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September  2021, 17(5): 2761-2782. doi: 10.3934/jimo.2020093

The $ F $-objective function method for differentiable interval-valued vector optimization problems

Tadeusz Antczak

Faculty of Mathematics and Computer Science, University of Łódź, Banacha 22, 90-238 Łódź, Poland

Received  July 2019 Revised  January 2020 Published  September 2021 Early access  May 2020

In this paper, a differentiable vector optimization problem with the multiple interval-valued objective function and with both inequality and equality constraints is considered. The Karush-Kuhn-Tucker necessary optimality conditions are established for such a differentiable interval-valued multiobjective programming problem. Further, a new approach, called $ F $-objective function method, is introduced for solving the considered differentiable vector optimization problem with the multiple interval-valued objective function. In this method, its associated vector optimization problem with the multiple interval-valued $ F $-objective function is constructed. Their equivalence is established under $ F $-convexity assumptions. It is shown that the introduced approach can be used to solve a nonlinear nonconvex interval-valued optimization problem. By using the introduced approximation method, it is also presented in some cases that a nonlinear nonconvex interval-valued optimization problem can be solved by the help of methods for solving linear interval-valued optimization problems.

Keywords: Differentiable interval-valued vector optimization, $ F $-objective function method, Karush-Kuhn-Tucker necessary optimality conditions, (weak) $ LU $-Pareto solution, $ F $-convex function.
Mathematics Subject Classification: 90C29, 90C30, 90C46, 90C59, 90C26.
Citation: Tadeusz Antczak. The $ F $-objective function method for differentiable interval-valued vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2761-2782. doi: 10.3934/jimo.2020093
References:
[1]

I. Ahmad, D. Singh and B. A. Dar, Optimality conditions in multiobjective programming problems with interval valued objective functions, Control Cybernet., 44 (2015), 19-45.   Google Scholar

[2]

I. Ahmad, D. Singh and B. A. Dar, Optimality conditions for invex interval valued nonlinear programming problems involving generalized $H$-derivative, Filomat, 30 (2016), 2121-2138.  doi: 10.2298/FIL1608121A.  Google Scholar

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M. Allahdadi and H. M. Nehi, The optimal solution set of the interval linear programming problems, Optim. Lett., 7 (2013), 1893-1911.  doi: 10.1007/s11590-012-0530-4.  Google Scholar

[5]

T. Antczak, A new approach to multiobjective programming with a modified objective function, J. Global Optim., 27 (2003), 485-495.  doi: 10.1023/A:1026080604790.  Google Scholar

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T. Antczak, An $\eta $-approximation method in vector optimization, Nonlinear Anal., 63 (2005), 225-236.  doi: 10.1016/j.na.2005.05.008.  Google Scholar

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A. K. Bhurjee and G. Panda, Efficient solution of interval optimization problem, Math. Method Oper. Res., 76 (2012), 273-288.  doi: 10.1007/s00186-012-0399-0.  Google Scholar

[8]

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 Optim. Decis. Mak., 12 (2013), 305-322.  doi: 10.1007/s10700-013-9156-y.  Google Scholar

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

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J. W. Chinneck and K. Ramadan, Linear programming with interval coefficients, JORS, 51 (1996), 209-220.   Google Scholar

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M. Ehrgott, Multicriteria Optimization, 2nd edition, Springer-Verlag, Berlin, 2005.  Google Scholar

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G. Eichfelder, Adaptive Scalarization Methods in Multiobjective Optimization, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-79159-1.  Google Scholar

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M. A. Hanson and B. Mond, Further generalizations of convexity in mathematical programming, J. Inform. Optim. Sci., 3 (1982), 25-32.  doi: 10.1080/02522667.1982.10698716.  Google Scholar

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M. Hladik, Interval Linear Programming: A Survey. Linear Programming-New Frontiers in Theory and Applications, Nova Science Publishers, New York, 2011. doi: 10.1016/j.ejor.2009.04.019.  Google Scholar

[15]

E. Hosseinzade and H. Hassanpour, The Karush-Kuhn-Tucker optimality conditions in interval-valued multiobjective programming problems, J. Appl. Math. Inform., 29 (2011), 1157-1165.   Google Scholar

[16]

M. Inuiguchi and Y. Kume, Minimax regret in linear programming problems with an interval objective function, in Multiple Criteria Decision Making, Springer-Verlag, New York, 1994, 65–74. doi: 10.1007/978-1-4612-2666-6_8.  Google Scholar

[17]

M. Inuiguchi and M. Sakawa, Minimax regret solution to linear programming problems with an interval objective function, European J. Oper. Res., 86 (1995), 526-536.  doi: 10.1016/0377-2217(94)00092-Q.  Google Scholar

[18]

H. Ishihuchi and M. Tanaka, Multiobjective programming in optimization of the interval objective function, European J. Oper. Res., 48 (1990), 219-225.   Google Scholar

[19]

J. Jahn, Vector Optimization: Theory, Applications, and Extensions, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-540-24828-6.  Google Scholar

[20]

M. Jana and G. Panda, Solution of nonlinear interval vector optimization problem, Oper. Res. Int. J., 1 (2014), 71-85.   Google Scholar

[21]

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

[22]

C. Jiang, X. Han, G. R. Liu and G. P. Liu, A nonlinear interval number programming method for uncertain optimization problems, European J. Oper. Res., 188 (2008), 1-13.  doi: 10.1016/j.ejor.2007.03.031.  Google Scholar

[23]

S. Karmakar and K. Bhunia, An alternative optimization technique for interval objective constrained optimization problems via multiobjective programming, J. Egypt. Math. Soc., 22 (2014), 292-303.  doi: 10.1016/j.joems.2013.07.002.  Google Scholar

[24]

D. S. Kim, Generalized convexity and duality for multiobjective optimization problems, J. Inform. Optim. Sci., 13 (1992), 383-390.  doi: 10.1080/02522667.1992.10699123.  Google Scholar

[25]

L. Li, S. Liu and J. Zhang, On interval-valued invex mappings and optimality conditions for interval-valued optimization problems, J. Inequal. Appl., (2015), No. 179, 19 pp. doi: 10.1186/s13660-015-0692-6.  Google Scholar

[26]

J. Lin, Maximal vectors and multi-objective optimization, J. Optim. Theory Appl., 18 (1976), 41-64.  doi: 10.1007/BF00933793.  Google Scholar

[27]

D. V. Luu and T. T. Mai, Optimality and duality in constrained interval-valued optimization, 4OR-Q J Oper Res., 16 (2018), 311-337.  doi: 10.1007/s10288-017-0369-8.  Google Scholar

[28]

O. L. Mangasarian, Nonlinear Programming, McGraw-Hill Book Co., New York-London-Sydney, 1969.  Google Scholar

[29]

K. Miettinen, Nonlinear Multiobjective Optimization. International Series in Operations Research & Management Science, Vol. 12, Kluwer Academic Publishers, Boston, MA, 2004.  Google Scholar

[30]

R. E. Moore, Method and Applications of Interval Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1979.  Google Scholar

[31]

R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009. doi: 10.1137/1.9780898717716.  Google Scholar

[32]

F. Mráz, Calculating the exact bounds of optimal values in LP with interval coefficients, Ann. Oper. Res., 81 (1998), 51-62.  doi: 10.1023/A:1018985914065.  Google Scholar

[33]

C. Oliveira and C. H. Antunes, Multiple objective linear programming models with interval coefficients - an illustrated overview, European J. Oper. Res., 181 (2007), 1434-1463.  doi: 10.1016/j.ejor.2005.12.042.  Google Scholar

[34]

R. Osuna-Gómez, B. Hernández-Jiménez, Y. Chalco-Cano and G. Ruiz-Garzón, New efficiency conditions for multiobjective interval - valued programming problems, Inform. Sci., 420 (2017), 235-248.  doi: 10.1016/j.ins.2017.08.022.  Google Scholar

[35]

V. Preda, On efficiency and duality for multiobjective programs, J. Math. Anal. Appl., 166 (1992), 365-377.  doi: 10.1016/0022-247X(92)90303-U.  Google Scholar

[36]

S. Ruzika and M. M. Wiecek, Approximation methods in multiobjective programming, J. Optim. Theory Appl., 126 (2005), 473-501.  doi: 10.1007/s10957-005-5494-4.  Google Scholar

[37]

D. Singh, B. A. Dar and A. Goyal, KKT optimality conditions for interval valued optimization problems, J. Nonlinear Anal. Optim., 5 (2014), 91-103.   Google Scholar

[38]

D. Singh, B. A. Dar and D. S. Kim, KKT optimality conditions in interval valued multiobjective programming with generalized differentiable functions, European J. Oper. Res., 254 (2016), 29-39.  doi: 10.1016/j.ejor.2016.03.042.  Google Scholar

[39]

R. E. Steuer, Algorithms for linear programming problems with interval objective function coefficients, Math. Oper. Res., 6 (1981), 333-348.  doi: 10.1287/moor.6.3.333.  Google Scholar

[40]

H. Suprajitno and I. bin Mohd, Linear programming with interval arithmetic, Int. J. Contemp. Math. Sci., 5 (2010), 323-332.   Google Scholar

[41]

B. Urli and R. Nadeau, An interactive method to multiobjective linear programming problems with interval coefficients, INFOR: Information Systems and Operational Research, 30 (1992), 127-137.  doi: 10.1080/03155986.1992.11732189.  Google Scholar

[42]

H.-C. Wu, On interval-valued nonlinear programming problems, J. Math. Anal. Appl., 338 (2008), 299-316.  doi: 10.1016/j.jmaa.2007.05.023.  Google Scholar

[43]

H.-C. Wu, The Karush-Kuhn-Tucker optimality conditions in multiobjective programming problems with interval-valued objective functions, European J. Oper. Res., 196 (2009), 49-60.  doi: 10.1016/j.ejor.2008.03.012.  Google Scholar

[44]

J. Zhang, S. Liu, L. Li and Q. Feng, The KKT optimality conditions in a class of generalized convex optimization problems with an interval-valued objective function, Optim. Lett., 8 (2014), 607-631.  doi: 10.1007/s11590-012-0601-6.  Google Scholar

[45]

H.-C. Zhou and Y-J. Wang, Optimality condition and mixed duality for interval-valued optimization, in Fuzzy Information and Engineering, Vol. 2, Springer, Berlin, Heidelberg, 2009, 1315–1323. doi: 10.1007/978-3-642-03664-4_140.  Google Scholar

show all references

References:
[1]

I. Ahmad, D. Singh and B. A. Dar, Optimality conditions in multiobjective programming problems with interval valued objective functions, Control Cybernet., 44 (2015), 19-45.   Google Scholar

[2]

I. Ahmad, D. Singh and B. A. Dar, Optimality conditions for invex interval valued nonlinear programming problems involving generalized $H$-derivative, Filomat, 30 (2016), 2121-2138.  doi: 10.2298/FIL1608121A.  Google Scholar

[3]

G. Alefeld and J. Herzberger, Introduction to Interval Computations, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983.  Google Scholar

[4]

M. Allahdadi and H. M. Nehi, The optimal solution set of the interval linear programming problems, Optim. Lett., 7 (2013), 1893-1911.  doi: 10.1007/s11590-012-0530-4.  Google Scholar

[5]

T. Antczak, A new approach to multiobjective programming with a modified objective function, J. Global Optim., 27 (2003), 485-495.  doi: 10.1023/A:1026080604790.  Google Scholar

[6]

T. Antczak, An $\eta $-approximation method in vector optimization, Nonlinear Anal., 63 (2005), 225-236.  doi: 10.1016/j.na.2005.05.008.  Google Scholar

[7]

A. K. Bhurjee and G. Panda, Efficient solution of interval optimization problem, Math. Method Oper. Res., 76 (2012), 273-288.  doi: 10.1007/s00186-012-0399-0.  Google Scholar

[8]

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 Optim. Decis. Mak., 12 (2013), 305-322.  doi: 10.1007/s10700-013-9156-y.  Google Scholar

[9]

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

[10]

J. W. Chinneck and K. Ramadan, Linear programming with interval coefficients, JORS, 51 (1996), 209-220.   Google Scholar

[11]

M. Ehrgott, Multicriteria Optimization, 2nd edition, Springer-Verlag, Berlin, 2005.  Google Scholar

[12]

G. Eichfelder, Adaptive Scalarization Methods in Multiobjective Optimization, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-79159-1.  Google Scholar

[13]

M. A. Hanson and B. Mond, Further generalizations of convexity in mathematical programming, J. Inform. Optim. Sci., 3 (1982), 25-32.  doi: 10.1080/02522667.1982.10698716.  Google Scholar

[14]

M. Hladik, Interval Linear Programming: A Survey. Linear Programming-New Frontiers in Theory and Applications, Nova Science Publishers, New York, 2011. doi: 10.1016/j.ejor.2009.04.019.  Google Scholar

[15]

E. Hosseinzade and H. Hassanpour, The Karush-Kuhn-Tucker optimality conditions in interval-valued multiobjective programming problems, J. Appl. Math. Inform., 29 (2011), 1157-1165.   Google Scholar

[16]

M. Inuiguchi and Y. Kume, Minimax regret in linear programming problems with an interval objective function, in Multiple Criteria Decision Making, Springer-Verlag, New York, 1994, 65–74. doi: 10.1007/978-1-4612-2666-6_8.  Google Scholar

[17]

M. Inuiguchi and M. Sakawa, Minimax regret solution to linear programming problems with an interval objective function, European J. Oper. Res., 86 (1995), 526-536.  doi: 10.1016/0377-2217(94)00092-Q.  Google Scholar

[18]

H. Ishihuchi and M. Tanaka, Multiobjective programming in optimization of the interval objective function, European J. Oper. Res., 48 (1990), 219-225.   Google Scholar

[19]

J. Jahn, Vector Optimization: Theory, Applications, and Extensions, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-540-24828-6.  Google Scholar

[20]

M. Jana and G. Panda, Solution of nonlinear interval vector optimization problem, Oper. Res. Int. J., 1 (2014), 71-85.   Google Scholar

[21]

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

[22]

C. Jiang, X. Han, G. R. Liu and G. P. Liu, A nonlinear interval number programming method for uncertain optimization problems, European J. Oper. Res., 188 (2008), 1-13.  doi: 10.1016/j.ejor.2007.03.031.  Google Scholar

[23]

S. Karmakar and K. Bhunia, An alternative optimization technique for interval objective constrained optimization problems via multiobjective programming, J. Egypt. Math. Soc., 22 (2014), 292-303.  doi: 10.1016/j.joems.2013.07.002.  Google Scholar

[24]

D. S. Kim, Generalized convexity and duality for multiobjective optimization problems, J. Inform. Optim. Sci., 13 (1992), 383-390.  doi: 10.1080/02522667.1992.10699123.  Google Scholar

[25]

L. Li, S. Liu and J. Zhang, On interval-valued invex mappings and optimality conditions for interval-valued optimization problems, J. Inequal. Appl., (2015), No. 179, 19 pp. doi: 10.1186/s13660-015-0692-6.  Google Scholar

[26]

J. Lin, Maximal vectors and multi-objective optimization, J. Optim. Theory Appl., 18 (1976), 41-64.  doi: 10.1007/BF00933793.  Google Scholar

[27]

D. V. Luu and T. T. Mai, Optimality and duality in constrained interval-valued optimization, 4OR-Q J Oper Res., 16 (2018), 311-337.  doi: 10.1007/s10288-017-0369-8.  Google Scholar

[28]

