In this paper, we focus our study on a multiobjective variational control problem and establish sufficient optimality conditions under the assumptions of
Citation: |
[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.![]() ![]() ![]() |