NEW ZEALAND JOURNAL OF MATHEMATICS Volume 34 (2005), 1–9 MULTIOBJECTIVE MIXED SYMMETRIC DUALITY WITH INVEXITY Izhar Ahmad (Received September 2003) Abstract. The usual duality results are established for mixed symmetric mul- tiobjective dual programs without nonnegativity constraints using the notion of invexity/ generalized invexity which has allowed weakening various types of convexity/ generalized convexity assumptions. This mixed symmetric dual formulation unifies two existing symmetric dual formulations in the literature. 1. Introduction A pair of dual problems is called symmetric if the dual of the dual is the primal problem. Symmetric duality in nonlinear programming was introduced by Dorn [5]. Dantzig et al .[4], Mond [8], Bazaraa and Goode [2] and Mond and Weir [10] etc. further developed the concept of symmetric duality. Hanson [7] introduced the concept of invexity. Since then many duality results which previously required convexity/generalized convexity assumptions have been extended by using invexity/generalized invexity. Mond and Hanson [9] applied invexity to symmetric dual programs of [4] with an additional assumption on the invexity. Later on in multiobjective programming, Weir and Mond [11] studied symmetric duality with convexity/ generalized convexity assumptions. Xu [12] introduced the mixed type duals in multiobjective programming and proved duality theorems. In this paper, we study invexity/generalized invexity for mixed type symmetric dual in multiobjective programming problems ignoring nonnegativity constraints of Bector et al .[1] but adjoining an additional condition on invexity/generalized invexity. Self duality for our programs is also incorporated. 2. Notations and Prerequistes Let R n denote the n–dimensional Euclidean space. The following conventions of vectors in R n will be followed throughout this paper: x ≦ y ⇔ x i ≦ y i , i =1, 2, ··· ,n; x ≤ y ⇔ x ≦ y and x = y; x<y ⇔ x i <y i , i =1, 2, ··· ,n. For N = {1, 2, ··· ,n} and M = {1, 2, ··· ,m} let J 1 ⊆ N,K 1 ⊆ M and J 2 = N \ J 1 and K 2 = M \ K 1 . Let |J 1 | denote the number of elements in the subset of J 1 . The other symbols |J 2 |, |K 1 | and |K 2 | are defined similarly. Let x 1 ∈ R |J1| , x 2 ∈ R |J2| , then any x ∈ R n can be written as (x 1 ,x 2 ). Similarly for y 1 ∈ R |K1| and y 2 ∈ R |K2| , y ∈ R m can be written as (y 1 ,y 2 ). Let f : R |J1| × R |K1| −→ R l and g : R |J2| × R |K2| −→ R l be twice differentiable functions and e = (1, 1, ··· , 1) ∈ R l . 1991 Mathematics Subject Classification 90C26, 90C29, 90C30,90C46. Key words and phrases: Mixed symmetric duality, differentiable programming, invexity, pseudoin- vexity, self duality.