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Optimality conditions and duality in nondifferentiable intervalvalued programming
1.  School of Mathematics and Physics, University of Science and Technology, Beijing, 100083, China 
2.  College of Science, China Agricultural University, 100083, China 
References:
[1] 
T. Fu. Liang and H. Wen. Cheng, Multiobjective aggregate production planning decisions using twophase fuzzy goal programming method, Journal of Industrial and Management Optimization, 7 (2011), 365383. 
[2] 
J. Ramík, Duality in fuzzy linear programming: some new concepts and results, Fuzzy Optimization and Decision Making, 4 (2005), 2539. 
[3] 
N. M. Amiri and S. H. Nasseri, Duality in fuzzy number linear programming by use of a certain linear ranking function, Applied Mathematics and Computation, 180 (2006), 206216. doi: 10.1016/j.amc.2005.11.161. 
[4] 
J. Ramík, Optimal solutions in optimization problem with objective function depending on fuzzy parameters, Fuzzy Sets and Systems, 158 (2007), 18731881. doi: 10.1016/j.fss.2007.04.003. 
[5] 
C. Zhang, X. Yuan and E. S. Lee, Duality theory in fuzzy mathematical programming problems with fuzzy coefficients, Computers and Mathematics with Applications, 49 (2005), 17091730. 
[6] 
H. C. Wu, Duality theory in fuzzy optimization problems formulated by the Wolfe's primal and dual pair, Fuzzy Optimization and Decision Making, 6 (2007), 179198. doi: 10.1007/s107000079014x. 
[7] 
H. C. Wu, Duality theorems in fuzzy mathematical programming problems based on the concept of necessity, Fuzzy Sets and Systems, 139 (2003), 363377. doi: 10.1016/S01650114(02)005754. 
[8] 
H. C. Wu, Duality theory in fuzzy optimization problems, Fuzzy Optimization and Decision Making, 3 (2004), 345365. 
[9] 
Z. T. Gong and H. X. Li, Saddle point optimality conditions in fuzzy optimization problems, Fuzzy Information and Engineering, AISC, 2 (2009), 714. 
[10] 
H. C. Wu, The optimality conditions for optimization problems with convex constraints and multiple fuzzyvalued objective functions, Fuzzy Optimization and Decision Making, 8 (2009), 295321. 
[11] 
D. Dentcheva and W. Römisch , Duality gaps in nonconvex stochastic optimization, Mathematical Programming, Ser. A, 101 (2004), 515535. 
[12] 
L. A. Korf, Stochastic programming duality: $L^{\propto}$ multipliers for unbounded constraints with an application to mathematical finance, Mathematical Programming, Ser. A, 99 (2004), 241259. 
[13] 
D. Dentcheva and A.Ruszczyński, Optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints, Mathematical Programming, Ser. A, 99 (2004), 329350. 
[14] 
A. Shapiro, Stochastic programming approach to optimization under uncertainty, Mathematical Programming, Ser. B, 112 (2008), 183220. 
[15] 
H. Zhu, Convex duality for finitefuel problems in singular stochastic control, Journal of Optimization Theory and Applications, 75 (1992), 154181. 
[16] 
H. C. Wu, Duality theory for optimization problems with intervalvalued objective functions, Journal of Optimization Theory and Applications, 144 (2010), 615628. 
[17] 
H. C. Wu, On intervalvalued nonlinear programming problems, Journal of Mathematical Analysis and Application, 338 (2008), 299316. 
[18] 
H. C. Wu, Wolfe duality for intervalvalued optimization, Journal of Optimization Theory and Applications, 138 (2008), 497509. 
[19] 
H. C. Zhou and Y. J. Wang, Optimality condition and mixed duality for intervalvalued optimization. Fuzzy Information and Engineering, AISC, 62 (2009), 13151323. 
[20] 
H. C. Wu, The KarushKuhnTucker optimality conditions in an optimization problem with intervalvalued objective function, European Journal of Operational Research, 176 (2007), 4659. 
[21] 
H. C. Wu, The KarushKuhnTucker optimality conditions in multiobjective programming problems with intervalvalued objective functions, European Journal of Operational Research, 196 (2009), 4960. 
[22] 
R. E. Moore, "Method and Applications of Interval Analysis," SIAM, Philadelphia, 1979. 
[23] 
A. Prékopa, "Stochastic Programming: Mathematics and Its Applications," Kluwer Academic Publishers Group, Dordrecht, 1995. 
[24] 
M. Schechter, More on subgradient duality, Journal of Mathematical Analysis and Application, 71 (1979), 251262. 
show all references
References:
[1] 
T. Fu. Liang and H. Wen. Cheng, Multiobjective aggregate production planning decisions using twophase fuzzy goal programming method, Journal of Industrial and Management Optimization, 7 (2011), 365383. 
[2] 
J. Ramík, Duality in fuzzy linear programming: some new concepts and results, Fuzzy Optimization and Decision Making, 4 (2005), 2539. 
[3] 
N. M. Amiri and S. H. Nasseri, Duality in fuzzy number linear programming by use of a certain linear ranking function, Applied Mathematics and Computation, 180 (2006), 206216. doi: 10.1016/j.amc.2005.11.161. 
[4] 
J. Ramík, Optimal solutions in optimization problem with objective function depending on fuzzy parameters, Fuzzy Sets and Systems, 158 (2007), 18731881. doi: 10.1016/j.fss.2007.04.003. 
[5] 
C. Zhang, X. Yuan and E. S. Lee, Duality theory in fuzzy mathematical programming problems with fuzzy coefficients, Computers and Mathematics with Applications, 49 (2005), 17091730. 
[6] 
H. C. Wu, Duality theory in fuzzy optimization problems formulated by the Wolfe's primal and dual pair, Fuzzy Optimization and Decision Making, 6 (2007), 179198. doi: 10.1007/s107000079014x. 
[7] 
H. C. Wu, Duality theorems in fuzzy mathematical programming problems based on the concept of necessity, Fuzzy Sets and Systems, 139 (2003), 363377. doi: 10.1016/S01650114(02)005754. 
[8] 
H. C. Wu, Duality theory in fuzzy optimization problems, Fuzzy Optimization and Decision Making, 3 (2004), 345365. 
[9] 
Z. T. Gong and H. X. Li, Saddle point optimality conditions in fuzzy optimization problems, Fuzzy Information and Engineering, AISC, 2 (2009), 714. 
[10] 
H. C. Wu, The optimality conditions for optimization problems with convex constraints and multiple fuzzyvalued objective functions, Fuzzy Optimization and Decision Making, 8 (2009), 295321. 
[11] 
D. Dentcheva and W. Römisch , Duality gaps in nonconvex stochastic optimization, Mathematical Programming, Ser. A, 101 (2004), 515535. 
[12] 
L. A. Korf, Stochastic programming duality: $L^{\propto}$ multipliers for unbounded constraints with an application to mathematical finance, Mathematical Programming, Ser. A, 99 (2004), 241259. 
[13] 
D. Dentcheva and A.Ruszczyński, Optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints, Mathematical Programming, Ser. A, 99 (2004), 329350. 
[14] 
A. Shapiro, Stochastic programming approach to optimization under uncertainty, Mathematical Programming, Ser. B, 112 (2008), 183220. 
[15] 
H. Zhu, Convex duality for finitefuel problems in singular stochastic control, Journal of Optimization Theory and Applications, 75 (1992), 154181. 
[16] 
H. C. Wu, Duality theory for optimization problems with intervalvalued objective functions, Journal of Optimization Theory and Applications, 144 (2010), 615628. 
[17] 
H. C. Wu, On intervalvalued nonlinear programming problems, Journal of Mathematical Analysis and Application, 338 (2008), 299316. 
[18] 
H. C. Wu, Wolfe duality for intervalvalued optimization, Journal of Optimization Theory and Applications, 138 (2008), 497509. 
[19] 
H. C. Zhou and Y. J. Wang, Optimality condition and mixed duality for intervalvalued optimization. Fuzzy Information and Engineering, AISC, 62 (2009), 13151323. 
[20] 
H. C. Wu, The KarushKuhnTucker optimality conditions in an optimization problem with intervalvalued objective function, European Journal of Operational Research, 176 (2007), 4659. 
[21] 
H. C. Wu, The KarushKuhnTucker optimality conditions in multiobjective programming problems with intervalvalued objective functions, European Journal of Operational Research, 196 (2009), 4960. 
[22] 
R. E. Moore, "Method and Applications of Interval Analysis," SIAM, Philadelphia, 1979. 
[23] 
A. Prékopa, "Stochastic Programming: Mathematics and Its Applications," Kluwer Academic Publishers Group, Dordrecht, 1995. 
[24] 
M. Schechter, More on subgradient duality, Journal of Mathematical Analysis and Application, 71 (1979), 251262. 
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