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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 2019 Early access  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 and 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.

[2]

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

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[9]

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

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[17]

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

[18]

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

[19]

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

[20]

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

[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.

[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.

[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.

[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.

[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.

[26]

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

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.

[2]

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

[3]

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

[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.

[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.

[6]

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

[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.

[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.

[9]

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

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[17]

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

[18]

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

[19]

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

[20]

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

[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.

[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.

[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.

[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.

[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.

[26]

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

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