]

X. M. YangK. L. Teo and X. Q. Yang, Symmetric duality for a class of nonlinear fractional programming problems, J. Math. Anal. Appl., 271 (2002), 7-15. doi: 10.1016/S0022-247X(02)00042-2. Google Scholar

[25]

X. M. YangX. Q. Yang and K. L. Teo, Duality and saddle-point type optimality for generalized nonlinear fractional programming, J. Math. Anal. Appl., 289 (2004), 100-109. doi: 10.1016/j.jmaa.2003.08.029. Google Scholar

[26]

C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific, London, 2002. doi: 10.1142/9789812777096. Google Scholar

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