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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 and 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.

[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.

[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.

[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.

[5]

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

[6]

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

[7]

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

[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.

[9]

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

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[5]

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

[6]

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

[7]

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

[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.

[9]

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

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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