The advantage of second-order duality is that if a feasible point of the primal is given and first-order duality conditions are not applicable (infeasible), then we may use second-order duality to provide a lower bound for the value of primal problem. Consequently, it is quite interesting to discuss the duality results for the case of second order. Thus, we focus our study on a discussion of duality relationships of a minimax fractional programming problem under the assumptions of second order B-(p, r)-invexity. Weak, strong and strict converse duality theorems are established in order to relate the primal and dual problems under the assumptions. An example of a non trivial function has been given to show the existence of second order B-(p, r)-invex functions.
The research of Dr. Vikas Sharma is supported by Thapar University, Patiala under Seed Money Project no. TU/DORSP/57/581. He gratefully acknowledges the support provided by the Thapar University to carry out this research.
Ahmad, I. (2013). Second order nondifferentiable minimax fractional programming with sqaure root terms. Filomat, 27(1), 135–142.CrossRefGoogle Scholar
Ahmad, I., & Husain, Z. (2006). Optimality conditions and duality in nondifferentiable minimax fractional programming with generalized convexity. Journal of Optimization Theory and Applications, 129, 255–275.CrossRefGoogle Scholar
Antczak, T. (2001). (p, r)-invex sets and functions. Journal of Mathematical Analysis and Applications, 263, 355–379.CrossRefGoogle Scholar
Craven, B. D. (1981). Invex functions and constrained local minima. Bulletin of the Australian Mathematical Society, 24, 357–366.CrossRefGoogle Scholar
Dangar, D., & Gupta, S. K. (2013). On second-order duality for a class of nondifferentiable minimax fractional programming problem with \((C, \alpha, \rho, d)\)-convexity. Journal of Applied Mathematics and Computing, 43, 11–30.CrossRefGoogle Scholar
Gulati, T. R., & Gupta, S. K. (2011). Nondifferentiable second-order minimax mixed integer symmetric duality. Journal of Korean Mathematical Society, 48, 13–21.CrossRefGoogle Scholar
Gupta, S. K., Dangar, D., & Kumar, S. (2012). Second-order duality for a nondifferentiable minimax fractional programming under generalized \(\alpha \)-univexity. Journal of Inequalities and Applications, 2012, 187.CrossRefGoogle Scholar
Hanson, M. A. (1981). On sufficiency of the Kuhn-Tucker conditions. Journal of Mathematical Analysis and Applications, 80, 545–550.CrossRefGoogle Scholar
Khan, M. A. (2013). Second-order duality for nondifferentiable minimax fractional programming problems with generalized convexity. Journal of Inequalities and Applications, 2013, 500.CrossRefGoogle Scholar
Lai, H. C., & Chen, H. M. (2010). Duality on a nondifferentiable minimax fractional programming. Journal of Global Optimization, doi:10.1007/s-10898-010-9631-8, published on line:18 December Springer.
Lai, H. C., & Ho, S. C. (2012). Optimality and duality for nonsmooth minimax fractional programming with exponential \((p, r)\)-invexity. Journal of Nonlinear and Convex Analysis, 13, 433–447.Google Scholar
Lai, H. C., & Ho, S. C. (2014). Duality for nonsmooth minimax fractional programming with exponential \((p, r)\)-invexity. Journal of Nonlinear and Convex Analysis, 15, 711–725.Google Scholar
Lai, H. C., & Lee, J. C. (2002). On duality theorems for a nondifferentiable minimax fractional programming. Journal of computational and Applied Mathematics, 146, 115–126.CrossRefGoogle Scholar
Lai, H. C., & Lee, J. C. (2005). Parameter-free dual models for fractional programming with generalized invexity. Annals of Operations Research, 133, 47–61.CrossRefGoogle Scholar
Lai, H. C., Liu, J. C., & Tanaka, K. (1999). Necessary and sufficient conditions for minimax fractional programming. Journal of Mathematical Analysis and Applications, 230, 311–328.CrossRefGoogle Scholar
Liu, J. C., & Wu, C. S. (1998). On minimax fractional optimality conditions with invexity. Journal of Mathematical Analysis and Applications, 219, 21–35.CrossRefGoogle Scholar
Mangasarian, O. L. (1975). Second and higher-order duality in nonlinear programming. Journal of Mathematical Analysis and Applications, 51, 607–620.CrossRefGoogle Scholar
Neumann, J. V. (1947). On a maximization problem. Princeton, New Jersey: Institute for advanced study.Google Scholar
Schmitendorf, W. E. (1977). Necessary conditions and sufficient conditions for static minmax problems. Journal of Mathematical Analysis and Applications, 57, 683–693.CrossRefGoogle Scholar