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Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints
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 |
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.
|
[2] |
J. Ramík, Duality in fuzzy linear programming: some new concepts and results,, Fuzzy Optimization and Decision Making, 4 (2005), 25.
|
[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. |
[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. |
[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.
|
[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. |
[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. |
[8] |
H. C. Wu, Duality theory in fuzzy optimization problems,, Fuzzy Optimization and Decision Making, 3 (2004), 345.
|
[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.
|
[11] |
D. Dentcheva and W. Römisch, Duality gaps in nonconvex stochastic optimization,, Mathematical Programming, 101 (2004), 515.
|
[12] |
L. A. Korf, Stochastic programming duality: $L^{\propto}$ multipliers for unbounded constraints with an application to mathematical finance,, Mathematical Programming, 99 (2004), 241.
|
[13] |
D. Dentcheva and A.Ruszczyński, Optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints,, Mathematical Programming, 99 (2004), 329.
|
[14] |
A. Shapiro, Stochastic programming approach to optimization under uncertainty,, Mathematical Programming, 112 (2008), 183.
|
[15] |
H. Zhu, Convex duality for finite-fuel problems in singular stochastic control,, Journal of Optimization Theory and Applications, 75 (1992), 154.
|
[16] |
H. C. Wu, Duality theory for optimization problems with interval-valued objective functions,, Journal of Optimization Theory and Applications, 144 (2010), 615.
|
[17] |
H. C. Wu, On interval-valued nonlinear programming problems,, Journal of Mathematical Analysis and Application, 338 (2008), 299.
|
[18] |
H. C. Wu, Wolfe duality for interval-valued optimization,, Journal of Optimization Theory and Applications, 138 (2008), 497.
|
[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.
|
[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.
|
[22] |
R. E. Moore, "Method and Applications of Interval Analysis,", SIAM, (1979).
|
[23] |
A. Prékopa, "Stochastic Programming: Mathematics and Its Applications,", Kluwer Academic Publishers Group, (1995).
|
[24] |
M. Schechter, More on subgradient duality,, Journal of Mathematical Analysis and Application, 71 (1979), 251.
|
show all references
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.
|
[2] |
J. Ramík, Duality in fuzzy linear programming: some new concepts and results,, Fuzzy Optimization and Decision Making, 4 (2005), 25.
|
[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. |
[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. |
[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.
|
[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. |
[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. |
[8] |
H. C. Wu, Duality theory in fuzzy optimization problems,, Fuzzy Optimization and Decision Making, 3 (2004), 345.
|
[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.
|
[11] |
D. Dentcheva and W. Römisch, Duality gaps in nonconvex stochastic optimization,, Mathematical Programming, 101 (2004), 515.
|
[12] |
L. A. Korf, Stochastic programming duality: $L^{\propto}$ multipliers for unbounded constraints with an application to mathematical finance,, Mathematical Programming, 99 (2004), 241.
|
[13] |
D. Dentcheva and A.Ruszczyński, Optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints,, Mathematical Programming, 99 (2004), 329.
|
[14] |
A. Shapiro, Stochastic programming approach to optimization under uncertainty,, Mathematical Programming, 112 (2008), 183.
|
[15] |
H. Zhu, Convex duality for finite-fuel problems in singular stochastic control,, Journal of Optimization Theory and Applications, 75 (1992), 154.
|
[16] |
H. C. Wu, Duality theory for optimization problems with interval-valued objective functions,, Journal of Optimization Theory and Applications, 144 (2010), 615.
|
[17] |
H. C. Wu, On interval-valued nonlinear programming problems,, Journal of Mathematical Analysis and Application, 338 (2008), 299.
|
[18] |
H. C. Wu, Wolfe duality for interval-valued optimization,, Journal of Optimization Theory and Applications, 138 (2008), 497.
|
[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.
|
[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.
|
[22] |
R. E. Moore, "Method and Applications of Interval Analysis,", SIAM, (1979).
|
[23] |
A. Prékopa, "Stochastic Programming: Mathematics and Its Applications,", Kluwer Academic Publishers Group, (1995).
|
[24] |
M. Schechter, More on subgradient duality,, Journal of Mathematical Analysis and Application, 71 (1979), 251.
|
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