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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

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  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.

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-CanoW. 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

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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

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

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

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

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

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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

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A. JayswalI. 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

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A. JayswalI. 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. JayswalI. Stancu-MinasianJ. 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. JayswalI. 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. JiangX. HanG. 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. LiG. ZhangJ. 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

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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. SanzM. GalarA. JurioA. BrugosM. 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

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

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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-CanoW. 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. JayswalI. 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. JayswalI. 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. JayswalI. Stancu-MinasianJ. 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. JayswalI. 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. JiangX. HanG. 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. LiG. ZhangJ. 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. SanzM. GalarA. JurioA. BrugosM. 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

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