Nondifferentiable Minimax Fractional Programming Research Articles

Sufficient optimality condition and duality of nondifferentiable minimax ratio constraint problems under (p, r)-ρ-(η, θ)-invexity

Abstract There are several classes of decision-making problems that explicitly or implicitly prompt fractional programming problems. Portfolio selection problems, agricultural planning, information transfer, numerical analysis of stochastic processes, and resource allocation problems are just a few examples. The huge number of applications of minimax fractional programming problems inspired us to work on this topic. This paper is concerned with a nondifferentiable minimax fractional programming problem. We study a parametric dual model, corresponding to the primal problem, and derive the sufficient optimality condition for an optimal solution to the considered problem. Further, we obtain the various duality results under (p, r)-ρ-(η, θ)-invexity assumptions. Also, we identify a function lying exclusively in the class of (−1, 1)-ρ-(η, θ)-invex functions but not in the class of (1, −1)-invex functions and convex function already existing in the literature. We have given a non-trivial model of nondifferentiable minimax problem and obtained its optimal solution using optimality results derived in this paper.

Nondifferentiable generalized minimax fractional programming under (Ф,ρ)-invexity

In this paper, a class of nonconvex nondifferentiable generalized minimax fractional programming problems is considered. Sufficient optimality conditions for the considered nondifferentiable generalized minimax fractional programming problem are established under the concept of (?, ?)-invexity. Further, two types of dual models are formulated and various duality theorems relating to the primal minimax fractional programming problem and dual problems are established. The results established in the paper generalize and extend several known results in the literature to a wider class of nondifferentiable minimax fractional programming problems. To the best of our knowledge, these results have not been established till now.

Higher-Order Non-Symmetric Duality for Nondifferentiable Minimax Fractional Programs with Square Root Terms

In this paper, we emphasize on a nondifferentiable minimax fractional programming (NMFP) problem and obtain appropriate duality results for higher-order dual model under higher-order B-(p,r)-invex functions. We provide a nontrivial illustration of a function which belongs to the class of higher-order B-(p,r)-invex but not in the class of second-order B-(p,r)-invex functions already existing in literature. An example of finding a minimax solution of NMFP problem by using higher-order B-(p,r)-invex functions has also been given. Various known results are discussed as particular cases.

Duality for Unified Higher-Order Minimax Fractional Programming with Support Function under Type-I Assumptions

This article is devoted to discussing the nondifferentiable minimax fractional programming problem with type-I functions. We focus our study on a nondifferentiable minimax fractional programming problem and formulate a higher-order dual model. Next, we establish weak, strong, and strict converse duality theorems under generalized higher-order strictly pseudo ( V , α , ρ , d ) -type-I functions. In the final section, we turn our focus to study a nondifferentiable unified minimax fractional programming problem and the results obtained in this paper naturally unify. Further, we extend some previously known results on nondifferentiable minimax fractional programming in the literature.

Duality for nondifferentiable minimax fractional programming problem involving higher order $$(\varvec{C},\varvec{\alpha}, \varvec{\rho}, \varvec{d})$$ ( C , α , ρ , d ) -convexity

In this paper, we present new class of higher-order \((C, \alpha , \rho , d)\)-convexity and formulate two types of higher-order duality for a nondifferentiable minimax fractional programming problem. Based on the higher-order \((C, \alpha , \rho , d)\)-convexity, we establish appropriate higher-order duality results. These results extend several known results to a wider class of programs.

Duality in Minimax Fractional Programming Problem Involving Nonsmooth Generalized (F,α,P,d)-Convexity

In this paper, we discuss nondifferentiable minimax fractional programming problem where the involved functions are locally Lipschitz. Furthermore, weak, strong and strict converse duality theorems are proved in the setting of Mond-Weir type dual under the assumption of generalized (F, α, ρ, d)-convexity.

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/s10957-006-9052-5

Second-order duality for nondifferentiable minimax fractional programming problems with generalized convexity

In the present paper, we are concerned with second-order duality for nondifferentiable minimax fractional programming under the second-order generalized convexity type assumptions. The weak, strong and converse duality theorems are proved. Results obtained in this paper extend some previously known results on nondifferentiable minimax fractional programming in the literature.

Non-differentiable minimax fractional programming with higher-order type Ifunctions

In this article, we are concerned with a class of non-differentiable minimaxfractional programming problems and their higher-order dual model. Weak, strongand converse duality theorems are discussed involving generalized higher-ordertype I functions. The presented results extend some previously known results onnon-differentiable minimax fractional programming.

