American Institute of Mathematical Sciences

Advanced
July  2018, 14(3): 1157-1178. doi: 10.3934/jimo.2018004

Solving the interval-valued optimization problems based on the concept of null set

Hsien-Chung Wu

Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 802, Taiwan

Received  April 2016 Revised  August 2017 Published  July 2018 Early access  January 2018

We introduce the concept of null set in the space of all bounded closed intervals. Based on this concept, we can define two partial orderings according to the substraction and Hukuhara difference between any two bounded closed intervals, which will be used to define the solution concepts of interval-valued optimization problems. On the other hand, we transform the interval-valued optimization problems into the conventional vector optimization problem. Under these settings, we can apply the technique of scalarization to solve this transformed vector optimization problem. Finally, we show that the optimal solution of the scalarized problem is also the optimal solution of the original interval-valued optimization problem.

Keywords: Minimal element, maximal element, Hukuhara difference, partial ordering, pointed cone.
Mathematics Subject Classification: Primary: 90C26, 90C30; Secondary: 65G30.
Citation: Hsien-Chung Wu. Solving the interval-valued optimization problems based on the concept of null set. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1157-1178. doi: 10.3934/jimo.2018004
References:
[1]

A. K. Bhurjee and G. Panda, Efficient solution of interval optimization problem, Mathematical Methods of Operations Research, 76 (2012), 273-288.  doi: 10.1007/s00186-012-0399-0.  Google Scholar

[2]

J. R. Birge and F. Louveaux, Introduction to Stochastic Programming, Physica-Verlag, NY, 1997.  Google Scholar

[3]

G. R. Bitran, Linear multiple objective problems with interval coefficients, Management Science, 26 (1980), 694-706.  doi: 10.1287/mnsc.26.7.694.  Google Scholar

[4]

Y. Chalco-Cano, W. A. Lodwick and A. Rufian-Lizana, Optimality conditions of type kkt for optimization problem with interval-valued objective function via generalized derivative, Fuzzy Optimization and Decision Making, 12 (2013), 305-322.  doi: 10.1007/s10700-013-9156-y.  Google Scholar

[5]

S. Chanas and D. Kuchta, Multiobjective programming in optimization of interval objective functions --a generalized approach, European Journal of Operational Research, 94 (1996), 594-598.  doi: 10.1016/0377-2217(95)00055-0.  Google Scholar

[6]

A. Charnes, F. Granot and F. Phillips, An algorithm for solving interval linear programming problems, Operations Research, 25 (1977), 688-695.  doi: 10.1287/opre.25.4.688.  Google Scholar

[7]

J. W. Chinneck and K. Ramadan, Linear programming with interval coefficients, The Journal of the Operational Research Society, 51 (2000), 209-220.   Google Scholar

[8]

M. Delgado, J. Kacprzyk, J. -L. Verdegay and M. A. Vila (eds. ), Fuzzy Optimization: Recent Advances, Physica-Verlag, NY, 1994.  Google Scholar

[9]

M. Inuiguchi and J. Ramík, Possibilistic linear programming: A brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem, Fuzzy Sets and Systems, 111 (2000), 3-28.  doi: 10.1016/S0165-0114(98)00449-7.  Google Scholar

[10]

A. Jayswal, I. Stancu-Minasian and I. Ahmad, On sufficiency and duality for a class of interval-valued programming problems, Applied Mathematics and Computation, 218 (2011), 4119-4127.  doi: 10.1016/j.amc.2011.09.041.  Google Scholar

[11]

P. Kall, Stochastic Linear Programming, Springer-Verlag, NY, 1976.  Google Scholar

[12]

R. Osuna-Gomez, Y. Chalco-Cano, B. Hernandez-Jimenez and G. Ruiz-Garzon, Optimality conditions for generalized differentiable interval-valued functions, Information Sciences, 321 (2015), 136-146.  doi: 10.1016/j.ins.2015.05.039.  Google Scholar

[13]

A. Prékopa, Stochastic Programming, Kluwer Academic Publishers, Boston, 1995. Google Scholar

[14]

R. S lowiński (ed. ), Fuzzy Sets in Decision Analysis, Operations Research and Statistics, Kluwer Academic Publishers, Boston, 1998. Google Scholar

[15]

R. S lowiński and J. Teghem (eds), Stochastic Versus Fuzzy Approaches to Multiobjective Mathematical Programming under Uncertainty, Kluwer Academic Publishers, Boston, 1990. Google Scholar

