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March  2017, 6(1): 93-109. doi: 10.3934/eect.2017006

Sufficiency and mixed type duality for multiobjective variational control problems involving α-V-univexity

Sarita Sharma 1,†,, , Anurag Jayswal 2, and  Sarita Choudhury 2,

1. 

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247 667, India
2. 

Department of Applied Mathematics, Indian School of Mines †, Dhanbad-826 004, Jharkhand, India

* Corresponding author: Sarita Sharma

Indian School of Mines has now been renamed as Indian Institute of Technology (Indian School of Mines)

Received  July 2015 Revised  July 2016 Published  December 2016

In this paper, we focus our study on a multiobjective variational control problem and establish sufficient optimality conditions under the assumptions of α-V-univex function. Furthermore, mixed type duality results are also discussed under the aforesaid assumption in order to relate the primal and dual problems. Examples are given to show the existence of α-V-univex function and to elucidate duality result.

Keywords: Multiobjective control problem, α-V-univexity, sufficiency, mixed type duality.
Mathematics Subject Classification: 90C30, 90C11, 90C20, 90C26.
Citation: Sarita Sharma, Anurag Jayswal, Sarita Choudhury. Sufficiency and mixed type duality for multiobjective variational control problems involving α-V-univexity. Evolution Equations & Control Theory, 2017, 6 (1) : 93-109. doi: 10.3934/eect.2017006
References:
[1]

I. Ahmad and T. R. Gulati, Mixed type duality for multiobjective variational problems with generalized (F, ρ)-convexity, J. Math. Anal. Appl., 306 (2005), 669-683.  doi: 10.1016/j.jmaa.2004.10.019.  Google Scholar

[2]

I. Ahmad and S. Sharma, Sufficiency and duality for multiobjective variational control problems with generalized (F, α, ρ, θ)-V-convexity, Nonlinear Anal., 72 (2010), 2564-2579.  doi: 10.1016/j.na.2009.11.005.  Google Scholar

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C. R. Bector, S. K. Suneja and S. Gupta, Univex functions and univex nonlinear programming, Proc. Admin. Sci. Asso. Canada, (1992), 115-124.  doi: 10.1007/978-3-642-46802-5_1.  Google Scholar

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B. D. Craven, Mathematical Programming and Control Theory Chapman & Hall, London, 1978.  Google Scholar

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D. Bhatia and P. Kumar, Multiobjective control problem with generalized invexity, J. Math. Anal. Appl., 189 (1995), 676-692.  doi: 10.1006/jmaa.1995.1045.  Google Scholar

[6]

T. R. Gulati, I. Husain and A. Ahmed, Optimality conditions and duality for multiobjective control problems, J. Appl. Anal., 11 (2005), 225-245.  doi: 10.1515/JAA.2005.225.  Google Scholar

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M. A. Hanson, Bounds for functionally convex optimal control problems, J. Math. Anal. Appl., 8 (1964), 84-89.  doi: 10.1016/0022-247X(64)90086-1.  Google Scholar

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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.  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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N. Kailey and S. K. Gupta, Duality for a class of symmetric nondifferentiable multiobjective fractional variational problems with generalized (F, α, ρ, d)-convexity, Math. Comput. Model., 57 (2013), 1453-1465.  doi: 10.1016/j.mcm.2012.12.007.  Google Scholar

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K. Khazafi and N. Rueda, Multiobjective variational programming under generalized type-Ⅰ univexity, J. Optim. Theory Appl., 142 (2009), 363-376.  doi: 10.1007/s10957-009-9526-3.  Google Scholar

[12]

K. Khazafi, N. Rueda and P. Enflo, Sufficiency and duality for multiobjective control problems under generalized (B, ρ)-type Ⅰ functions, J. Global Optim., 46 (2010), 111-132.  doi: 10.1007/s10898-009-9412-4.  Google Scholar

[13]

