# American Institute of Mathematical Sciences

January  2013, 9(1): 131-142. doi: 10.3934/jimo.2013.9.131

## Optimality conditions and duality in nondifferentiable interval-valued programming

 1 School of Mathematics and Physics, University of Science and Technology, Beijing, 100083, China 2 College of Science, China Agricultural University, 100083, China

Received  March 2012 Revised  May 2012 Published  December 2012

In this paper, we define the concepts of $LU$ optimal solution to interval-valued programming problem. By considering the concepts of $LU$ optimal solution, the Fritz John type and Kuhn-Tucker type necessary and sufficient optimality conditions for nondifferentiable interval programming are derived. Further, we establish the dual problem and prove the duality theorems.
Citation: 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
##### References:
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##### References:
 [1] T. Fu. Liang and H. Wen. Cheng, Multi-objective aggregate production planning decisions using two-phase fuzzy goal programming method, Journal of Industrial and Management Optimization, 7 (2011), 365-383.  Google Scholar [2] J. Ramík, Duality in fuzzy linear programming: some new concepts and results, Fuzzy Optimization and Decision Making, 4 (2005), 25-39.  Google Scholar [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), 206-216. doi: 10.1016/j.amc.2005.11.161.  Google Scholar [4] J. Ramík, Optimal solutions in optimization problem with objective function depending on fuzzy parameters, Fuzzy Sets and Systems, 158 (2007), 1873-1881. doi: 10.1016/j.fss.2007.04.003.  Google Scholar [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), 1709-1730.  Google Scholar [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), 179-198. doi: 10.1007/s10700-007-9014-x.  Google Scholar [7] H. C. Wu, Duality theorems in fuzzy mathematical programming problems based on the concept of necessity, Fuzzy Sets and Systems, 139 (2003), 363-377. doi: 10.1016/S0165-0114(02)00575-4.  Google Scholar [8] H. C. Wu, Duality theory in fuzzy optimization problems, Fuzzy Optimization and Decision Making, 3 (2004), 345-365.  Google Scholar [9] Z. T. Gong and H. X. Li, Saddle point optimality conditions in fuzzy optimization problems, Fuzzy Information and Engineering, AISC, 2 (2009), 7-14. Google Scholar [10] H. C. Wu, The optimality conditions for optimization problems with convex constraints and multiple fuzzy-valued objective functions, Fuzzy Optimization and Decision Making, 8 (2009), 295-321.  Google Scholar [11] D. Dentcheva and W. Römisch , Duality gaps in nonconvex stochastic optimization, Mathematical Programming, Ser. A, 101 (2004), 515-535.  Google Scholar [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), 241-259.  Google Scholar [13] D. Dentcheva and A.Ruszczyński, Optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints, Mathematical Programming, Ser. A, 99 (2004), 329-350.  Google Scholar [14] A. Shapiro, Stochastic programming approach to optimization under uncertainty, Mathematical Programming, Ser. B, 112 (2008), 183-220.  Google Scholar [15] H. Zhu, Convex duality for finite-fuel problems in singular stochastic control, Journal of Optimization Theory and Applications, 75 (1992), 154-181.  Google Scholar [16] H. C. Wu, Duality theory for optimization problems with interval-valued objective functions, Journal of Optimization Theory and Applications, 144 (2010), 615-628.  Google Scholar [17] H. C. Wu, On interval-valued nonlinear programming problems, Journal of Mathematical Analysis and Application, 338 (2008), 299-316.  Google Scholar [18] H. C. Wu, Wolfe duality for interval-valued optimization, Journal of Optimization Theory and Applications, 138 (2008), 497-509.  Google Scholar [19] H. C. Zhou and Y. J. Wang, Optimality condition and mixed duality for interval-valued optimization. Fuzzy Information and Engineering, AISC, 62 (2009), 1315-1323. Google Scholar [20] H. C. Wu, The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective function, European Journal of Operational Research, 176 (2007), 46-59.  Google Scholar [21] H. C. Wu, The Karush-Kuhn-Tucker optimality conditions in multiobjective programming problems with interval-valued objective functions, European Journal of Operational Research, 196 (2009), 49-60.  Google Scholar [22] R. E. Moore, "Method and Applications of Interval Analysis," SIAM, Philadelphia, 1979.  Google Scholar [23] A. Prékopa, "Stochastic Programming: Mathematics and Its Applications," Kluwer Academic Publishers Group, Dordrecht, 1995.  Google Scholar [24] M. Schechter, More on subgradient duality, Journal of Mathematical Analysis and Application, 71 (1979), 251-262.  Google Scholar
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