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Optimality Conditions and Duality of Three Kinds of Nonlinear Fractional Programming Problems

DOI: 10.1155/2013/708979

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Some assumptions for the objective functions and constraint functions are given under the conditions of convex and generalized convex, which are based on the -convex, -convex, and -convex. The sufficiency of Kuhn-Tucker optimality conditions and appropriate duality results are proved involving -convex, -convex, and generalized -convex functions. 1. Introduction Multiobjective optimization theory is a development of numerical optimization and related to many subjects, such as nonsmooth analysis, convex analysis, nonlinear analysis, and the theory of set value. It has a wide range of applications in the fields of industrial design, economics, engineering, military, management sciences, financial investment, transport, and so forth, and now it is an interdisciplinary science branch between applied mathematics and decision sciences. Convexity plays an important role in optimization theory, and it becomes an important theoretical basis and useful tool for mathematical programming and optimization theory. Convex function theory can be traced back to the works of Holder, Jensen, and Minkowski in the beginning of this century, but the real work that caught the attention of people is the research on game theory and mathematical programming by von Neumann and Morgenstern [1], Dantzing, and Kuhn and Tucker in the forties to fifties, and people have done a lot of intensive research about convex functions from the fifties to sixties. In the middle of the sixties convex analysis was produced, and the concept of convex function is promoted in a variety of ways, and the notion of generalized convex is given. Fractional programming has an important significance in the optimization problems; for instance, in order to measure the production or the efficiency of a system, we should minimize a ratio of functions between a given period of time and a utilized resource in engineering and economics. Preda [2] has established the concept of -convex based on -convex [3] and -convex [4] and obtained some results, which are the expansion of -convex and -convex. Motivated by various concepts of convexity, Liang et al. [5] have put forward a generalized convexity, which was called -convex, which extended -convex, and Liang et al. [6], Weir and Mond [7], Weir [8], Jeyakumar and Mond [9], Egudo [10], Preda [2], and Gulati and Islam [3] obtained some corresponding optimality conditions and applied these optimality conditions to define dual problems and derived duality theorems for single objective fractional problems and multiobjective problems. Then the definition of generalized

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