Z. A. Liang, H. X. Huang and P. M. Pardalos, Optimality conditions and duality for a class of nonlinear fractional programming problems, J. Optim. Theory Appl., 110 (2001), 611-619.  doi: 10.1023/A:1017540412396.  Google Scholar

[14]

Z. A. Liang, H. X. Huang and P. M. Pardalos, Efficient conditions and duality for a class of multiobjective programming problems, J. Global Optim., 27 (2003), 447-471.  doi: 10.1023/A:1026041403408.  Google Scholar

[15]

B. Mond and I. Smart, Duality and sufficiency in control problems with invexity, J. Math. Anal. Appl., 136 (1988), 325-333.  doi: 10.1016/0022-247X(88)90135-7.  Google Scholar

[16]

C. Nahak and S. Nanda, On efficiency and duality for multiobjective variational control problems with (F, $ρ$)-convexity, J. Math. Anal. Appl., 209 (1997), 415-434.  doi: 10.1006/jmaa.1997.5332.  Google Scholar

[17]

C. Nahak and S. Nanda, Sufficient optimality criteria and duality for multiobjective variational control problems with $V$-invexity, Nonlinear Anal., 66 (2007), 1513-1525.  doi: 10.1016/j.na.2006.02.006.  Google Scholar

[18]

M. A. Noor, On generalized preinvex functions and monotonicities, J. Inequal. Pure Appl. Math. \textbf{5} (2004), Article 110, 9 pp. (electronic).  Google Scholar

[19]

V. Preda, I. M. Minasian, M. Beldiman and A. M. Stancu, Generalized V-univexity type Ⅰ for multiobjective programming with $n$-set functions, J. Global Optim., 44 (2009), 131-148.  doi: 10.1007/s10898-008-9315-9.  Google Scholar

[20]

R. T. Rockafellar, Conjugate convex functions in optimal control and the calculus of variations, J. Math. Anal. Appl., 32 (1970), 174-222.  doi: 10.1016/0022-247X(70)90324-0.  Google Scholar

[21]

R. T. Rockafellar, Convex integral functionals and duality, in Contributions to Nonlinear Functional Analysis (E. Zarantonello, ed.), Academic Press, (1971), 215-236.   Google Scholar

[22]

S. Sharma, Duality for higher order variational control programming problems Int. Trans. Oper. Res. (2015). doi: 10.1111/itor.12192.  Google Scholar

[23]

Z. Xu, Mixed type duality in multiobjective programming problems, J. Math. Anal. Appl., 198 (1996), 621-635.  doi: 10.1006/jmaa.1996.0103.  Google Scholar

show all references

References:
[1]

I. Ahmad and T. R. Gulati, Mixed type duality for multiobjective variational problems with generalized (F, ρ)-convexity, J. Math. Anal. Appl., 306 (2005), 669-683.  doi: 10.1016/j.jmaa.2004.10.019.  Google Scholar

[2]

I. Ahmad and S. Sharma, Sufficiency and duality for multiobjective variational control problems with generalized (F, α, ρ, θ)-V-convexity, Nonlinear Anal., 72 (2010), 2564-2579.  doi: 10.1016/j.na.2009.11.005.  Google Scholar

[3]

C. R. Bector, S. K. Suneja and S. Gupta, Univex functions and univex nonlinear programming, Proc. Admin. Sci. Asso. Canada, (1992), 115-124.  doi: 10.1007/978-3-642-46802-5_1.  Google Scholar

[4]

B. D. Craven, Mathematical Programming and Control Theory Chapman & Hall, London, 1978.  Google Scholar

[5]

D. Bhatia and P. Kumar, Multiobjective control problem with generalized invexity, J. Math. Anal. Appl., 189 (1995), 676-692.  doi: 10.1006/jmaa.1995.1045.  Google Scholar

[6]