[16]

I. M. Stancu-Minasian, Stochastic Programming with Multiple Objective Functions, D. Reidel Publishing Company, 1984.  Google Scholar

[17] S. Vajda, Probabilistic Programming, Academic Press,, NY, 1972.   Google Scholar
[18]

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.  doi: 10.1016/j.ejor.2005.09.007.  Google Scholar

[19]

H.-C. Wu, On Interval-valued nonlinear programming problems, Journal of Mathematical Analysis and Applications, 338 (2008), 299-316.  doi: 10.1016/j.jmaa.2007.05.023.  Google Scholar

[20]

H.-C. Wu, Wolfe duality for interval-valued optimization, Journal of Optimization Theory and Applications, 138 (2008), 497-509.  doi: 10.1007/s10957-008-9396-0.  Google Scholar

[21]

H.-C. Wu, Duality theory for optimization problems with interval-valued objective functions, Journal of Optimization Theory and Applications, 144 (2010), 615-628.  doi: 10.1007/s10957-009-9613-5.  Google Scholar

show all references

References:
[1]

A. K. Bhurjee and G. Panda, Efficient solution of interval optimization problem, Mathematical Methods of Operations Research, 76 (2012), 273-288.  doi: 10.1007/s00186-012-0399-0.  Google Scholar

[2]

J. R. Birge and F. Louveaux, Introduction to Stochastic Programming, Physica-Verlag, NY, 1997.  Google Scholar

[3]

G. R. Bitran, Linear multiple objective problems with interval coefficients, Management Science, 26 (1980), 694-706.  doi: 10.1287/mnsc.26.7.694.  Google Scholar

[4]

Y. Chalco-Cano, W. A. Lodwick and A. Rufian-Lizana, Optimality conditions of type kkt for optimization problem with interval-valued objective function via generalized derivative, Fuzzy Optimization and Decision Making, 12 (2013), 305-322.  doi: 10.1007/s10700-013-9156-y.  Google Scholar

[5]

S. Chanas and D. Kuchta, Multiobjective programming in optimization of interval objective functions --a generalized approach, European Journal of Operational Research, 94 (1996), 594-598.  doi: 10.1016/0377-2217(95)00055-0.  Google Scholar

[6]

A. Charnes, F. Granot and F. Phillips, An algorithm for solving interval linear programming problems, Operations Research, 25 (1977), 688-695.  doi: 10.1287/opre.25.4.688.  Google Scholar

[7]

J. W. Chinneck and K. Ramadan, Linear programming with interval coefficients, The Journal of the Operational Research Society, 51 (2000), 209-220.   Google Scholar

[8]

M. Delgado, J. Kacprzyk, J. -L. Verdegay and M. A. Vila (eds. ), Fuzzy Optimization: Recent Advances, Physica-Verlag, NY, 1994.  Google Scholar

[9]

M. Inuiguchi and J. Ramík, Possibilistic linear programming: A brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem, Fuzzy Sets and Systems, 111 (2000), 3-28.  doi: 10.1016/S0165-0114(98)00449-7.  Google Scholar

[10]

A. Jayswal, I. Stancu-Minasian and I. Ahmad, On sufficiency and duality for a class of interval-valued programming problems, Applied Mathematics and Computation, 218 (2011), 4119-4127.  doi: 10.1016/j.amc.2011.09.041.  Google Scholar

[11]

P. Kall, Stochastic Linear Programming, Springer-Verlag, NY, 1976.  Google Scholar

[12]

R. Osuna-Gomez, Y. Chalco-Cano, B. Hernandez-Jimenez and G. Ruiz-Garzon, Optimality conditions for generalized differentiable interval-valued functions, Information Sciences, 321 (2015), 136-146.  doi: 10.1016/j.ins.2015.05.039.  Google Scholar

[13]

A. Prékopa, Stochastic Programming, Kluwer Academic Publishers, Boston, 1995. Google Scholar

[14]

R. S lowiński (ed. ), Fuzzy Sets in Decision Analysis, Operations Research and Statistics, Kluwer Academic Publishers, Boston, 1998. Google Scholar

[15]

R. S lowiński and J. Teghem (eds), Stochastic Versus Fuzzy Approaches to Multiobjective Mathematical Programming under Uncertainty, Kluwer Academic Publishers, Boston, 1990. Google Scholar

[16]

I. M. Stancu-Minasian, Stochastic Programming with Multiple Objective Functions, D. Reidel Publishing Company, 1984.  Google Scholar

[17] S. Vajda, Probabilistic Programming, Academic Press,, NY, 1972.   Google Scholar
[18]

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.  doi: 10.1016/j.ejor.2005.09.007.  Google Scholar

[19]

H.-C. Wu, On Interval-valued nonlinear programming problems, Journal of Mathematical Analysis and Applications, 338 (2008), 299-316.  doi: 10.1016/j.jmaa.2007.05.023.  Google Scholar

[20]

H.-C. Wu, Wolfe duality for interval-valued optimization, Journal of Optimization Theory and Applications, 138 (2008), 497-509.  doi: 10.1007/s10957-008-9396-0.  Google Scholar

[21]

H.-C. Wu, Duality theory for optimization problems with interval-valued objective functions, Journal of Optimization Theory and Applications, 144 (2010), 615-628.  doi: 10.1007/s10957-009-9613-5.  Google Scholar

[1]

Gonzalo Galiano, Julián Velasco. Finite element approximation of a population spatial adaptation model. Mathematical Biosciences & Engineering, 2013, 10 (3) : 637-647. doi: 10.3934/mbe.2013.10.637
[2]

P. K. Jha, R. Lipton. Finite element approximation of nonlocal dynamic fracture models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1675-1710. doi: 10.3934/dcdsb.2020178
[3]

Cornel M. Murea, H. G. E. Hentschel. A finite element method for growth in biological development. Mathematical Biosciences & Engineering, 2007, 4 (2) : 339-353. doi: 10.3934/mbe.2007.4.339
[4]

Eduardo Casas, Mariano Mateos, Arnd Rösch. Finite element approximation of sparse parabolic control problems. Mathematical Control & Related Fields, 2017, 7 (3) : 393-417. doi: 10.3934/mcrf.2017014
[5]

Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59
[6]

Kersten Schmidt, Ralf Hiptmair. Asymptotic boundary element methods for thin conducting sheets. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 619-647. doi: 10.3934/dcdss.2015.8.619
[7]

Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095
[8]

Robert Jensen, Andrzej Świech. Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE. Communications on Pure & Applied Analysis, 2005, 4 (1) : 199-207. doi: 10.3934/cpaa.2005.4.187
[9]

Sören Bartels, Marijo Milicevic. Iterative finite element solution of a constrained total variation regularized model problem. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1207-1232. doi: 10.3934/dcdss.2017066
[10]

Eduardo Casas, Boris Vexler, Enrique Zuazua. Sparse initial data identification for parabolic PDE and its finite element approximations. Mathematical Control & Related Fields, 2015, 5 (3) : 377-399. doi: 10.3934/mcrf.2015.5.377
[11]

Fang Liu, Aihui Zhou. Localizations and parallelizations for two-scale finite element discretizations. Communications on Pure & Applied Analysis, 2007, 6 (3) : 757-773. doi: 10.3934/cpaa.2007.6.757
[12]

Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie. A combined finite volume - finite element scheme for a dispersive shallow water system. Networks & Heterogeneous Media, 2016, 11 (1) : 1-27. doi: 10.3934/nhm.2016.11.1
[13]

Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641
[14]

Binjie Li, Xiaoping Xie, Shiquan Zhang. New convergence analysis for assumed stress hybrid quadrilateral finite element method. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2831-2856. doi: 10.3934/dcdsb.2017153
[15]

Wolf-Jüergen Beyn, Janosch Rieger. Galerkin finite element methods for semilinear elliptic differential inclusions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 295-312. doi: 10.3934/dcdsb.2013.18.295
[16]

Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216
[17]

Kun Wang, Yinnian He, Yueqiang Shang. Fully discrete finite element method for the viscoelastic fluid motion equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 665-684. doi: 10.3934/dcdsb.2010.13.665
[18]

Junjiang Lai, Jianguo Huang. A finite element method for vibration analysis of elastic plate-plate structures. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 387-419. doi: 10.3934/dcdsb.2009.11.387
[19]

Chunjuan Hou, Yanping Chen, Zuliang Lu. Superconvergence property of finite element methods for parabolic optimal control problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 927-945. doi: 10.3934/jimo.2011.7.927
[20]

So-Hsiang Chou. An immersed linear finite element method with interface flux capturing recovery. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2343-2357. doi: 10.3934/dcdsb.2012.17.2343

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (349)
  • HTML views (895)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences