• Previous Article
    The optimal pricing and ordering policy for temperature sensitive products considering the effects of temperature on demand
  • JIMO Home
  • This Issue
  • Next Article
    Single-machine bi-criterion scheduling with release times and exponentially time-dependent learning effects
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

[1]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[2]

Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023

[3]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088

[4]

Habib Ammari, Josselin Garnier, Vincent Jugnon. Detection, reconstruction, and characterization algorithms from noisy data in multistatic wave imaging. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 389-417. doi: 10.3934/dcdss.2015.8.389

[5]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[6]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

[7]

Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113

[8]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233

[9]

Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397

[10]

Mansour Shrahili, Ravi Shanker Dubey, Ahmed Shafay. Inclusion of fading memory to Banister model of changes in physical condition. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 881-888. doi: 10.3934/dcdss.2020051

[11]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223

[12]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[13]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311

[14]

Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024

[15]

Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, Fernanda Pambianco. Linear nonbinary covering codes and saturating sets in projective spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 119-147. doi: 10.3934/amc.2011.5.119

[16]

W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349

[17]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[18]

Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81

[19]

Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055

[20]

Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (272)
  • HTML views (1122)
  • Cited by (1)

Other articles
by authors

[Back to Top]