Abstract:
Convexity and generalized convexity play important roles in optimization theory. With the development of programming problem, there has been a growing interest in the higher-order dual problem and a lot of related generalized convexities are given. In this paper, we give the convexity of (F, α ,p ,d ,b , φ )_{β} vector-pseudo- quasi-Type I and formulate a higher-order duality for minimax fractional type programming involving symmetric matrices, and give the weak, strong and strict converse duality theorems under the condition of higher-order (F, α ,p ,d ,b , φ )_{β} vector-pseudoquasi-Type I.

Abstract:
Necessary and sufficient optimality conditions are established for a class of nondifferentiable minimax fractional programming problems with square root terms. Subsequently, we apply the optimality conditions to formulate a parametric dual problem and we prove some duality results.

Abstract:
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.

Abstract:
We start our discussion with a class of nondifferentiable minimax programming problems in complex space and establish sufficient optimality conditions under generalized convexity assumptions. Furthermore, we derive weak, strong, and strict converse duality theorems for the two types of dual models in order to prove that the primal and dual problems will have no duality gap under the framework of generalized convexity for complex functions. 1. Introduction The literature of the mathematical programming is crowded with necessary and sufficient conditions for a point to be an optimal solution to the optimization problem. Levinson [1] was the first to study mathematical programming in complex space who extended the basic theorems of linear programming over complex space. In particular, using a variant of the Farkas lemma from real space to complex space, he generalized duality theorems from real linear programming. Since then, linear fractional, nonlinear, and nonlinear fractional complex programming problems were studied by many researchers (see [2–5]). Minimax problems are encountered in several important contexts. One of the major context is zero sum games, where the objective of the first player is to minimize the amount given to the other player and the objective of the second player is to maximize this amount. Ahmad and Husain [6] established sufficient optimality conditions for a class of nondifferentiable minimax fractional programming problems involving -convexity. Later on, Jayswal et al. [7] extended the work of Ahmad and Husain [6] to establish sufficient optimality conditions and duality theorems for the nondifferentiable minimax fractional problem under the assumptions of generalized -convexity. Recently, Jayswal and Kumar [8] established sufficient optimality conditions and duality theorems for a class of nondifferentiable minimax fractional programming problems under the assumptions of -convexity. Lai et al. [9] established several sufficient optimality conditions for minimax programming in complex spaces under the assumptions of generalized convexity of complex functions. Subsequently, they applied the optimality conditions to formulate parametric dual and derived weak, strong, and strict converse duality theorems. The first work on fractional programming in complex space appeared in 1970, when Swarup and Sharma [10] generalized the results of Charnes and Cooper [11] to the complex space. Lai and Huang [12] showed that a minimax fractional programming problem is equivalent to a minimax nonfractional parametric problem for a given parameter

Abstract:
In this present article we have given some mathematical fractional programming problems with their symmetric duals and have derived weak and strong duality results with respect to such programs. Moreover, we have also used most general type of convexity assumptions involved with the functions which are related to the programming problems. It is to be pointed out that the objective functions in such programs contain terms like support functions which in turn are able to give results on particular classes of programs involving quadratic terms. Our results in particular give as of special cases some eariler results symmetric duals given in the current literature. All discussion goes to complex spaces.