T. R. Gulati, I. Husain and A. Ahmed, Optimality conditions and duality for multiobjective control problems, J. Appl. Anal., 11 (2005), 225-245.  doi: 10.1515/JAA.2005.225.  Google Scholar

[7]

M. A. Hanson, Bounds for functionally convex optimal control problems, J. Math. Anal. Appl., 8 (1964), 84-89.  doi: 10.1016/0022-247X(64)90086-1.  Google Scholar

[8]

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.  Google Scholar

[9]

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

[10]

N. Kailey and S. K. Gupta, Duality for a class of symmetric nondifferentiable multiobjective fractional variational problems with generalized (F, α, ρ, d)-convexity, Math. Comput. Model., 57 (2013), 1453-1465.  doi: 10.1016/j.mcm.2012.12.007.  Google Scholar

[11]

K. Khazafi and N. Rueda, Multiobjective variational programming under generalized type-Ⅰ univexity, J. Optim. Theory Appl., 142 (2009), 363-376.  doi: 10.1007/s10957-009-9526-3.  Google Scholar

[12]

K. Khazafi, N. Rueda and P. Enflo, Sufficiency and duality for multiobjective control problems under generalized (B, ρ)-type Ⅰ functions, J. Global Optim., 46 (2010), 111-132.  doi: 10.1007/s10898-009-9412-4.  Google Scholar

[13]

Z. A. Liang, H. X. Huang and P. M. Pardalos, Optimality conditions and duality for a class of nonlinear fractional programming problems, J. Optim. Theory Appl., 110 (2001), 611-619.  doi: 10.1023/A:1017540412396.  Google Scholar

[14]

Z. A. Liang, H. X. Huang and P. M. Pardalos, Efficient conditions and duality for a class of multiobjective programming problems, J. Global Optim., 27 (2003), 447-471.  doi: 10.1023/A:1026041403408.  Google Scholar

[15]

B. Mond and I. Smart, Duality and sufficiency in control problems with invexity, J. Math. Anal. Appl., 136 (1988), 325-333.  doi: 10.1016/0022-247X(88)90135-7.  Google Scholar

[16]

C. Nahak and S. Nanda, On efficiency and duality for multiobjective variational control problems with (F, $ρ$)-convexity, J. Math. Anal. Appl., 209 (1997), 415-434.  doi: 10.1006/jmaa.1997.5332.  Google Scholar

[17]

C. Nahak and S. Nanda, Sufficient optimality criteria and duality for multiobjective variational control problems with $V$-invexity, Nonlinear Anal., 66 (2007), 1513-1525.  doi: 10.1016/j.na.2006.02.006.  Google Scholar

[18]

M. A. Noor, On generalized preinvex functions and monotonicities, J. Inequal. Pure Appl. Math. \textbf{5} (2004), Article 110, 9 pp. (electronic).  Google Scholar

[19]

V. Preda, I. M. Minasian, M. Beldiman and A. M. Stancu, Generalized V-univexity type Ⅰ for multiobjective programming with $n$-set functions, J. Global Optim., 44 (2009), 131-148.  doi: 10.1007/s10898-008-9315-9.  Google Scholar

[20]

R. T. Rockafellar, Conjugate convex functions in optimal control and the calculus of variations, J. Math. Anal. Appl., 32 (1970), 174-222.  doi: 10.1016/0022-247X(70)90324-0.  Google Scholar

[21]

R. T. Rockafellar, Convex integral functionals and duality, in Contributions to Nonlinear Functional Analysis (E. Zarantonello, ed.), Academic Press, (1971), 215-236.   Google Scholar

[22]

S. Sharma, Duality for higher order variational control programming problems Int. Trans. Oper. Res. (2015). doi: 10.1111/itor.12192.  Google Scholar

[23]

Z. Xu, Mixed type duality in multiobjective programming problems, J. Math. Anal. Appl., 198 (1996), 621-635.  doi: 10.1006/jmaa.1996.0103.  Google Scholar

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