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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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[21]

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

[22]

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

[23]

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

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[21]

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

[22]

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

[23]

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

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