In this paper，we first introduce new classes of generalized convex type-I functions by relaxing definitions of arcwise connected function and type-I function.We present some sufficient optimality conditions and dual theorems for non-differential multi-objective programming problem under various generalized convex type-I functions assumptions.This paper is divided into four sections.Section 2recalls some definitions and related results which will be used in later sections，and introduces new classes generalized convex type-I functions.In section 3and section 4，the sufficient optimality conditions and Mond-Weir type duality results are established for non-differential multi-objective programming problem involving these generalized convex functions，respectively.
2.Preliminaries
In this section，we first recall some concepts and results related arcwise connected functions.Let Rn be the n-dimensional Euclidean space and R1be the set of all real numbers.Throughout this paper，the following convention for vectors in Rn will be followed：
x＜y if and only if xi＜yi，i=1，2，…，n，
x≦y if and only if xi≦yi，i=1，2，…，n，
x≤y if and only if xi≦yi i=1，2，…，n，but x≠y，
x≮y is the negation of x＜y.
Definition 2.1.（See ［15］）A subsetis said to be an arcwise connected（AC）set，if for every x∈X，u∈X，there exists a continuous vector-valued functions Hx，u：［0，1］→X，called an arc，such that
Hx，u（0）=x，Hx，u（1）=u.
Definition 2.2.（See［15］）Let f be a real-valued function defined on an AC set.Then f is said to be an arcwise connected function （CN）if，for every x∈X，u∈X，there exists an arc Hx，u such that
f（Hx，u（θ））≦（1-θ）f（x）+θf（u），for 0≦θ≦1
Definition 2.3.（See ［13-14］）Let be an AC set，and Let f be a real-valued function defined on X.For any x∈X，u∈X，the directional derivative of f with respect to Hx，u at θ=0is defined as
provided the limit exists and is denoted by f +（Hx，u（0））.
If
exists and it is denoted by H+x，u（0），then vector H+x，u（0）is called directional derivative of Hx，u at θ=0.
Consider the following multiobjective programming problem：
（MP）min f（x）
4.Duality Results
Now，in relation to （MP）we consider the following Mond and Weir type dual under the CN-d-type-I and generalized CN-d-type-I assumptions.
satisfied.This leads to the similar contradiction as in the proof of Theorem 4.1.
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1Yu Guolin，male，Ph.D.，associate professor，supervisor of postgraduate.
