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 & 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

[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

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

[1]

Xinmin Yang, Xiaoqi Yang. A note on mixed type converse duality in multiobjective programming problems. Journal of Industrial & 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 & 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 & Continuous Dynamical Systems - B, 2010, 14 (2) : 587-602. doi: 10.3934/dcdsb.2010.14.587
[4]

Deepak Singh, Bilal Ahmad Dar, Do Sang Kim. Sufficiency and duality in non-smooth interval valued programming problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 647-665. doi: 10.3934/jimo.2018063
[5]

Xinmin Yang. On second order symmetric duality in nondifferentiable multiobjective programming. Journal of Industrial & Management Optimization, 2009, 5 (4) : 697-703. doi: 10.3934/jimo.2009.5.697
[6]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355
[7]

Xinmin Yang, Xiaoqi Yang, Kok Lay Teo. Higher-order symmetric duality in multiobjective programming with invexity. Journal of Industrial & Management Optimization, 2008, 4 (2) : 385-391. doi: 10.3934/jimo.2008.4.385
[8]

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

Xian-Jun Long, Nan-Jing Huang, Zhi-Bin Liu. Optimality conditions, duality and saddle points for nondifferentiable multiobjective fractional programs. Journal of Industrial & Management Optimization, 2008, 4 (2) : 287-298. doi: 10.3934/jimo.2008.4.287
[10]

Liping Tang, Xinmin Yang, Ying Gao. Higher-order symmetric duality for multiobjective programming with cone constraints. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-12. doi: 10.3934/jimo.2019033
[11]

Lars Grüne, Marleen Stieler. Multiobjective model predictive control for stabilizing cost criteria. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3905-3928. doi: 10.3934/dcdsb.2018336
[12]

Regina Sandra Burachik, Alex Rubinov. On the absence of duality gap for Lagrange-type functions. Journal of Industrial & Management Optimization, 2005, 1 (1) : 33-38. doi: 10.3934/jimo.2005.1.33
[13]

Yibing Lv, Zhongping Wan. Linear bilevel multiobjective optimization problem: Penalty approach. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1213-1223. doi: 10.3934/jimo.2018092
[14]

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 & Continuous Dynamical Systems - A, 2011, 29 (2) : 505-522. doi: 10.3934/dcds.2011.29.505
[15]

Georg Vossen, Torsten Hermanns. On an optimal control problem in laser cutting with mixed finite-/infinite-dimensional constraints. Journal of Industrial & Management Optimization, 2014, 10 (2) : 503-519. doi: 10.3934/jimo.2014.10.503
[16]

Leszek Gasiński. Optimal control problem of Bolza-type for evolution hemivariational inequality. Conference Publications, 2003, 2003 (Special) : 320-326. doi: 10.3934/proc.2003.2003.320
[17]

Annamaria Barbagallo, Rosalba Di Vincenzo, Stéphane Pia. On strong Lagrange duality for weighted traffic equilibrium problem. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1097-1113. doi: 10.3934/dcds.2011.31.1097
[18]

Xinmin Yang. On symmetric and self duality in vector optimization problem. Journal of Industrial & Management Optimization, 2011, 7 (3) : 523-529. doi: 10.3934/jimo.2011.7.523
[19]

Naïla Hayek. Infinite-horizon multiobjective optimal control problems for bounded processes. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1121-1141. doi: 10.3934/dcdss.2018064
[20]

Hongyu Liu, Jun Zou. Uniqueness in determining multiple polygonal scatterers of mixed type. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 375-396. doi: 10.3934/dcdsb.2008.9.375

2018 Impact Factor: 1.048

Tools

Metrics

  • PDF downloads (12)
  • HTML views (98)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences