Article Contents
Article Contents

# Sufficiency and duality in non-smooth interval valued programming problems

• * Corresponding author: Do Sang Kim
• In this paper a non-smooth optimization problem is studied in an uncertain environment. The objective function of this problem is interval valued function. We introduce the class of $LU-(p,r)-[ρ^L,ρ^U]-(η, θ)$-invex interval valued functions about the Clarke generalized gradient. Then, through non trivial examples, we illustrate that the class of functions introduced exists. Based upon the proposed invexity assumptions, the sufficient optimality conditions are established. Further, we derive weak, strong and strict converse duality theorems for Mond-Weir type and Wolfe type dual programs. Some examples are also given in order to illustrate our results.

Mathematics Subject Classification: Primary: 90C30, 90C46; Secondary: 49N15.

 Citation:

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Table 1.  Summary of Example 7.1 and Remark 7.2

 Functions Valued of $\rho$ Domain $f^L(x)$ $\rho^L=\frac{-0.1}{|x+0.01|^4}$ $x\in (-0.4599,1.0766)$ $f^U(x)$ $\rho^U=\frac{-0.09}{|x+0.01|^4}$ $x\in (-0.2362,1.1438)$ $\sum_{j=1}^2\mu_jg_j(x)$ $\rho=4$ $x\in (-0.5172,0.5172)$ $[f^L(x),f^U(x)]$ $[\rho^L,\rho^U]=\left[\frac{-0.1}{|x+0.01|^4},\frac{-0.09}{|x+0.01|^4}\right]$ $x\in(-0.4599,1.0766)\cap (-0.2362,1.1438)$ $f^L_0(x)$ $\rho^L_0=\frac{-0.13}{|x+0.01|^4}$ $x\in(-1.0567,1.0567)$ $[f^L_0(x),f^U(x)]$ $[\rho^L_0,\rho^U]=\left[\frac{-0.13}{|x+0.01|^4},\frac{-0.09}{|x+0.01|^4}\right]$ $x\in(-1.0567,1.0567)\cap(-0.2362,1.1438)$
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