Abstract
In this paper, we establish the sufficient Karush–Kuhn–Tucker (KKT) conditions for the existence of minimizers of a set-valued minimax fractional programming problem. The duals of Mond–Weir, Wolfe, and mixed types of the said problem are also formulated and the duality results are proved.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahmad, I.: Optimality conditions and duality in fractional minimax programming involving generalized \(\rho \)-invexity. Int. J. Manag. Syst. 19, 165–180 (2003)
Ahmad, I., Husain, Z.: Optimality conditions and duality in nondifferentiable minimax fractional programming with generalized convexity. J. Optim. Theory Appl. 129(2), 255–275 (2006)
Aubin, J.P.: Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions. In: Nachbin, L. (ed.) Mathematical Analysis and Applications, Part A, pp. 160–229. Academic Press, New York (1981)
Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birhäuser, Boston (1990)
Bector, C.R., Bhatia, B.L.: Sufficient optimality conditions and duality for a minmax problem. Util. Math. 27, 229–247 (1985)
Borwein, J.: Multivalued convexity and optimization: a unified approach to inequality and equality constraints. Math. Program. 13(1), 183–199 (1977)
Chandra, S., Kumar, V.: Duality in fractional minimax programming. J. Austral. Math. Soc. (Ser. A.) 58, 376–386 (1995)
Corley, H.W.: Existence and Lagrangian duality for maximizations of set-valued functions. J. Optim. Theory Appl. 54(3), 489–501 (1987)
Das, K., Nahak, C.: Sufficient optimality conditions and duality theorems for set-valued optimization problem under generalized cone convexity. Rend. Circ. Mat. Palermo 63(3), 329–345 (2014)
Das, K., Nahak, C.: Set-valued fractional programming problems under generalized cone convexity. Opsearch 53, 157–177 (2016)
Das, K., Nahak, C.: Set-valued minimax programming problems under generalized cone convexity. Rend. Circ. Mat. Palermo II. Ser. 66(3), 361–374 (2017)
Jahn, J., Rauh, R.: Contingent epiderivatives and set-valued optimization. Math. Method Oper. Res. 46(2), 193–211 (1997)
Lai, H.C., Lee, J.C.: On duality theorems for nondifferentiable minimax fractional programming. J. Comput. Appl. Math. 146, 115–126 (2002)
Lai, H.C., Liu, J.C., Tanaka, K.: Necessary and sufficient conditions for minimax fractional programming. J. Math. Anal. Appl. 230, 311–328 (1999)
Liang, Z.A., Shi, Z.W.: Optimality conditions and duality for minimax fractional programming with generalized convexity. J. Math. Anal. Appl. 277, 474–488 (2003)
Liu, J.C., Wu, C.S.: On minimax fractional optimality conditions with \((\rm F, \rho )\)-convexity. J. Math. Anal. Appl. 219, 36–51 (1998)
Weir, T.: Pseudoconvex minimax programming. Util. Math. 42, 234–240 (1992)
Yadav, S.R., Mukherjee, R.N.: Duality for fractional minimax programming problems. J. Austral. Math. Soc. (Ser. B.) 31, 484–492 (1990)
Zalmai, G.J.: Optimality criteria and duality for a class of minimax programming problems with generalized invexity conditions. Util. Math. 32, 35–57 (1987)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Das, K., Nahak, C. Optimality conditions for set-valued minimax fractional programming problems. SeMA 77, 161–179 (2020). https://doi.org/10.1007/s40324-019-00209-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40324-019-00209-7