On optimality conditions and duality for non-differentiable interval-valued programming problems with the generalized (F, ρ)-convexity

Xiuhong Chen; Zhihua Li

doi:10.3934/jimo.2017081

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2018, Volume 14, Issue 3: 895-912. Doi: 10.3934/jimo.2017081

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On optimality conditions and duality for non-differentiable interval-valued programming problems with the generalized (F, ρ)-convexity

Xiuhong Chen^1, , and
Zhihua Li^2,

1.
School of Digital Media, Jiangnan University, Wuxi 214122, Jiangsu, China
2.
School of Internet of Things, Jiangnan University, Wuxi 214122, Jiangsu, China

* Corresponding author: Xiuhong Chen
* Corresponding author: Xiuhong Chen

Received: July 31, 2017

Early access: August 2017

Published: June 2018

The first author is supported in part by the National Natural Foundation of China under grant:61373055 and Jiangsu Key Laboratory of Media Design and Software Technology(Jiangnan University).

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Abstract

Because interval-valued programming problem is used to tackle interval uncertainty that appears in many mathematical or computer models of some deterministic real-world phenomena, this paper considers a non-differentiable interval-valued optimization problem in which objective and all constraint functions are interval-valued functions, and the involved endpoint functions in interval-valued functions are locally Lipschitz and Clarke sub-differentiable. A necessary optimality condition is first established. Some sufficient optimality conditions of the considered problem are derived for a feasible solution to be an efficient solution under the $G-(F, ρ)$ convexity assumption. Weak, strong, and converse duality theorems for Wolfe and Mond-Weir type duals are also obtained in order to relate the efficient solution of primal and dual inter-valued programs.

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Mathematics Subject Classification: Primary: 26A27, 26A51, 65G40, 90C30, 90C46; Secondary: 49N15, 90C26, 90C29.

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