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

[2]

J. Ramík, Duality in fuzzy linear programming: some new concepts and results, Fuzzy Optimization and Decision Making, 4 (2005), 25-39.

[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.

[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.

[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.

[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.

[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.

[8]

H. C. Wu, Duality theory in fuzzy optimization problems, Fuzzy Optimization and Decision Making, 3 (2004), 345-365.

[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.

[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.

[11]

D. Dentcheva and W. Römisch , Duality gaps in nonconvex stochastic optimization, Mathematical Programming, Ser. A, 101 (2004), 515-535.

[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.

[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.

[14]

A. Shapiro, Stochastic programming approach to optimization under uncertainty, Mathematical Programming, Ser. B, 112 (2008), 183-220.

[15]

H. Zhu, Convex duality for finite-fuel problems in singular stochastic control, Journal of Optimization Theory and Applications, 75 (1992), 154-181.

[16]

H. C. Wu, Duality theory for optimization problems with interval-valued objective functions, Journal of Optimization Theory and Applications, 144 (2010), 615-628.

[17]

H. C. Wu, On interval-valued nonlinear programming problems, Journal of Mathematical Analysis and Application, 338 (2008), 299-316.

[18]

H. C. Wu, Wolfe duality for interval-valued optimization, Journal of Optimization Theory and Applications, 138 (2008), 497-509.

[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.

[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.

[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.

[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), 251-262.

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), 365-383.

[2]

J. Ramík, Duality in fuzzy linear programming: some new concepts and results, Fuzzy Optimization and Decision Making, 4 (2005), 25-39.

[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.

[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.

[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.

[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.

[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.

[8]

H. C. Wu, Duality theory in fuzzy optimization problems, Fuzzy Optimization and Decision Making, 3 (2004), 345-365.

[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.

[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.

[11]

D. Dentcheva and W. Römisch , Duality gaps in nonconvex stochastic optimization, Mathematical Programming, Ser. A, 101 (2004), 515-535.

[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.

[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.

[14]

A. Shapiro, Stochastic programming approach to optimization under uncertainty, Mathematical Programming, Ser. B, 112 (2008), 183-220.

[15]

H. Zhu, Convex duality for finite-fuel problems in singular stochastic control, Journal of Optimization Theory and Applications, 75 (1992), 154-181.

[16]

H. C. Wu, Duality theory for optimization problems with interval-valued objective functions, Journal of Optimization Theory and Applications, 144 (2010), 615-628.

[17]

H. C. Wu, On interval-valued nonlinear programming problems, Journal of Mathematical Analysis and Application, 338 (2008), 299-316.

[18]

H. C. Wu, Wolfe duality for interval-valued optimization, Journal of Optimization Theory and Applications, 138 (2008), 497-509.

[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.

[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.

[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.

[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), 251-262.

[1]

Xiuhong Chen, Zhihua Li. On optimality conditions and duality for non-differentiable interval-valued programming problems with the generalized (F, ρ)-convexity. Journal of Industrial and Management Optimization, 2018, 14 (3) : 895-912. doi: 10.3934/jimo.2017081

[2]

Tadeusz Antczak, Najeeb Abdulaleem. Optimality conditions for $ E $-differentiable vector optimization problems with the multiple interval-valued objective function. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2971-2989. doi: 10.3934/jimo.2019089

[3]

Bhawna Kohli. Sufficient optimality conditions using convexifactors for optimistic bilevel programming problem. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3209-3221. doi: 10.3934/jimo.2020114

[4]

Xian-Jun Long, Jing Quan. Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 361-370. doi: 10.3934/naco.2011.1.361

[5]

Xiao-Bing Li, Qi-Lin Wang, Zhi Lin. Optimality conditions and duality for minimax fractional programming problems with data uncertainty. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1133-1151. doi: 10.3934/jimo.2018089

[6]

M. Soledad Aronna. Second order necessary and sufficient optimality conditions for singular solutions of partially-affine control problems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1233-1258. doi: 10.3934/dcdss.2018070

[7]

Nazih Abderrazzak Gadhi, Fatima Zahra Rahou. Sufficient optimality conditions and Mond-Weir duality results for a fractional multiobjective optimization problem. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021216

[8]

Piernicola Bettiol, Nathalie Khalil. Necessary optimality conditions for average cost minimization problems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2093-2124. doi: 10.3934/dcdsb.2019086

[9]

Shahlar F. Maharramov. Necessary optimality conditions for switching control problems. Journal of Industrial and Management Optimization, 2010, 6 (1) : 47-55. doi: 10.3934/jimo.2010.6.47

[10]

Jiang-Xia Nan, Deng-Feng Li. Linear programming technique for solving interval-valued constraint matrix games. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1059-1070. doi: 10.3934/jimo.2014.10.1059

[11]

Ram U. Verma. General parametric sufficient optimality conditions for multiple objective fractional subset programming relating to generalized $(\rho,\eta,A)$ -invexity. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 333-339. doi: 10.3934/naco.2011.1.333

[12]

Nazih Abderrazzak Gadhi. A note on the paper "Sufficient optimality conditions using convexifactors for optimistic bilevel programming problem". Journal of Industrial and Management Optimization, 2022, 18 (5) : 3073-3081. doi: 10.3934/jimo.2021103

[13]

Geng-Hua Li, Sheng-Jie Li. Unified optimality conditions for set-valued optimizations. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1101-1116. doi: 10.3934/jimo.2018087

[14]

Mansoureh Alavi Hejazi, Soghra Nobakhtian. Optimality conditions for multiobjective fractional programming, via convexificators. Journal of Industrial and Management Optimization, 2020, 16 (2) : 623-631. doi: 10.3934/jimo.2018170

[15]

Gang Li, Yinghong Xu, Zhenhua Qin. Optimality conditions for composite DC infinite programming problems. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022064

[16]

Deepak Singh, Bilal Ahmad Dar, Do Sang Kim. Sufficiency and duality in non-smooth interval valued programming problems. Journal of Industrial and Management Optimization, 2019, 15 (2) : 647-665. doi: 10.3934/jimo.2018063

[17]

Nuno R. O. Bastos, Rui A. C. Ferreira, Delfim F. M. Torres. Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 417-437. doi: 10.3934/dcds.2011.29.417

[18]

Sofia O. Lopes, Fernando A. C. C. Fontes, Maria do Rosário de Pinho. On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 559-575. doi: 10.3934/dcds.2011.29.559

[19]

Monika Dryl, Delfim F. M. Torres. Necessary optimality conditions for infinite horizon variational problems on time scales. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 145-160. doi: 10.3934/naco.2013.3.145

[20]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control and Related Fields, 2021, 11 (4) : 739-769. doi: 10.3934/mcrf.2020045

2021 Impact Factor: 1.411

Metrics

  • PDF downloads (286)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]