On second-order duality for nondifferentiable minimax fractional programming

Two types of second-order dual models for a nondifferentiable minimax fractional programming problem are formulated and proved weak, strong, strict converse duality theorems using η-bonvexity assumptions. Special cases are also discussed to show that this work extends some known results of the literature.

On second-order duality for a class of nondifferentiable minimax fractional programming problem with (C,α,ρ,d)-convexity

In this paper, we derive appropriate duality theorems for three second-order dual models of a nondifferentiable minimax fractional programming problem under second-order (C,α,ρ,d)-convexity assumptions. A nontrivial example has also been exemplified to show the existence of second-order (C,α,ρ,d)-convex functions. Several known results including many recent works are obtained as special cases.

On nondifferentiable minimax fractional programming involving higher order generalized convexity

In this article, we focus our study on a nondifferentiable minimax fractional programming problem and establish weak, strong and strict converse duality theorems under generalized higher order (F, α, p, d)-Type I assumptions. Our results extend and unify some of the known results in the literature.

Second order nondifferentiable minimax fractional programming with square root terms

In this paper, we study a nondifferentiable minimax fractional programming problem under the assumptions of generalized second order (F , α, ρ,d)−convexity. A second order parametric dual is formulated. Weak, strong and converse duality theorems are established in order to relate the primal and dual problems under the afore-said assumptions.

Higher-order duality in nondifferentiable minimax fractional programming involving generalized convexity

A higher-order dual for a non-differentiable minimax fractional programming problem is formulated. Using the generalized higher-order η-convexity assumptions on the functions involved, weak, strong and strict converse duality theorems are established in order to relate the primal and dual problems. Results obtained in this paper naturally unify and extend some previously known results on non-differentiable minimax fractional programming in the literature.

Minimax fractional programming problem involving nonsmooth generalized ?-univex functions

In this paper, we introduce a new class of generalized ?-univex functions where the involved functions are locally Lipschitz. We extend the concept of ?-type I invex [S. K. Mishra, J. S. Rautela, On nondifferentiable minimax fractional programming under generalized ?-type I invexity, J. Appl. Math. Comput. 31 (2009) 317-334] to ?-univexity and an example is provided to show that there exist functions that are ?-univex but not ?-type I invex. Furthermore, Karush-Kuhn-Tucker-type sufficient optimality conditions and duality results for three different types of dual models are obtained for nondifferentiable minimax fractional programming problem involving generalized ?-univex functions. The results in this paper extend some known results in the literature.

Second-order duality for a nondifferentiable minimax fractional programming under generalized α-univexity

In this paper, we concentrate our study to derive appropriate duality theorems for two types of second-order dual models of a nondifferentiable minimax fractional programming problem involving second-order α-univex functions. Examples to show the existence of α-univex functions have also been illustrated. Several known results including many recent works are obtained as special cases.

Second-order duality for non-differentiable minimax fractional programming

In this paper, a second-order duality for a non-differentiable minimax fractional programming is formulated by generalizing the one developed by Husian et al. [Second order duality for minimax fractional programming, Optim. Lett. 3 (2009), pp. 277–286], for the minimax fractional programming. The weak, strong and strict converse duality theorems are proved for these programs under generalized η-bonvexity assumptions.

Duality in nondifferentiable minimax fractional programming with B-(p, r)- invexity

In this article, we are concerned with a nondifferentiable minimax fractional programming problem. We derive the sufficient condition for an optimal solution to the problem and then establish weak, strong, and strict converse duality theorems for the problem and its dual problem under B-(p, r)-invexity assumptions. Examples are given to show that B-(p, r)-invex functions are generalization of (p, r)-invex and convex functions AMS Subject Classification: 90C32; 90C46; 49J35.

Optimality and Duality for Nondifferentiable Minimax Fractional Programming with Generalized Convexity

We establish several sufficient optimality conditions for a class of nondifferentiable minimax fractional programming problems from a view point of generalized convexity. Subsequently, these optimality criteria are utilized as a basis for constructing dual models, and certain duality results have been derived in the framework of generalized convex functions. Our results extend and unify some known results on minimax fractional programming problems.

Nondifferentiable minimax fractional programming in complex spaces with parametric duality

The purpose of this paper is to study a duality programming problem for a complex nondifferentiable minimax fractional programming with complex variables. The necessary and sufficient optimality conditions are established by Lai and Huang (J Math Anal Appl 359:229---239, 2009). Employing the optimality conditions, we can constitute a parametric dual model problem. Moreover, we establish the duality theorem and prove that optimal values of the duality problem as well as the primary problem are equal under some reasonable assumptions. That is, there are no duality gap between the primary problem and its duality problem.