American Institute of Mathematical Sciences

Advanced
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 and 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.

[1]

Xinmin Yang, Xiaoqi Yang. A note on mixed type converse duality in multiobjective programming problems. Journal of Industrial and Management Optimization, 2010, 6 (3) : 497-500. doi: 10.3934/jimo.2010.6.497
[2]

Xian-Jun Long, Jing Quan. Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 361-370. doi: 10.3934/naco.2011.1.361
[3]

Victor Kozyakin. Polynomial reformulation of the Kuo criteria for v- sufficiency of map-germs. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 587-602. doi: 10.3934/dcdsb.2010.14.587
[4]

Nazih Abderrazzak Gadhi, Fatima Zahra Rahou. Sufficient optimality conditions and Mond-Weir duality results for a fractional multiobjective optimization problem. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021216
[5]

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
[6]

Najeeb Abdulaleem. $ V $-$ E $-invexity in $ E $-differentiable multiobjective programming. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 427-443. doi: 10.3934/naco.2021014
[7]

Najeeb Abdulaleem. Optimality and duality for $ E $-differentiable multiobjective programming problems involving $ E $-type Ⅰ functions. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022004
[8]

Xinmin Yang. On second order symmetric duality in nondifferentiable multiobjective programming. Journal of Industrial and Management Optimization, 2009, 5 (4) : 697-703. doi: 10.3934/jimo.2009.5.697
[9]

Xinmin Yang, Xiaoqi Yang, Kok Lay Teo. Higher-order symmetric duality in multiobjective programming with invexity. Journal of Industrial and Management Optimization, 2008, 4 (2) : 385-391. doi: 10.3934/jimo.2008.4.385
[10]

Xinmin Yang, Jin Yang, Heung Wing Joseph Lee. Strong duality theorem for multiobjective higher order nondifferentiable symmetric dual programs. Journal of Industrial and Management Optimization, 2013, 9 (3) : 525-530. doi: 10.3934/jimo.2013.9.525
[11]

Liping Tang, Xinmin Yang, Ying Gao. Higher-order symmetric duality for multiobjective programming with cone constraints. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1873-1884. doi: 10.3934/jimo.2019033
[12]

Xian-Jun Long, Nan-Jing Huang, Zhi-Bin Liu. Optimality conditions, duality and saddle points for nondifferentiable multiobjective fractional programs. Journal of Industrial and Management Optimization, 2008, 4 (2) : 287-298. doi: 10.3934/jimo.2008.4.287
[13]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems and Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355
[14]

Lars Grüne, Marleen Stieler. Multiobjective model predictive control for stabilizing cost criteria. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3905-3928. doi: 10.3934/dcdsb.2018336
[15]

Yibing Lv, Zhongping Wan. Linear bilevel multiobjective optimization problem: Penalty approach. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1213-1223. doi: 10.3934/jimo.2018092
[16]

Regina Sandra Burachik, Alex Rubinov. On the absence of duality gap for Lagrange-type functions. Journal of Industrial and Management Optimization, 2005, 1 (1) : 33-38. doi: 10.3934/jimo.2005.1.33
[17]

Maria do Rosário de Pinho, Ilya Shvartsman. Lipschitz continuity of optimal control and Lagrange multipliers in a problem with mixed and pure state constraints. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 505-522. doi: 10.3934/dcds.2011.29.505
[18]

Georg Vossen, Torsten Hermanns. On an optimal control problem in laser cutting with mixed finite-/infinite-dimensional constraints. Journal of Industrial and Management Optimization, 2014, 10 (2) : 503-519. doi: 10.3934/jimo.2014.10.503
[19]

Annamaria Barbagallo, Rosalba Di Vincenzo, Stéphane Pia. On strong Lagrange duality for weighted traffic equilibrium problem. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1097-1113. doi: 10.3934/dcds.2011.31.1097
[20]

Xinmin Yang. On symmetric and self duality in vector optimization problem. Journal of Industrial and Management Optimization, 2011, 7 (3) : 523-529. doi: 10.3934/jimo.2011.7.523

2021 Impact Factor: 1.169

Tools

Metrics

  • PDF downloads (187)
  • HTML views (144)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy