BULL. AUSTRAL. MATH. SOC. 90C29 VOL. 74 (2006) [207-218] APPROXIMATE CONVEXITY IN VECTOR OPTIMISATION ANJANA GUPTA, APARNA MEHRA AND DAVINDER BHATIA Approximate convex functions are characterised in terms of Clarke generalised gradient. We apply this characterisation to derive optimality conditions for quasi efficient solutions of nonsmooth vector optimisation problems. Two new classes of generalised approximate convex functions are defined and mixed duality results are obtained. 1. INTRODUCTION There have been several studies in the past to demonstrate the key role played by 'duality' in Economics and Optimisation Theory. Many dual models have been proposed for the constrained vector optimisation problems and corresponding duality results have been investigated. Among them the two dual models namely Wolfe dual model and Mond-Weir dual model have been widely studied both for smooth as well as nonsmooth vector optimisation problems ([1, 2, 4, 6, 7, 8, 10] and references cited therein). Later, combining the two dual models a mixed dual model was proposed and duality results were obtained by Xu [12]. In order to have a deeper insight of the mixed dual model Bector, Chandra and Abha ([1, 2]) defined the notion of incomplete Lagrange function and observed that the Mond-Weir dual is connected to the incomplete Lagrange function exactly in the same manner as the Wolfe dual is connected to the usual Lagrange function. This inspired them to study mixed duality for various classes of nonlinear scalar-valued programming problems. It is worth to note that the notions of convexity and generalised convexity play a crucial role in establishing the primal-dual relationships. Moreover, advances in nonsmooth analysis and nonsmooth subdifferenital calculus rules led various authors to search for the class of nonconvex functions possessing properties that are similar to convex functions and also satisfy the basic subdifferential calculus rules. In this context, Ngai, Luc and Thera [9] defined a new class of approximate convex functions and showed that functions belonging to this class enjoy many of the desired properties. In this article we intend to use the notion of approximate convexity to develop mixed duality results for nonsmooth vector optimisation problems. The structure of the paper is as follows. In section 2 we present a characterisation of approximate convex function Received 4th April, 2006 Copyright Clearance Centre, Inc. Serial-fee code: 0004-9727/06 SA2.00+0.00. 207 https://doi.org/10.1017/S0004972700035656 Published online by Cambridge University Press