April  2017, 13(2): 803-824. doi: 10.3934/jimo.2016047

Some characterizations of robust optimal solutions for uncertain fractional optimization and applications

1. 

College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China

2. 

School of Management, Southwest University of Political Science and Law, Chongqing 401120, China

3. 

College of Mathematics and Statistics, Chongqing JiaoTong University, Chongqing 400074, China

* Corresponding author: Hong-Yong Fu

Received  August 2015 Revised  January 2016 Published  August 2016

Fund Project: This research was supported by the National Natural Science Foundation of China (11301570,11471059,71501162), the Basic and Advanced Research Project of CQ CSTC (cstc2015jcyjA00002, cstc2015jcyjB00001, cstc2015jcyjBX0131), the Education Committee Project Research Foundation of Chongqing (KJ1500626), the Southwest University of Political Science and Law (2014XZQN-17), and the China Postdoctoral Science Foundation (2015M580770).

In this paper, following the framework of robust optimization, we consider robust optimal solutions for a fractional optimization problem in the face of data uncertainty both in the objective and constraints. To this end, by using the properties of the subdifferential sum formulae, we first introduce some robust basic subdifferential constraint qualifications, and then obtain some completely characterizations of the robust optimal solutions of this uncertain fractional optimization problem. We show that our results encompass as special cases some optimization problems considered in the recent literature. Moreover, as applications, the proposed approach is applied to investigate weakly robust efficient solutions for multi-objective fractional optimization problems in the face of data uncertainty both in the objective and constraints.

Citation: 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
References:
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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

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

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D. BertsimasD. S. Brown and C. Caramanis, Theory and applications of robust optimization, SIAM Rev., 53 (2011), 464-501.  doi: 10.1137/080734510.  Google Scholar

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J. R. Birge and F. Louveaux, Introduction to Stochastic Programming, Springer, New York, 1997.  Google Scholar

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

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R. I. BoţS. 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

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R. I. BoţI. 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

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K. Deb and H. Gupta, Introducing robustness in multi-objective optimization, Evol. Comput., 14 (2006), 463-494.  doi: 10.1162/evco.2006.14.4.463.  Google Scholar

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B. L. Gorissen, Robust fractional programming, J. Optim. Theory Appl., 166 (2015), 508-528.  doi: 10.1007/s10957-014-0633-4.  Google Scholar

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X. L. Guo and S. J. Li, Optimality conditions for vector optimization problems with difference of convex maps, J. Optim. Theory Appl., 162 (2014), 821-844.  doi: 10.1007/s10957-013-0327-3.  Google Scholar

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A. JayswalA. K. Prasad and I. Ahmad, On minimax fractional programming problems involving generalized ($H_p$, r)-invex functions, J. Ind. Manag. Optim., 10 (2014), 1001-1018.  doi: 10.3934/jimo.2014.10.1001.  Google Scholar

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V. JeyakumarG. M. Lee and G. Y. Li, Characterizing robust solution sets of convex programs under data uncertainty, J. Optim. Theory Appl., 164 (2015), 407-435.  doi: 10.1007/s10957-014-0564-0.  Google Scholar

[13]

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

[14]

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

[15]

D. Kuroiwa and G. M. Lee, On robust convex multiobjective optimization, J. Nonlinear Convex Anal., 15 (2014), 1125-1136.   Google Scholar

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J. H. Lee and G. M. Lee, On $\varepsilon$-solutions for convex optimization problems with uncertainty data, Positivity, 16 (2012), 509-526.  doi: 10.1007/s11117-012-0186-4.  Google Scholar

[17]

Z. A. LiangH. X. Huang and P. M. Pardalos, Optimality conditions and duality for a class of nonlinear fractional programming problems, J. Optim. Theory Appl., 110 (2001), 611-619.  doi: 10.1023/A:1017540412396.  Google Scholar

[18]

J. Y. LinH. J. Chen and R. L. Sheu, Augmented Lagrange primal-dual approach for generalized fractional programming problems, J. Ind. Manag. Optim., 9 (2013), 723-741.  doi: 10.3934/jimo.2013.9.723.  Google Scholar

[19]

J. C. Liu and K. Yokoyama, $\varepsilon$-optimality and duality for fractional programming, Taiwan. J. Math., 3 (1999), 311-322.   Google Scholar

[20]

O. L. Mangasarian, Nonlinear Programming, McGraw-Hill Inc, New York, 1969.  Google Scholar

[21]

S. Schaible, Bibliography in fractional programming, Zeitschrift für Oper. Res., 26 (1982), 211-241.   Google Scholar

[22]

S. Schaible and T. Ibaraki, Fractional programming, Eur. J. Oper. Res., 12 (1983), 325-338.  doi: 10.1016/0377-2217(83)90153-4.  Google Scholar

[23]

A. Shapiro, D. Dentcheva and A. Ruszczynski, Lectures on Stochastic Programming: Modeling and Theory, SIAM, Philadelphia, 2009. doi: 10.1137/1.9780898718751.  Google Scholar

[24]

I. M. Stancu-Minasian, A seventh bibliography of fractional programming, Adv. Model. Optim., 15 (2013), 309-386.   Google Scholar

[25]

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

[26]

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

[27]

X. K. SunX. J. Long and Y. Chai, Sequential optimality conditions for fractional optimization with applications to vector optimization, J. Optim. Theory Appl., 164 (2015), 479-499.  doi: 10.1007/s10957-014-0578-7.  Google Scholar

[28]

