Multiobjective Fractional Programming Problems and Second Order Generalized Hybrid Invexity Frameworks
Abstract
In this paper, first generalized sufficient efficiency conditions for multiobjective fractional programming based on the generalized hybrid invexities are developed , and then efficient solutions to multiobjective fractional programming problems are established. The obtained results generalize and unify a wide range of investigations in the literature.References
A. Ben-Israel and B. Mond, What is the invexity? Journal of Australian Mathematical Society Ser. B 28 (1986), 1 - 9.
L. Caiping and Y. Xinmin, Generalized (ρ,θ,η)−invariant monotonicity and generalized (ρ,θ,η)−invexity of non-differentiable functions, Journal of Inequalities and Applications Vol. 2009(2009), Article ID # 393940, 16 pages.
M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, Journal of Mathematical Analysis and Applications 80 (1981), 545 - 550.
V. Jeyakumar, Strong and weak invexity in mathematical programming, Methods Oper. Res.
(1985), 109–125.
H. Kawasaki, Second-order necessary conditions of the Kuhn-Tucker type under new constraint qualifications, Journal of Optimization Theory and Applications 57 (2) (1988), 253 - 264.
M.H.Kim,G.S.KimandG.M.Lee,On ϵ−optimality conditions for multiobjective fractional optimization problems, Fixed Point Theory and Applications 2011:6 doi:10.1186/1687-1812-2011-6.
J. C. Liu, Second order duality for minimax programming, Utilitas Math. 56 (1999), 53 - 63.
O. L. Mangasarian, Second- and higher-order duality theorems in nonlinear programming, J. Math. Anal. Appl. 51 (1975), 607 - 620.
S. K. Mishra, Second order generalized invexity and duality in mathematical programming,
Optimization 42 (1997), 51 - 69.
S. K. Mishra, Second order symmetric duality in mathematical programming with F-convexity, European J. Oper. Res. 127 (2000), 507 - 518.
S. K. Mishra and N. G. Rueda, Higher-order generalized invexity and duality in mathematical programming, J. Math. Anal. Appl. 247 (2000), 173 - 182.
S. K. Mishra and N. G. Rueda, Second-order duality for nondifferentiable minimax programming involving generalized type I functions, J. Optim. Theory Appl. 130 (2006), 477- 486.
S. K. Mishra, V. Laha and R. U. Verma, Generalized vector variational-like inequalities and nonsmooth vector optimization of radially continuous functions, Advances in Nonlinear Variational Inequalities 14 (2)(2011), 1 - 18.
B. Mond and T. Weir, Generalized convexity and higher-order duality, J. Math. Sci. 16-18 (1981-1983), 74 - 94.
B. Mond and J. Zhang, Duality for multiobjective programming involving second-order V-invex functions, in Proceedings of the Optimization Miniconference II (B. M. Glover and V. Jeyakumar, eds.), University of New South Wales, Sydney, Australia, 1995, pp. 89 - 100.
B. Mond and J. Zhang, Higher order invexity and duality in mathemaical programming, in Generalized Convexity, Generalized Monotonicity : Recent Results (J. P. Crouzeix, et al.,eds.), Kluwer Academic Publishers, printed in the Netherlands, 1998, pp. 357 - 372.
R. B. Patel, Second order duality in multiobjective fractional programming, Indian J. Math. 38 (1997), 39 - 46.
M. K. Srivastava and M. Bhatia, Symmetric duality for multiobjective programming using second order (F,ρ)-convexity, Opsearch 43 (2006), 274 - 295.
K. K. Srivastava and M. G. Govil, Second order duality for multiobjective programming involving (F,ρ,σ)-type I functions, Opsearch 37 (2000), 316 - 326.
S. K. Suneja, C. S. Lalitha, and S. Khurana, Second order symmetric duality in multiobjective programming, European J. Oper. Res. 144 (2003), 492 - 500.
M. N. Vartak and I. Gupta, Duality theory for fractional programming problems under η-convexity, Opsearch 24 (1987), 163 - 174.
R. U. Verma, Weak ϵ− efficiency conditions for multiobjective fractional programming, Applied Mathematics and Computation 219 (2013), 6819 - 6827.