O. L. Mangasarian, Nonlinear Programming, McGraw-Hill Book Co., New York-London-Sydney, 1969.  Google Scholar

[29]

K. Miettinen, Nonlinear Multiobjective Optimization. International Series in Operations Research & Management Science, Vol. 12, Kluwer Academic Publishers, Boston, MA, 2004.  Google Scholar

[30]

R. E. Moore, Method and Applications of Interval Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1979.  Google Scholar

[31]

R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009. doi: 10.1137/1.9780898717716.  Google Scholar

[32]

F. Mráz, Calculating the exact bounds of optimal values in LP with interval coefficients, Ann. Oper. Res., 81 (1998), 51-62.  doi: 10.1023/A:1018985914065.  Google Scholar

[33]

C. Oliveira and C. H. Antunes, Multiple objective linear programming models with interval coefficients - an illustrated overview, European J. Oper. Res., 181 (2007), 1434-1463.  doi: 10.1016/j.ejor.2005.12.042.  Google Scholar

[34]

R. Osuna-Gómez, B. Hernández-Jiménez, Y. Chalco-Cano and G. Ruiz-Garzón, New efficiency conditions for multiobjective interval - valued programming problems, Inform. Sci., 420 (2017), 235-248.  doi: 10.1016/j.ins.2017.08.022.  Google Scholar

[35]

V. Preda, On efficiency and duality for multiobjective programs, J. Math. Anal. Appl., 166 (1992), 365-377.  doi: 10.1016/0022-247X(92)90303-U.  Google Scholar

[36]

S. Ruzika and M. M. Wiecek, Approximation methods in multiobjective programming, J. Optim. Theory Appl., 126 (2005), 473-501.  doi: 10.1007/s10957-005-5494-4.  Google Scholar

[37]

D. Singh, B. A. Dar and A. Goyal, KKT optimality conditions for interval valued optimization problems, J. Nonlinear Anal. Optim., 5 (2014), 91-103.   Google Scholar

[38]

D. Singh, B. A. Dar and D. S. Kim, KKT optimality conditions in interval valued multiobjective programming with generalized differentiable functions, European J. Oper. Res., 254 (2016), 29-39.  doi: 10.1016/j.ejor.2016.03.042.  Google Scholar

[39]

R. E. Steuer, Algorithms for linear programming problems with interval objective function coefficients, Math. Oper. Res., 6 (1981), 333-348.  doi: 10.1287/moor.6.3.333.  Google Scholar

[40]

H. Suprajitno and I. bin Mohd, Linear programming with interval arithmetic, Int. J. Contemp. Math. Sci., 5 (2010), 323-332.   Google Scholar

[41]

B. Urli and R. Nadeau, An interactive method to multiobjective linear programming problems with interval coefficients, INFOR: Information Systems and Operational Research, 30 (1992), 127-137.  doi: 10.1080/03155986.1992.11732189.  Google Scholar

[42]

H.-C. Wu, On interval-valued nonlinear programming problems, J. Math. Anal. Appl., 338 (2008), 299-316.  doi: 10.1016/j.jmaa.2007.05.023.  Google Scholar

[43]

H.-C. Wu, The Karush-Kuhn-Tucker optimality conditions in multiobjective programming problems with interval-valued objective functions, European J. Oper. Res., 196 (2009), 49-60.  doi: 10.1016/j.ejor.2008.03.012.  Google Scholar

[44]

J. Zhang, S. Liu, L. Li and Q. Feng, The KKT optimality conditions in a class of generalized convex optimization problems with an interval-valued objective function, Optim. Lett., 8 (2014), 607-631.  doi: 10.1007/s11590-012-0601-6.  Google Scholar

[45]

H.-C. Zhou and Y-J. Wang, Optimality condition and mixed duality for interval-valued optimization, in Fuzzy Information and Engineering, Vol. 2, Springer, Berlin, Heidelberg, 2009, 1315–1323. doi: 10.1007/978-3-642-03664-4_140.  Google Scholar

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