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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 |
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.
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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