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Two penalized mixed–integer nonlinear programming approaches to tackle multicollinearity and outliers effects in linear regression models
The $ F $-objective function method for differentiable interval-valued vector optimization problems
Faculty of Mathematics and Computer Science, University of Łódź, Banacha 22, 90-238 Łódź, Poland |
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.
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.
|
[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. |
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[Harcourt Brace Jovanovich, Publishers], New York, 1983.
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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.
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[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. |
[6] |
T. Antczak,
An $\eta $-approximation method in vector optimization, Nonlinear Anal., 63 (2005), 225-236.
doi: 10.1016/j.na.2005.05.008. |
[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. |
[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. |
[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. |
[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. |
[12] |
G. Eichfelder, Adaptive Scalarization Methods in Multiobjective Optimization, Springer-Verlag, Berlin, 2008.
doi: 10.1007/978-3-540-79159-1. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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.
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[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. |
[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. |
[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. |
[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. |
[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. |
[26] |
J. Lin,
Maximal vectors and multi-objective optimization, J. Optim. Theory Appl., 18 (1976), 41-64.
doi: 10.1007/BF00933793. |
[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. |
[28] |
O. L. Mangasarian, Nonlinear Programming, McGraw-Hill Book Co., New York-London-Sydney, 1969. |
[29] |
K. Miettinen, Nonlinear Multiobjective Optimization. International Series in Operations Research & Management Science, Vol. 12, Kluwer Academic Publishers, Boston, MA, 2004. |
[30] |
R. E. Moore, Method and Applications of Interval Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1979. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[40] |
H. Suprajitno and I. bin Mohd,
Linear programming with interval arithmetic, Int. J. Contemp. Math. Sci., 5 (2010), 323-332.
|
[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. |
[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. |
[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. |
[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. |
[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. |
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.
|
[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. |
[3] |
G. Alefeld and J. Herzberger, Introduction to Interval Computations, Academic Press, Inc.
[Harcourt Brace Jovanovich, Publishers], New York, 1983.
doi: doi. |
[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. |
[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. |
[6] |
T. Antczak,
An $\eta $-approximation method in vector optimization, Nonlinear Anal., 63 (2005), 225-236.
doi: 10.1016/j.na.2005.05.008. |
[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. |
[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. |
[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. |
[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. |
[12] |
G. Eichfelder, Adaptive Scalarization Methods in Multiobjective Optimization, Springer-Verlag, Berlin, 2008.
doi: 10.1007/978-3-540-79159-1. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[26] |
J. Lin,
Maximal vectors and multi-objective optimization, J. Optim. Theory Appl., 18 (1976), 41-64.
doi: 10.1007/BF00933793. |
[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. |
[28] |
O. L. Mangasarian, Nonlinear Programming, McGraw-Hill Book Co., New York-London-Sydney, 1969. |
[29] |
K. Miettinen, Nonlinear Multiobjective Optimization. International Series in Operations Research & Management Science, Vol. 12, Kluwer Academic Publishers, Boston, MA, 2004. |
[30] |
R. E. Moore, Method and Applications of Interval Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1979. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[40] |
H. Suprajitno and I. bin Mohd,
Linear programming with interval arithmetic, Int. J. Contemp. Math. Sci., 5 (2010), 323-332.
|
[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. |
[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. |
[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. |
[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. |
[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. |
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