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  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]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196
[2]

Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222
[3]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021014
[4]

Ondrej Budáč, Michael Herrmann, Barbara Niethammer, Andrej Spielmann. On a model for mass aggregation with maximal size. Kinetic & Related Models, 2011, 4 (2) : 427-439. doi: 10.3934/krm.2011.4.427
[5]

Kiyoshi Igusa, Gordana Todorov. Picture groups and maximal green sequences. Electronic Research Archive, , () : -. doi: 10.3934/era.2021025
[6]

Sel Ly, Nicolas Privault. Stochastic ordering by g-expectations. Probability, Uncertainty and Quantitative Risk, 2021, 6 (1) : 61-98. doi: 10.3934/puqr.2021004
[7]

Li Chu, Bo Wang, Jie Zhang, Hong-Wei Zhang. Convergence analysis of a smoothing SAA method for a stochastic mathematical program with second-order cone complementarity constraints. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1863-1886. doi: 10.3934/jimo.2020050
[8]

Guiyang Zhu. Optimal pricing and ordering policy for defective items under temporary price reduction with inspection errors and price sensitive demand. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021060
[9]

Nouressadat Touafek, Durhasan Turgut Tollu, Youssouf Akrour. On a general homogeneous three-dimensional system of difference equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2021017
[10]

Ruchika Sehgal, Aparna Mehra. Worst-case analysis of Gini mean difference safety measure. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1613-1637. doi: 10.3934/jimo.2020037
[11]

Leon Mons. Partial regularity for parabolic systems with VMO-coefficients. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021041
[12]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209
[13]

Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021020
[14]

Marcel Braukhoff, Ansgar Jüngel. Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3335-3355. doi: 10.3934/dcdsb.2020234
[15]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175
[16]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[17]

Qiao Liu. Partial regularity and the Minkowski dimension of singular points for suitable weak solutions to the 3D simplified Ericksen–Leslie system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021041
[18]

Dariusz Idczak. A Gronwall lemma for functions of two variables and its application to partial differential equations of fractional order. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021019
[19]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023
[20]

Jinye Shen, Xian-Ming Gu. Two finite difference methods based on an H2N2 interpolation for two-dimensional time fractional mixed diffusion and diffusion-wave equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021086

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (254)
  • HTML views (891)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences