American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems
  • JIMO Home
  • This Issue
  • Next Article
    Optimal production schedule in a single-supplier multi-manufacturer supply chain involving time delays in both levels
July  2018, 14(3): 895-912. doi: 10.3934/jimo.2017081

On optimality conditions and duality for non-differentiable interval-valued programming problems with the generalized (F, ρ)-convexity

Xiuhong Chen 1,, and  Zhihua Li 2,

1. 

School of Digital Media, Jiangnan University, Wuxi 214122, Jiangsu, China
2. 

School of Internet of Things, Jiangnan University, Wuxi 214122, Jiangsu, China

* Corresponding author: Xiuhong Chen

Received  August 2017 Published  July 2018 Early access  September 2017

Fund Project: The first author is supported in part by the National Natural Foundation of China under grant:61373055 and Jiangsu Key Laboratory of Media Design and Software Technology(Jiangnan University).

Because interval-valued programming problem is used to tackle interval uncertainty that appears in many mathematical or computer models of some deterministic real-world phenomena, this paper considers a non-differentiable interval-valued optimization problem in which objective and all constraint functions are interval-valued functions, and the involved endpoint functions in interval-valued functions are locally Lipschitz and Clarke sub-differentiable. A necessary optimality condition is first established. Some sufficient optimality conditions of the considered problem are derived for a feasible solution to be an efficient solution under the $G-(F, ρ)$ convexity assumption. Weak, strong, and converse duality theorems for Wolfe and Mond-Weir type duals are also obtained in order to relate the efficient solution of primal and dual inter-valued programs.

Keywords: Interval-valued function, Clarke sub-differentiability, G-(F, ρ) convexity, optimality condition, duality.
Mathematics Subject Classification: Primary: 26A27, 26A51, 65G40, 90C30, 90C46; Secondary: 49N15, 90C26, 90C29.
Citation: Xiuhong Chen, Zhihua Li. On optimality conditions and duality for non-differentiable interval-valued programming problems with the generalized (F, ρ)-convexity. Journal of Industrial & Management Optimization, 2018, 14 (3) : 895-912. doi: 10.3934/jimo.2017081
References:
[1]

A. K. Bhurjee and S. K. Padhan, Optimality conditions and duality results for non-differentiable interval optimization problems, J. Appl. Math. Comput., 50 (2016), 59-71.  doi: 10.1007/s12190-014-0858-2.  Google Scholar

[2]

Y. Chalco-Cano, W. A. Lodwick and A. Rufian-Lizana, Optimality conditions of type KKT for optimization problem with interval-valued object function via generalized derivative, Fuzzy Optim. Decis. Making, 12 (2013), 305-322.  doi: 10.1007/s10700-013-9156-y.  Google Scholar

[3]

S. Chanas and D. Kuchta, Multiobjective programming in optimization of interval objective functions-A generalized approach, Eur. J. Oper. Res., 94 (1996), 594-598.  doi: 10.1016/0377-2217(95)00055-0.  Google Scholar

[4]

X. H. Chen, Optimality and duality for the multiobjective fractional programming with the generalized $(F, ρ)$ convexity, J. Math. Anal. Appl., 273 (2002), 190-205.  doi: 10.1016/S0022-247X(02)00248-2.  Google Scholar

[5]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, 1983.  Google Scholar

[6]

B. D. Craven, Invex functions and constrained local minima, Bull. Austral. Math. Soc., 24 (1981), 357-366.  doi: 10.1017/S0004972700004895.  Google Scholar

[7]

I. V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, Springer-Verlag, New York, 1972.  Google Scholar

[8]

M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl., 80 (1981), 545-550.  doi: 10.1016/0022-247X(81)90123-2.  Google Scholar

[9]

M. Hukuhara, Integration des applications mesurables dont la valeur est un compact convexe, Funkcialaj Ekvacioj, 10 (1967), 205-223.   Google Scholar

[10]

H. Ishibuchi and H. Tanaka, Multiobjective programming in optimization of the interval objective function, Eur. J. Oper. Res., 48 (1990), 219-225.  doi: 10.1016/0377-2217(90)90375-L.  Google Scholar

[11]

A. Jayswal, I. Ahmad and J. Banerjee, Nonsmooth interval-valued optimization and saddle-point optimality criteria, Bull. Malays. Math. Sci. Soc., 39 (2016), 1391-1411.  doi: 10.1007/s40840-015-0237-7.  Google Scholar

[12]

A. Jayswal, I. Stancu-Minasian and I. Ahmad, On sufficiency and duality for a class of interval-valued programming problems, Appl. Math. Comput., 218 (2011), 4119-4127.  doi: 10.1016/j.amc.2011.09.041.  Google Scholar

[13]

A. Jayswal, I. Stancu-Minasian, J. Banerjee and A. M. Stancu, Sufficiency and duality for optimization problems involving interval-valued invex functions in parametric form, An. Inter. J. Oper. Res., 15 (2015), 137-161.  doi: 10.1007/s12351-015-0172-2.  Google Scholar

[14]

A. Jayswal, I. Stancu-Minasian and J. Banerjee, On interval-valued programming problem with invex functions, J. Nonlin. and Conv. Anal., 17 (2016), 549-567.   Google Scholar

[15]

C. Jiang, X. Han, G. R. Liu and G. P. Liu, A nonlinear interval number programming method for uncertain optimization problems, Eur. J. Oper. Res., 188 (2008), 1-13.  doi: 10.1016/j.ejor.2007.03.031.  Google Scholar

[16]

C. Li, G. Zhang, J. Yi and M. Wang, Uncertainty degree and modeling of interval type-2 fuzzy sets: definition, method and application, Comput. Math. Appl., 66 (2013), 1822-1835.  doi: 10.1016/j.camwa.2013.07.021.  Google Scholar

[17]

F. Mráz, Calculating the exact bounds of optimal values in LP with interval coefficients, Ann. Oper. Res., 81 (1998), 51-62.  doi: 10.1023/A:1018985914065.  Google Scholar

[18]

J. A. Sanz, M. Galar, A. Jurio, A. Brugos, M. Pagola and H. Bustince, Medical diagnosis of cardiovascular diseases using an interval-valued fuzzy rule-based classification system, Appl. Soft Comput., 20 (2014), 103-111.  doi: 10.1016/j.asoc.2013.11.009.  Google Scholar

[19]

M. Schechter, More on subgradient duality, J. Math. Anal. Appl., 71 (1979), 251-262.  doi: 10.1016/0022-247X(79)90228-2.  Google Scholar

[20]

A. Sengupta and T. K. Pal, On comparing interval numbers, Eur. J. Oper. Res., 127 (2000), 28-43.  doi: 10.1016/S0377-2217(99)00319-7.  Google Scholar

[21]

A. L. Soyster, Inexact linear programming with generalized resource sets, Eur. J. Oper. Res., 3 (1979), 316-321.  doi: 10.1016/0377-2217(79)90227-3.  Google Scholar

[22]

A. M. Stancu, Mathematical Programming with Type-Ⅰ Functions, Matrix Romania, Bucharest, 2013. Google Scholar

[23]

R. E. Steuer, Algorithms for linear programming problems with interval objective function coefficients, Math. Oper. Res., 6 (1981), 333-348.  doi: 10.1287/moor.6.3.333.  Google Scholar

[24]

Y. Sun and L. Wang, Optimality conditions and duality in nondifferentiable interval-valued programming, J. Ind. Manag. Optim., 9 (2013), 131-142.  doi: 10.3934/jimo.2013.9.131.  Google Scholar

[25]

B. Urli and R. Nadeau, PROMISE/scenarios: An interactive method for multiobjective stochastic linear programming under partial uncertainty, Eur. J. Oper. Res., 155 (2004), 361-372.  doi: 10.1016/S0377-2217(02)00859-7.  Google Scholar

[26]

