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.   Google Scholar [2] J. Ramík, Duality in fuzzy linear programming: some new concepts and results,, Fuzzy Optimization and Decision Making, 4 (2005), 25.   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.  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.  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.   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.  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.  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.   Google Scholar [9] Z. T. Gong and H. X. Li, Saddle point optimality conditions in fuzzy optimization problems,, Fuzzy Information and Engineering, 2 (2009), 7.   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.   Google Scholar [11] D. Dentcheva and W. Römisch, Duality gaps in nonconvex stochastic optimization,, Mathematical Programming, 101 (2004), 515.   Google Scholar [12] L. A. Korf, Stochastic programming duality: $L^{\propto}$ multipliers for unbounded constraints with an application to mathematical finance,, Mathematical Programming, 99 (2004), 241.   Google Scholar [13] D. Dentcheva and A.Ruszczyński, Optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints,, Mathematical Programming, 99 (2004), 329.   Google Scholar [14] A. Shapiro, Stochastic programming approach to optimization under uncertainty,, Mathematical Programming, 112 (2008), 183.   Google Scholar [15] H. Zhu, Convex duality for finite-fuel problems in singular stochastic control,, Journal of Optimization Theory and Applications, 75 (1992), 154.   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.   Google Scholar [17] H. C. Wu, On interval-valued nonlinear programming problems,, Journal of Mathematical Analysis and Application, 338 (2008), 299.   Google Scholar [18] H. C. Wu, Wolfe duality for interval-valued optimization,, Journal of Optimization Theory and Applications, 138 (2008), 497.   Google Scholar [19] H. C. Zhou and Y. J. Wang, Optimality condition and mixed duality for interval-valued optimization., Fuzzy Information and Engineering, 62 (2009), 1315.   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.   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.   Google Scholar [22] R. E. Moore, "Method and Applications of Interval Analysis,", SIAM, (1979).   Google Scholar [23] A. Prékopa, "Stochastic Programming: Mathematics and Its Applications,", Kluwer Academic Publishers Group, (1995).   Google Scholar [24] M. Schechter, More on subgradient duality,, Journal of Mathematical Analysis and Application, 71 (1979), 251.   Google Scholar
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