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J Glob Optim (2007) 38:653666
DOI 10.1007/s10898-006-9103-3
ORIGINAL PAPER
Sanming Liu Enmin Feng
Received: 6 October 2005 / Accepted: 8 October 2006 / Published online: 6 February 2007 Springer Science+Business Media B.V. 2006
Abstract A class of multi-objective fractional programming problems (MFP) are considered where the involved functions are locally Lipschitz. In order to deduce our main results, we give the denition of the generalized (F, , , d)-convex class about the Clarkes generalized gradient. Under the above generalized convexity assumption, necessary and sufcient conditions for optimality are given. Finally, a dual problem corresponding to (MFP) is formulated, appropriate dual theorems are proved.
Keywords Multi-objective fractional programming Sublinear functions Generalized convex functions Optimality conditions Duality AMS Subject Classication 90C29, 90C32, 90C46
1 Introduction
Convexity plays a very important role in optimization theory. But for many mathematical models used in decision sciences, economics, management sciences, stochastics, applied mathematics and engineering, the notion of convexity does no longer sufce. To relax convexity assumptions imposed on the functions in theorems on optimality and duality, several denitions extending the concept of convexity of a function have been introduced. Hanson [1] introduced the concept of invexity, generalizing the difference x y in the denition of convex function to any function (x, y).He established KarushKuhnTucker type sufcient optimality conditions for the scalar
S. Liu (B)
Department of Mathematics and Physics, Jiangsu University of Science and Technology, Zhenjiang, 212003, Chinae-mail: [email protected]
E. Feng
Department of Applied Mathematics, Dalian University of Technology, Dalian, China e-mail: [email protected]
Optimality conditions and duality for a class of nondifferentiable multi-objective fractional programming problems
654 J Glob Optim (2007) 38:653666
optimization problem. During the last 20 years, numerous articles have appeared in the literature reecting further generalizations and applications in this category (see, e.g. [210]). Under various kinds of generalized convexities, some results for optimality conditions and duality in multi-objective fractional programming problems (MFP) have been obtained (see [1120]). In particular, Bector et al. [11] derived Fritz John and KarushKuhnTucker necessary and sufcient optimality conditions for a class of nondifferentiable convex (MFP), and they established some duality theorems. Following the approaches of Bector et al. [11], Liu [14,15] obtained necessary and sufcient conditions and derived duality theorem for a class of nonsmooth (MFP) involving either pseudoinvex functions or...