H. C. Wu, The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective function, Eur. J. Oper. Res, 176 (2007), 46-59.  doi: 10.1016/j.ejor.2005.09.007.  Google Scholar

[27]

H. C. Wu, The Karush-Kuhn-Tucker optimality conditions in multiobjective programming problems with intervalued objective functions, Eur. J. Oper. Res., 196 (2009), 49-60.  doi: 10.1016/j.ejor.2008.03.012.  Google Scholar

[28]

H. C. Wu, On interval-valued nonlinear programming problems, J. Math. Anal. Appl., 338 (2008), 299-316.  doi: 10.1016/j.jmaa.2007.05.023.  Google Scholar

[29]

H. C. Wu, Wolfe duality for interval-valued optimization, J. Optim. Theory Appl., 138 (2008), 497-509.  doi: 10.1007/s10957-008-9396-0.  Google Scholar

show all references

References:
[1]

A. K. Bhurjee and S. K. Padhan, Optimality conditions and duality results for non-differentiable interval optimization problems, J. Appl. Math. Comput., 50 (2016), 59-71.  doi: 10.1007/s12190-014-0858-2.  Google Scholar

[2]

Y. Chalco-Cano, W. A. Lodwick and A. Rufian-Lizana, Optimality conditions of type KKT for optimization problem with interval-valued object function via generalized derivative, Fuzzy Optim. Decis. Making, 12 (2013), 305-322.  doi: 10.1007/s10700-013-9156-y.  Google Scholar

[3]

S. Chanas and D. Kuchta, Multiobjective programming in optimization of interval objective functions-A generalized approach, Eur. J. Oper. Res., 94 (1996), 594-598.  doi: 10.1016/0377-2217(95)00055-0.  Google Scholar

[4]

X. H. Chen, Optimality and duality for the multiobjective fractional programming with the generalized $(F, ρ)$ convexity, J. Math. Anal. Appl., 273 (2002), 190-205.  doi: 10.1016/S0022-247X(02)00248-2.  Google Scholar

[5]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, 1983.  Google Scholar

[6]

B. D. Craven, Invex functions and constrained local minima, Bull. Austral. Math. Soc., 24 (1981), 357-366.  doi: 10.1017/S0004972700004895.  Google Scholar

[7]

I. V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, Springer-Verlag, New York, 1972.  Google Scholar

[8]

M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl., 80 (1981), 545-550.  doi: 10.1016/0022-247X(81)90123-2.  Google Scholar

[9]

M. Hukuhara, Integration des applications mesurables dont la valeur est un compact convexe, Funkcialaj Ekvacioj, 10 (1967), 205-223.   Google Scholar

[10]

H. Ishibuchi and H. Tanaka, Multiobjective programming in optimization of the interval objective function, Eur. J. Oper. Res., 48 (1990), 219-225.  doi: 10.1016/0377-2217(90)90375-L.  Google Scholar

[11]

A. Jayswal, I. Ahmad and J. Banerjee, Nonsmooth interval-valued optimization and saddle-point optimality criteria, Bull. Malays. Math. Sci. Soc., 39 (2016), 1391-1411.  doi: 10.1007/s40840-015-0237-7.  Google Scholar

[12]

A. Jayswal, I. Stancu-Minasian and I. Ahmad, On sufficiency and duality for a class of interval-valued programming problems, Appl. Math. Comput., 218 (2011), 4119-4127.  doi: 10.1016/j.amc.2011.09.041.  Google Scholar

[13]

A. Jayswal, I. Stancu-Minasian, J. Banerjee and A. M. Stancu, Sufficiency and duality for optimization problems involving interval-valued invex functions in parametric form, An. Inter. J. Oper. Res., 15 (2015), 137-161.  doi: 10.1007/s12351-015-0172-2.  Google Scholar

[14]

A. Jayswal, I. Stancu-Minasian and J. Banerjee, On interval-valued programming problem with invex functions, J. Nonlin. and Conv. Anal., 17 (2016), 549-567.   Google Scholar

[15]