Abstract:
We establish properly efficient necessary and sufficient optimality conditions for multiobjective fractional programming involving nonsmooth generalized -univex functions. Utilizing the necessary optimality conditions, we formulate the parametric dual model and establish some duality results in the framework of generalized -univex functions. 1. Introduction In this paper, we consider the following nondifferentiable nonconvex multiobjective fractional programming problem: where (a1) , , and , are Lipschitz on , and is an open subset of ; (a2) , , .Minimize means obtaining efficient solution in the following sense. A point is said to be an efficient solution for (MFP) if there is no such that with at least one strict inequality. A point is said to be a？？properly efficient solution for (MFP) which was introduced by Geoffrion [1] if and only if (a) is an efficient solution for (MFP), (b) there exists a scalar such that for each , we have ？for some such that , whenever and . An efficient point for (MFP) that is not properly efficient is said to be improperly efficient. Thus, for to be improperly efficient for (MFP) means that to every scalar , there is a point and an such that and for all such that . Many papers have been devoted to the multiobjective fractional programming problem in recent decades; see for example [1–13]. In [8], Preda introduced generalized -convexity, an extension of -convexity and generalized -convexity defined by Vial [14, 15]. Bhatia and Jain [2] defined generalized -convexity for nonsmooth functions, an extension of generalized -convexity defined by Preda [8], and they derived some duality theorems for nonsmooth multiobjective programs. In [5, 6], Liu also established the Kuhn-Tucker type necessary and sufficient optimality conditions for multiobjective fractional programming involving nonsmooth pseudoinvex functions in [5] or -convex functions in [6] and considered the parameter dual problem in the framework of generalized convex functions. Recently, Zalmai [13, 16, 17] introduced generalized -univex -set functions and he also established sufficient efficiency conditions in multiobjective fractional subset programming [13] and sufficient optimality conditions in minimax fractional subset programming [16, 17] under various generalized -univexity assumptions. In [10], Preda et al. obtained duality results for a dual model of Zalmai [13] replacing the assumptions of sublinearity or convexity by that of quasiconvexity in the third argument. In this paper, we are inspired to consider the optimality and duality of properly efficient for

Abstract:
In this paper we introduce a new dual program, which is representable as a semi-definite linear programming problem, for a primal convex minimax programming model problem and show that there is no duality gap between the primal and the dual whenever the functions involved are SOS-convex polynomials. Under a suitable constraint qualification, we derive strong duality results for this class of minimax problems. Consequently, we present applications of our results to robust SOS-convex programming problems under data uncertainty and to minimax fractional programming problems with SOS-convex polynomials. We obtain these results by first establishing sum of squares polynomial representations of non-negativity of a convex max function over a system of SOS-convex constraints. The new class of SOS-convex polynomials is an important subclass of convex polynomials and it includes convex quadratic functions and separable convex polynomials. The SOS-convexity of polynomials can numerically be checked by solving semi-definite programming problems whereas numerically verifying convexity of polynomials is generally very hard.

Abstract:
We study a nonlinear multiple objective fractional programming with inequality constraints where each component of functions occurring in the problem is considered semidifferentiable along its own direction instead of the same direction. New Fritz John type necessary and Karush-Kuhn-Tucker type necessary and sufficient efficiency conditions are obtained for a feasible point to be weakly efficient or efficient. Furthermore, a general Mond-Weir dual is formulated and weak and strong duality results are proved using concepts of generalized semilocally V-type I-preinvex functions. This contribution extends earlier results of Preda (2003), Mishra et al. (2005), Niculescu (2007), and Mishra and Rautela (2009), and generalizes results obtained in the literature on this topic. 1. Introduction Because of many practical optimization problems where the objective functions are quotients of two functions, multiobjective fractional programming has received much interest and has grown significantly in different directions in the setting of efficiency conditions and duality theory these later years. The field of multiobjective fractional optimization has been naturally enriched by the introductions and applications of various types of convexity theory, with and without differentiability assumptions, and in the framework of symmetric duality, variational problems, minimax programming, continuous time programming, and so forth. More specifically, works in the area of nonsmooth setting can be found in Chen [1], Kim et al. [2], Kuk et al. [3], Mishra and Rautela [4], Mishra et al. [5], Niculescu [6], Preda [7], and Soleimani-damaneh [8]. Efficiency conditions and duality models for multiobjective fractional subset programming problems are studied by Preda et al. [9], Verma [10], and Zalmai [11–13]. Higher order duality in multiobjective fractional programming is discussed in Gulati and Geeta [14] and Suneja et al. [15]. Solving nonlinear multiobjective fractional programming problems by a modified objective function method is the subject matter of Antczak [16]. Further works on multiobjective fractional programming are established by Chinchuluun et al. [17], J.-C. Liu and C.-Y. Liu [18], Mishra et al. [19], Verma [20], Zhang and Wu [21], and others. The common point in all of these developments is the convexity theory that does not stop extending itself in different directions with new variants of generalized convexity and various applications to nonlinear programming problems in different settings. The concept of invexity introduced by Hanson [22] is a generalization of