On second order duality for nondifferentiable minimax fractional programming problems involving type-I functions

We introduce second order $(C,\alpha ,\rho ,d)$ type-I functions and formulate a second order dual model for a nondifferentiable minimax fractional programming problem. The usual duality relations are established under second order $(F,\alpha ,\rho ,d)/(C,\alpha ,\rho ,d)$ type-I assumptions. By citing a nontrivial example, it is shown that a second order $(C,\alpha ,\rho ,d)$ type-I function need not be $(F,\alpha ,\rho ,d)$ type-I. Several known results are obtained as special cases. References Ahmad, I., Husain, Z., Optimality conditions and duality in nondifferentiable minimax fractional programming with generalized convexity. J. Optimiz. Theory Appl. 129:255–275, 2006. doi:10.1007/s10957-006-9057-0 Ahmad, I., Husain, Z., Sharma, S., Second-order duality in nondifferentiable minmax programming involving type-I functions. J. Comput. Appl. Math. 215:91–102, 2008. doi:10.1016/j.cam.2007.03.022 Antczak, T., Generalized fractional minimax programming with $B$-$(p, r)$-invexity. Comput. Math. Appl. 56:1505–1525, 2008. doi:10.1016/j.camwa.2008.02.039 Chinchuluun, A., Yuan, D. H., Pardalos, P. M., Optimality conditions and duality for nondifferentiable multiobjective fractional programming with generalized convexity. Ann. Oper. Res. 154:133–147, 2007. doi:10.1007/s10479-007-0180-6 Du, D.-Z., Pardalos, P. M., Minimax and applications, Kluwer Academic Publishers, Dordrecht, 1995. http://vlsicad.eecs.umich.edu/BK/Slots/cache/www.wkap.nl/prod/b/0-7923-3615-1 Hachimi, M., Aghezzaf, B., Second order duality in multiobjective programming involving generalized type I functions. Numer. Funct. Anal. Optimiz. 25:725–736, 2005. doi:10.1081/NFA-200045804 Husain, Z., Ahmad, I., Sharma, S., Second order duality for minmax fractional programming. Optimiz. Lett. 3:277–286, 2009. doi:10.1007/s11590-008-0107-4 Hu, Q., Yang, G., Jian, J., On second order duality for minimax fractional programming. Nonlinear Anal. 12:3509–3514, 2011. doi:10.1016/j.nonrwa.2011.06.011 Lai, H. C., Lee, J. C., On duality theorems for a nondifferentiable minimax fractional programming. J. Comput. Appl. Math. 146:115–126, 2002. doi:10.1016/S0377-0427(02)00422-3 Lai, H. C., Liu, J. C., Tanaka, K., Necessary and sufficient conditions for minimax fractional programming. J. Math. Anal. Appl. 230:311–328, 1999. doi:10.1006/jmaa.1998.6204 Liu, J. C., Wu, C. S., On minimax fractional optimality conditions with invexity. J. Math. Anal. Appl. 219:21–35, 1998. doi:10.1006/jmaa.1997.5786 Long, X. J., Optimality conditions and duality for nondifferentiable multiobjective fractional programming problems with $(C,\alpha ,\rho ,d)$-convexity. J. Optimiz. Theory Appl. 148:197–208, 2011. doi:10.1007/s10957-010-9740-z Schmitendorf, W. E., Necessary conditions and sufficient conditions for static minmax problems. J. Math. Anal. Appl. 57:683–693, 1977. doi:10.1016/0022-247X(77)90255-4 Sharma, S., Gulati, T. R., Second order duality in minmax fractional programming with generalized univexity. J. Glob. Optimiz. 52:161–169, 2012. doi:10.1007/s10898-011-9694-1 Yuan, D. H., Liu, X. L., Chinchuluun, A., Pardalos, P. M., Nondifferentiable minimax fractional programming problems with $(C,\alpha , \rho , d)$-convexity. J. Optimiz. Theory Appl. 129:185–199, 2006. doi:10.1007/s10957-006-9052-

On Higher-order Duality in Nondifferentiable Minimax Fractional Programming

In this paper, we consider a nondifferentiable minimax fractional programming problem with continuously differentiable functions and formulated two types of higher-order dual models for such optimization problem.Weak, strong and strict converse duality theorems are derived under higherorder generalized invexity

On Minimax Fractional Optimality Conditions with Invexity

AbstractUnder different forms of invexity conditions, sufficient Kuhn–Tucker conditions and three dual models are presented for the minimax fractional programming

Minmax fractional programming problem involving generalized convex functions

In the present study we focus our attention on a minmax fractional programming problem and its second order dual problem. Duality results are obtained for the considered dual problem under the assumptions of second order $\left( {F,\alpha ,\rho ,d}\right)$ -type I functions

Bibliography on Nondifferentiable Optimization

This is a research bibliography with all the advantages and shortcomings that this implies. The author has used it as a bibliographical data base when writing papers, and it is therefore largely a reflection of his own personal research interests. However, it is hoped that this bibliography will nevertheless be of use to others interested in nondifferentiable optimization

Duality in mathematical programming.

In this thesis entitled, “Duality in Mathematical Programming”, the emphasis is given on formulation and conceptualization of the concepts of second-order duality, second-order mixed duality, second-order symmetric duality in a variety of nondifferentiable nonlinear programming under suitable second-order convexity/second-order invexity and generalized second-order convexity / generalized second-order invexity. Throughout the thesis nondifferentiablity occurs due to square root function and support functions. A support function which is more general than square root of a positive definite quadratic form. This thesis also addresses second-order duality in variational problems under suitable second-order invexity/secondorder generalized invexity. The duality results obtained for the variational problems are shown to be a dynamic generalization for thesis of nonlinear programming problem.Digital copy of Thesis.University of Kashmir