In this paper, a new class of (G,C,r)- type I functions and their generalizations are introduced. We consider a class of differentiable multiobjective optimization problems and establish sufficient optimality conditions. The results of the paper are more general than those existing in the literature. | Yugoslav Journal on Operations Research 23(2013) Number 2, 299–310 DOI: ON SUFFICIENCY IN MULTIOBJECTIVE PROGRAMMING INVOLVING GENERALIZED (G,C, ρ )-TYPE I FUNCTIONS Yadvendra SINGH, Amod KUMAR, B. B. UPADHYAY and Vinay SINGH Department of Mathematics Faculty of Science Banaras Hindu University,Varanasi, India ysinghze@ Received: January, 2013 / Accepted: March, 2013 Abstract: In this paper, a new class of (G,C, ρ )-type I functions and their generalizations are introduced. We consider a class of differentiable multiobjective optimization problems and establish sufficient optimality conditions. The results of the paper are more general than those existing in the literature. Keywords: Multiobjective programming, (G,C, ρ )-convexity, efficient solution, type I functions, generalized convexity. MSC: 90C46; 52A01 1. INTRODUCTION Convexity plays an important role in optimization theory as it extends the validity of a local solution of a minimization problem to a global one. But in several real world problems, the notion of convexity is no longer sufficient, which motivated the introduction of various generalizations of convex functions. It has been found that only a few properties of convex functions are needed for establishing sufficiency and duality theorems. Hanson 299 300 Yadvendra Singh et al. / On Sufficiency In Multiobjective [11] introduced the concept of differentiable invexity, which is a generalization of the concept of convexity. After the work of Hanson, other classes of differentiable nonconvex functions have been introduced to generalize the class of invex functions from different points of view, see the in [7-9, 12, 13, 16, 21, 22 ]. Later, Kaul and Kaur [14] presented strictly pseudoinvex, pseudoinvex and quasiinvex functions. Hanson and Mond [12] defined two new classes of functions called type I and type II functions. Rueda and Hanson [23] have introduced pseudo type I and and quasi type I functions. Mishra [24] .
