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April  2019, 15(2): 647-665. doi: 10.3934/jimo.2018063

Sufficiency and duality in non-smooth interval valued programming problems

Deepak Singh 1, , Bilal Ahmad Dar 2, and  Do Sang Kim 3,,

1. 

Department of Applied Sciences, NITTTR (under Ministry of HRD, Govt. of India), Bhopal, M.P., India
2. 

Department of Mathematics, Rajiv Gandhi Proudyogiki Vishwavidyalaya, (State Technological University of M.P.), Bhopal, M.P., India
3. 

Department of Applied Mathematics, Pukyong National University, Busan, Korea

* Corresponding author: Do Sang Kim

Received  October 2015 Revised  March 2018 Published  April 2019 Early access  June 2018

In this paper a non-smooth optimization problem is studied in an uncertain environment. The objective function of this problem is interval valued function. We introduce the class of $LU-(p,r)-[ρ^L,ρ^U]-(η, θ)$-invex interval valued functions about the Clarke generalized gradient. Then, through non trivial examples, we illustrate that the class of functions introduced exists. Based upon the proposed invexity assumptions, the sufficient optimality conditions are established. Further, we derive weak, strong and strict converse duality theorems for Mond-Weir type and Wolfe type dual programs. Some examples are also given in order to illustrate our results.

Keywords: Interval valued functions, LU-optimal solutions, Clarke generalized gradient, $LU-(p,r)-[ρ^L,ρ^U]-(η, θ)$-invex functions, sufficiency, duality.
Mathematics Subject Classification: Primary: 90C30, 90C46; Secondary: 49N15.
Citation: 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
References:
[1]

I. Ahmad, A. Jayswal and J. Banerjee, On interval-valued optimization problems with generalized invex functions, J. Inequal. Appl., 2013 (2013), 14pp. doi: 10.1186/1029-242X-2013-313.

[2]

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

[3]

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

[4]

T. Antczak, $(p, r)$-invex sets and functions, J. Math. Anal. Appl., 263 (2001), 355-379.  doi: 10.1006/jmaa.2001.7574.

[5]

A. Ben-Israel and P. D. Robers, A decomposition method for interval linear programming, Manage. Sci., 16 (1969/1970), 374-387.  doi: 10.1287/mnsc.16.5.374.

[6]

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

[7]

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. Ma., 12 (2013), 305-322.  doi: 10.1007/s10700-013-9156-y.

[8]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley-Interscience, New-York, 1983.

[9]

M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl., 80 (1981), 545-550.  doi: 10.1016/0022-247X(81)90123-2.

[10]

H. Ishibuchi and H. Tanaka, Multiobjective programming in optimization of interval valued objective functions, Eur. J. Oper. Res., 48 (1990), 219-225. 

[11]

M. Jana and G. Panda, Solution of nonlinear interval vector optimization problem, Oper. Res. Int. J., 14 (2014), 71-85. 

[12]

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.

[13]

A. Jayswal, A. K. Prasad and I. Stancu-Minasian, On nonsmooth multiobjective fractional programming problems involving $(p, r)-ρ- (η, θ)$ -invex functions, Yugosl. J. Oper. Res., 23 (2013), 367-386.  doi: 10.2298/YJOR130131012J.

[14]

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

[15]

P. Mandal and C. Nahak, Symmetric duality with $(p, r)-ρ-(η, θ)$-invexity, Appl. Math. Comput., 217 (2011), 8141-8148.  doi: 10.1016/j.amc.2011.02.068.

[16]

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

[17]

Y. Sun and L. Wang, Optimality conditions and duality in nondifferentiable interval valued programming, J. Ind. Manag. Optim., 9 (2013), 131-142.  doi: 10.3934/jimo.2013.9.131.

[18]

Y. Sun and L. Wang, Mond Weir's type duality for interval valued programming, Computer Science and Automation Engineering (CSAE), IEEE International Conference, 3 (2012), 27-30.  doi: 10.1109/CSAE.2012.6272900.

[19]

Y. Sun, X. Xu and L. Wang, Duality and saddle point type optimality for interval valued programming, Optim. Lett., 8 (2014), 1077-1091.  doi: 10.1007/s11590-013-0640-7.

[20]

H. C. Wu, The Karush Kuhn Tuker optimality conditions in an optimization problem with interval valued objective functions, Eur. J. Oper. Res., 176 (2007), 46-59.  doi: 10.1016/j.ejor.2005.09.007.

[21]

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.

[22]

H. C. Wu, Wolfe duality for interval valued optimization, J. Optimiz. Theory App., 138 (2008), 497-509.  doi: 10.1007/s10957-008-9396-0.

[23]

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

[24]

G. J. Zalmai, Generalized sufficiency criteria in continuous-time programming with application to a class of variational-type inequalities, J. Math. Anal. Appl., 153 (1990), 331-355.  doi: 10.1016/0022-247X(90)90217-4.

[25]

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.

[26]

J. Zhang, Optimality condition and wolfe duality for invex interval-valued nonlinear programming problems, J. Appl. Math., (2013), Article ID 641345, 11 pages.

[27]

H. C. Zhou and Y. J. Wang, Optimality conditions and mixed duality for interval valued optimization, Fuzzy Info. and Eng., 2 (2009), 1315-1323.  doi: 10.1007/978-3-642-03664-4_140.

show all references

References:
[1]

I. Ahmad, A. Jayswal and J. Banerjee, On interval-valued optimization problems with generalized invex functions, J. Inequal. Appl., 2013 (2013), 14pp. doi: 10.1186/1029-242X-2013-313.

[2]

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

[3]

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

[4]

T. Antczak, $(p, r)$-invex sets and functions, J. Math. Anal. Appl., 263 (2001), 355-379.  doi: 10.1006/jmaa.2001.7574.

[5]

A. Ben-Israel and P. D. Robers, A decomposition method for interval linear programming, Manage. Sci., 16 (1969/1970), 374-387.  doi: 10.1287/mnsc.16.5.374.

[6]

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

[7]

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. Ma., 12 (2013), 305-322.  doi: 10.1007/s10700-013-9156-y.

[8]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley-Interscience, New-York, 1983.

[9]

M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl., 80 (1981), 545-550.  doi: 10.1016/0022-247X(81)90123-2.

[10]

H. Ishibuchi and H. Tanaka, Multiobjective programming in optimization of interval valued objective functions, Eur. J. Oper. Res., 48 (1990), 219-225. 

[11]

M. Jana and G. Panda, Solution of nonlinear interval vector optimization problem, Oper. Res. Int. J., 14 (2014), 71-85. 

[12]

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.

[13]

A. Jayswal, A. K. Prasad and I. Stancu-Minasian, On nonsmooth multiobjective fractional programming problems involving $(p, r)-ρ- (η, θ)$ -invex functions, Yugosl. J. Oper. Res., 23 (2013), 367-386.  doi: 10.2298/YJOR130131012J.

[14]

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

[15]

P. Mandal and C. Nahak, Symmetric duality with $(p, r)-ρ-(η, θ)$-invexity, Appl. Math. Comput., 217 (2011), 8141-8148.  doi: 10.1016/j.amc.2011.02.068.

[16]

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

[17]

Y. Sun and L. Wang, Optimality conditions and duality in nondifferentiable interval valued programming, J. Ind. Manag. Optim., 9 (2013), 131-142.  doi: 10.3934/jimo.2013.9.131.

[18]

Y. Sun and L. Wang, Mond Weir's type duality for interval valued programming, Computer Science and Automation Engineering (CSAE), IEEE International Conference, 3 (2012), 27-30.  doi: 10.1109/CSAE.2012.6272900.

[19]

Y. Sun, X. Xu and L. Wang, Duality and saddle point type optimality for interval valued programming, Optim. Lett., 8 (2014), 1077-1091.  doi: 10.1007/s11590-013-0640-7.

[20]

H. C. Wu, The Karush Kuhn Tuker optimality conditions in an optimization problem with interval valued objective functions, Eur. J. Oper. Res., 176 (2007), 46-59.  doi: 10.1016/j.ejor.2005.09.007.

[21]

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.

[22]

H. C. Wu, Wolfe duality for interval valued optimization, J. Optimiz. Theory App., 138 (2008), 497-509.  doi: 10.1007/s10957-008-9396-0.

[23]

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

[24]

G. J. Zalmai, Generalized sufficiency criteria in continuous-time programming with application to a class of variational-type inequalities, J. Math. Anal. Appl., 153 (1990), 331-355.  doi: 10.1016/0022-247X(90)90217-4.

[25]

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.

[26]

J. Zhang, Optimality condition and wolfe duality for invex interval-valued nonlinear programming problems, J. Appl. Math., (2013), Article ID 641345, 11 pages.

[27]

H. C. Zhou and Y. J. Wang, Optimality conditions and mixed duality for interval valued optimization, Fuzzy Info. and Eng., 2 (2009), 1315-1323.  doi: 10.1007/978-3-642-03664-4_140.

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Table 1.  Summary of Example 7.1 and Remark 7.2
Functions Valued of $\rho$ Domain
$f^L(x)$ $\rho^L=\frac{-0.1}{|x+0.01|^4}$ $x\in (-0.4599,1.0766)$
$f^U(x)$ $\rho^U=\frac{-0.09}{|x+0.01|^4}$ $x\in (-0.2362,1.1438)$
$\sum_{j=1}^2\mu_jg_j(x)$ $\rho=4$ $x\in (-0.5172,0.5172)$
$[f^L(x),f^U(x)]$ $[\rho^L,\rho^U]=\left[\frac{-0.1}{|x+0.01|^4},\frac{-0.09}{|x+0.01|^4}\right]$ $x\in(-0.4599,1.0766)\cap (-0.2362,1.1438)$
$f^L_0(x)$ $\rho^L_0=\frac{-0.13}{|x+0.01|^4}$ $x\in(-1.0567,1.0567)$
$[f^L_0(x),f^U(x)]$ $[\rho^L_0,\rho^U]=\left[\frac{-0.13}{|x+0.01|^4},\frac{-0.09}{|x+0.01|^4}\right]$ $x\in(-1.0567,1.0567)\cap(-0.2362,1.1438)$
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Functions Valued of $\rho$ Domain
$f^L(x)$ $\rho^L=\frac{-0.1}{|x+0.01|^4}$ $x\in (-0.4599,1.0766)$
$f^U(x)$ $\rho^U=\frac{-0.09}{|x+0.01|^4}$ $x\in (-0.2362,1.1438)$
$\sum_{j=1}^2\mu_jg_j(x)$ $\rho=4$ $x\in (-0.5172,0.5172)$
$[f^L(x),f^U(x)]$ $[\rho^L,\rho^U]=\left[\frac{-0.1}{|x+0.01|^4},\frac{-0.09}{|x+0.01|^4}\right]$ $x\in(-0.4599,1.0766)\cap (-0.2362,1.1438)$
$f^L_0(x)$ $\rho^L_0=\frac{-0.13}{|x+0.01|^4}$ $x\in(-1.0567,1.0567)$
$[f^L_0(x),f^U(x)]$ $[\rho^L_0,\rho^U]=\left[\frac{-0.13}{|x+0.01|^4},\frac{-0.09}{|x+0.01|^4}\right]$ $x\in(-1.0567,1.0567)\cap(-0.2362,1.1438)$
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