R. U. Verma, New ϵ−optimality conditions for multiobjective fractional subset programming
problems, Transactions on Mathematical Programming and Applications 1 (1)(2013), 69 - 89.
R. U. Verma, Second-order (Φ,η,ρ,θ)−invexities and parameter-free ϵ−efficiency conditions
for multiobjective discrete minmax fractional programming problems, Advances in Nonlinear Variational Inequalities 17 (1)(2014), 27 - 46.
X. M. Yang, Second order symmetric duality for nonlinear programs, Opsearch 32 (1995), 205 - 209.
X. M. Yang, On second order symmetric duality in nondifferentiable multiobjective programming, J. Ind. Manag. Optim. 5 (2009), 697 - 703.
X. M. Yang and S. H. Hou, Second-order symmetric duality in multiobjective programming, Appl. Math. Lett. 14 (2001), 587 - 592.
X. M. Yang, K. L. Teo and X. Q. Yang, Higher-order generalized convexity and duality in nondifferentiable multiobjective mathematical programming, J. Math. Anal. Appl. 29 (2004), 48 - 55.
X. M. Yang, X. Q. Yang and K. L. Teo, Nondifferentiable second order symmetric duality in mathematical programming with F-convexity, European J. Oper. Res. 144 (2003), 554 - 559.
X.M.Yang,X.Q.YangandK.L.Teo,Huard type second-order converse duality for nonlinear programming, Appl. Math. Lett. 18 (2005), 205 - 208.
X. M. Yang, X. Q. Yang and K. L. Teo, Higher-order symmetric duality in multiobjective programming with invexity, J. Ind. Manag. Optim. 4 (2008), 385 - 391.
X. M. Yang, X. Q. Yang, K. L. Teo and S. H. Hou, Second order duality for nonlinear programming, Indian J. Pure Appl. Math. 35 (2004), 699 - 708.
K. Yokoyama, Epsilon approximate solutions for multiobjective programming problems, Journal of Mathematical Analysis and Applications 203 (1) (1996), 142 - 149.
G. J. Zalmai, Global parametric sufficient optimality conditions for discrete minmax fractional
programmingproblemscontaininggeneralized (θ,η,ρ)-V-invexfunctionsandarbitrarynorms, Journal of Applied Mathematics & Computing 23 (1-2) (2007), 1 - 23.
G. J. Zalmai, Hanson-Antczak-type generalized (α,β,γ,ξ,η,ρ,θ)-V-invex functions in
semiinfinite multiobjective fractional programming, I : Sufficient efficiency conditions, Advances in Nonlinear Variational Inequalities 16 (1) (2013), 91 - 114.
G. J. Zalmai, Hanson-Antczak-type generalized (α,β,γ,ξ,η,ρ,θ)-V-invex functions in
semiinfinite multiobjective fractional programming, II : First-order parametric duality models,
Advances in Nonlinear Variational Inequalities 16 (2) (2013), 61 - 90.
G. J. Zalmai, Hanson-Antczak-type generalized (α,β,γ,ξ,η,ρ,θ)-V-invex functions in semiinfinite multiobjective fractional programming, III: Second-order parametric duality models, Advances in Nonlinear Variational Inequalities 16 (2) (2013), 91 - 126.
G. J. Zalmai and Q. Zhang, Global nonparametric sufficient optimality conditions for semi-infinite discrete minmax fractional programming problems involving generalized (ρ,θ)−invex functions, Numerical Functional Analysis and Optimization 28(1-2) (2007), 173 - 209.
J. Zhang and B. Mond, Second order b-invexity and duality in mathematical programming, Utilitas Math. 50 (1996), 19 - 31.
J. Zhang and B. Mond, Second order duality for multiobjective nonlinear programming
involving generalized convexity, in Proceedings of the Optimization Miniconference III (B. M. Glover, B. D. Craven, and D. Ralph, eds.), University of Ballarat, 1997, pp. 79 - 95.
E.Zeidler,NonlinearFunctionalAnalysisanditsApplicationsIII,Springer-Verlag,NewYork,
New York, 1985.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
