Univex Interval-Valued Mapping with Differentiability and Its Application in Nonlinear Programming

Lifeng Li,Sanyang Liu,Jianke Zhang

doi:10.1155/2013/383692

Abstract

Interval-valued univex functions are introduced for differentiable programming problems. Optimality and duality results are derived for a class of generalized convex optimization problems with interval-valued univex functions.

Highlights

Imposing the uncertainty upon the optimization problems is an interesting research topic
The randomness occurring in the optimization problems is categorized as the stochastic optimization problems, and the imprecision occurring in the optimization problems is categorized as the fuzzy optimization problems
Wu has extended the concept of convexity for real-valued functions to LU-convexity for interval-valued functions, he has established the Karush-Tucker conditions [4,5,6] for an optimization problem with intervalvalued objective functions under the assumption of LUconvexity

Summary

Introduction

Imposing the uncertainty upon the optimization problems is an interesting research topic. Wu has extended the concept of convexity for real-valued functions to LU-convexity for interval-valued functions, he has established the Karush-Tucker conditions [4,5,6] for an optimization problem with intervalvalued objective functions under the assumption of LUconvexity. In 1981, Hanson [10] introduced the concept of invexity and established Karush-Tucker type sufficient optimality conditions for a nonlinear programming problem. In [11], Kaul et al considered a differentiable multiobjective programming problem involving generalized type I functions They investigated Karush-Tucker type necessary and sufficient conditions and obtained duality results under generalized type I functions. Gulati et al [18] introduced the concept of (F, α, ρ, d)-V-type I functions and studied sufficiency optimality conditions and duality multiobjective programming problems. The Karush-Tucker optimality conditions are proposed for an interval-valued function under the assumption of interval-valued univexity

Preliminaries

Interval-Valued Univex Functions

Optimality Criteria

Duality

Numerical Example