C. Jiang, X. Han, G. R. Liu and G. P. Liu, A nonlinear interval number programming method for uncertain optimization problems, Eur. J. Oper. Res., 188 (2008), 1-13.  doi: 10.1016/j.ejor.2007.03.031.  Google Scholar

[16]

C. Li, G. Zhang, J. Yi and M. Wang, Uncertainty degree and modeling of interval type-2 fuzzy sets: definition, method and application, Comput. Math. Appl., 66 (2013), 1822-1835.  doi: 10.1016/j.camwa.2013.07.021.  Google Scholar

[17]

F. Mráz, Calculating the exact bounds of optimal values in LP with interval coefficients, Ann. Oper. Res., 81 (1998), 51-62.  doi: 10.1023/A:1018985914065.  Google Scholar

[18]

J. A. Sanz, M. Galar, A. Jurio, A. Brugos, M. Pagola and H. Bustince, Medical diagnosis of cardiovascular diseases using an interval-valued fuzzy rule-based classification system, Appl. Soft Comput., 20 (2014), 103-111.  doi: 10.1016/j.asoc.2013.11.009.  Google Scholar

[19]

M. Schechter, More on subgradient duality, J. Math. Anal. Appl., 71 (1979), 251-262.  doi: 10.1016/0022-247X(79)90228-2.  Google Scholar

[20]

A. Sengupta and T. K. Pal, On comparing interval numbers, Eur. J. Oper. Res., 127 (2000), 28-43.  doi: 10.1016/S0377-2217(99)00319-7.  Google Scholar

[21]

A. L. Soyster, Inexact linear programming with generalized resource sets, Eur. J. Oper. Res., 3 (1979), 316-321.  doi: 10.1016/0377-2217(79)90227-3.  Google Scholar

[22]

A. M. Stancu, Mathematical Programming with Type-Ⅰ Functions, Matrix Romania, Bucharest, 2013. Google Scholar

[23]

R. E. Steuer, Algorithms for linear programming problems with interval objective function coefficients, Math. Oper. Res., 6 (1981), 333-348.  doi: 10.1287/moor.6.3.333.  Google Scholar

[24]

Y. Sun and L. Wang, Optimality conditions and duality in nondifferentiable interval-valued programming, J. Ind. Manag. Optim., 9 (2013), 131-142.  doi: 10.3934/jimo.2013.9.131.  Google Scholar

[25]

B. Urli and R. Nadeau, PROMISE/scenarios: An interactive method for multiobjective stochastic linear programming under partial uncertainty, Eur. J. Oper. Res., 155 (2004), 361-372.  doi: 10.1016/S0377-2217(02)00859-7.  Google Scholar

[26]

H. C. Wu, The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective function, Eur. J. Oper. Res, 176 (2007), 46-59.  doi: 10.1016/j.ejor.2005.09.007.  Google Scholar

[27]

H. C. Wu, The Karush-Kuhn-Tucker optimality conditions in multiobjective programming problems with intervalued objective functions, Eur. J. Oper. Res., 196 (2009), 49-60.  doi: 10.1016/j.ejor.2008.03.012.  Google Scholar

[28]

H. C. Wu, On interval-valued nonlinear programming problems, J. Math. Anal. Appl., 338 (2008), 299-316.  doi: 10.1016/j.jmaa.2007.05.023.  Google Scholar

[29]

H. C. Wu, Wolfe duality for interval-valued optimization, J. Optim. Theory Appl., 138 (2008), 497-509.  doi: 10.1007/s10957-008-9396-0.  Google Scholar

[1]

Yuhua Sun, Laisheng Wang. Optimality conditions and duality in nondifferentiable interval-valued programming. Journal of Industrial & Management Optimization, 2013, 9 (1) : 131-142. doi: 10.3934/jimo.2013.9.131
[2]

Tadeusz Antczak. The $ F $-objective function method for differentiable interval-valued vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2761-2782. doi: 10.3934/jimo.2020093
[3]