H. J. Wang and C. Z. Cheng, Duality and Farkas-type results for DC fractional programming with DC constraints, Math. Comput. Modelling, 53 (2011), 1026-1034.  doi: 10.1016/j.mcm.2010.11.059.  Google Scholar

[29]

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

[30]

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

[31]

H. Yu and H. M. Liu, Robust multiple objective game theory, J. Optim. Theory Appl., 159 (2013), 272-280.  doi: 10.1007/s10957-012-0234-z.  Google Scholar

[32]

X. H. Zhang and C. Z. Cheng, Some Farkas-type results for fractional programming with DC functions, Nonlinear Anal. Real World Appl., 10 (2009), 1679-1690.  doi: 10.1016/j.nonrwa.2008.02.006.  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 Mathematics, 2009. doi: 10.1515/9781400831050.  Google Scholar

[3]

D. BertsimasD. S. Brown and C. Caramanis, Theory and applications of robust optimization, SIAM Rev., 53 (2011), 464-501.  doi: 10.1137/080734510.  Google Scholar

[4]

J. R. Birge and F. Louveaux, Introduction to Stochastic Programming, Springer, New York, 1997.  Google Scholar

[5]

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

[6]

R. I. BoţS. 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

[7]

R. I. BoţI. 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

[8]

K. Deb and H. Gupta, Introducing robustness in multi-objective optimization, Evol. Comput., 14 (2006), 463-494.  doi: 10.1162/evco.2006.14.4.463.  Google Scholar

[9]

B. L. Gorissen, Robust fractional programming, J. Optim. Theory Appl., 166 (2015), 508-528.  doi: 10.1007/s10957-014-0633-4.  Google Scholar

[10]

X. L. Guo and S. J. Li, Optimality conditions for vector optimization problems with difference of convex maps, J. Optim. Theory Appl., 162 (2014), 821-844.  doi: 10.1007/s10957-013-0327-3.  Google Scholar

[11]

A. JayswalA. K. Prasad and I. Ahmad, On minimax fractional programming problems involving generalized ($H_p$, r)-invex functions, J. Ind. Manag. Optim., 10 (2014), 1001-1018.  doi: 10.3934/jimo.2014.10.1001.  Google Scholar

[12]

V. JeyakumarG. M. Lee and G. Y. Li, Characterizing robust solution sets of convex programs under data uncertainty, J. Optim. Theory Appl., 164 (2015), 407-435.  doi: 10.1007/s10957-014-0564-0.  Google Scholar

[13]

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

[14]

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

[15]

D. Kuroiwa and G. M. Lee, On robust convex multiobjective optimization, J. Nonlinear Convex Anal., 15 (2014), 1125-1136.   Google Scholar

[16]

J. H. Lee and G. M. Lee, On $\varepsilon$-solutions for convex optimization problems with uncertainty data, Positivity, 16 (2012), 509-526.  doi: 10.1007/s11117-012-0186-4.  Google Scholar

[17]

Z. A. LiangH. X. Huang and P. M. Pardalos, Optimality conditions and duality for a class of nonlinear fractional programming problems, J. Optim. Theory Appl., 110 (2001), 611-619.  doi: 10.1023/A:1017540412396.  Google Scholar

[18]

J. Y. LinH. J. Chen and R. L. Sheu, Augmented Lagrange primal-dual approach for generalized fractional programming problems, J. Ind. Manag. Optim., 9 (2013), 723-741.  doi: 10.3934/jimo.2013.9.723.  Google Scholar

[19]

J. C. Liu and K. Yokoyama, $\varepsilon$-optimality and duality for fractional programming, Taiwan. J. Math., 3 (1999), 311-322.   Google Scholar

[20]

O. L. Mangasarian, Nonlinear Programming, McGraw-Hill Inc, New York, 1969.  Google Scholar

[21]

S. Schaible, Bibliography in fractional programming, Zeitschrift für Oper. Res., 26 (1982), 211-241.   Google Scholar

[22]

S. Schaible and T. Ibaraki, Fractional programming, Eur. J. Oper. Res., 12 (1983), 325-338.  doi: 10.1016/0377-2217(83)90153-4.  Google Scholar

[23]

A. Shapiro, D. Dentcheva and A. Ruszczynski, Lectures on Stochastic Programming: Modeling and Theory, SIAM, Philadelphia, 2009. doi: 10.1137/1.9780898718751.  Google Scholar

[24]

I. M. Stancu-Minasian, A seventh bibliography of fractional programming, Adv. Model. Optim., 15 (2013), 309-386.   Google Scholar

[25]

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

[26]

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

[27]

X. K. SunX. J. Long and Y. Chai, Sequential optimality conditions for fractional optimization with applications to vector optimization, J. Optim. Theory Appl., 164 (2015), 479-499.  doi: 10.1007/s10957-014-0578-7.  Google Scholar

[28]

H. J. Wang and C. Z. Cheng, Duality and Farkas-type results for DC fractional programming with DC constraints, Math. Comput. Modelling, 53 (2011), 1026-1034.  doi: 10.1016/j.mcm.2010.11.059.  Google Scholar

[29]

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

[30]

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

[31]

H. Yu and H. M. Liu, Robust multiple objective game theory, J. Optim. Theory Appl., 159 (2013), 272-280.  doi: 10.1007/s10957-012-0234-z.  Google Scholar

[32]

X. H. Zhang and C. Z. Cheng, Some Farkas-type results for fractional programming with DC functions, Nonlinear Anal. Real World Appl., 10 (2009), 1679-1690.  doi: 10.1016/j.nonrwa.2008.02.006.  Google Scholar

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