Tadeusz Antczak, Najeeb Abdulaleem. Optimality conditions for $ E $-differentiable vector optimization problems with the multiple interval-valued objective function. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2971-2989. doi: 10.3934/jimo.2019089
[4]

Deng-Feng Li, Yin-Fang Ye, Wei Fei. Extension of generalized solidarity values to interval-valued cooperative games. Journal of Industrial & Management Optimization, 2020, 16 (2) : 919-931. doi: 10.3934/jimo.2018185
[5]

Jiang-Xia Nan, Deng-Feng Li. Linear programming technique for solving interval-valued constraint matrix games. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1059-1070. doi: 10.3934/jimo.2014.10.1059
[6]

Hsien-Chung Wu. Solving the interval-valued optimization problems based on the concept of null set. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1157-1178. doi: 10.3934/jimo.2018004
[7]

Deepak Singh, Bilal Ahmad Dar, Do Sang Kim. Sufficiency and duality in non-smooth interval valued programming problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 647-665. doi: 10.3934/jimo.2018063
[8]

Harish Garg, Kamal Kumar. Group decision making approach based on possibility degree measure under linguistic interval-valued intuitionistic fuzzy set environment. Journal of Industrial & Management Optimization, 2020, 16 (1) : 445-467. doi: 10.3934/jimo.2018162
[9]

Harish Garg. Some robust improved geometric aggregation operators under interval-valued intuitionistic fuzzy environment for multi-criteria decision-making process. Journal of Industrial & Management Optimization, 2018, 14 (1) : 283-308. doi: 10.3934/jimo.2017047
[10]

Jianshe Yu, Honghua Bin, Zhiming Guo. Periodic solutions for discrete convex Hamiltonian systems via Clarke duality. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 939-950. doi: 10.3934/dcds.2006.15.939
[11]

Ruonan Liu, Run Xu. Hermite-Hadamard type inequalities for harmonical $ (h1,h2)- $convex interval-valued functions. Mathematical Foundations of Computing, 2021, 4 (2) : 89-103. doi: 10.3934/mfc.2021005
[12]

Baojun Bian, Pengfei Guan. A structural condition for microscopic convexity principle. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 789-807. doi: 10.3934/dcds.2010.28.789
[13]

Alexandre Rocha, Mário Jorge Dias Carneiro. A dynamical condition for differentiability of Mather's average action. Journal of Geometric Mechanics, 2014, 6 (4) : 549-566. doi: 10.3934/jgm.2014.6.549
[14]

Tone-Yau Huang, Tamaki Tanaka. Optimality and duality for complex multi-objective programming. Numerical Algebra, Control & Optimization, 2022, 12 (1) : 121-134. doi: 10.3934/naco.2021055
[15]

Geng-Hua Li, Sheng-Jie Li. Unified optimality conditions for set-valued optimizations. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1101-1116. doi: 10.3934/jimo.2018087
[16]

Lijia Yan. Some properties of a class of $(F,E)$-$G$ generalized convex functions. Numerical Algebra, Control & Optimization, 2013, 3 (4) : 615-625. doi: 10.3934/naco.2013.3.615
[17]

Yong-Kui Chang, Xiaojing Liu. Time-varying integro-differential inclusions with Clarke sub-differential and non-local initial conditions: existence and approximate controllability. Evolution Equations & Control Theory, 2020, 9 (3) : 845-863. doi: 10.3934/eect.2020036
[18]

Anca Croitoru, Alina GavriluŢ, Alina Iosif, Anna Rita Sambucini. A note on convergence results for varying interval valued multisubmeasures. Mathematical Foundations of Computing, 2021, 4 (4) : 299-310. doi: 10.3934/mfc.2021020
[19]

Xian-Jun Long, Nan-Jing Huang, Zhi-Bin Liu. Optimality conditions, duality and saddle points for nondifferentiable multiobjective fractional programs. Journal of Industrial & Management Optimization, 2008, 4 (2) : 287-298. doi: 10.3934/jimo.2008.4.287
[20]

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

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (196)
  • HTML views (885)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy