Local Sharp Vector Variational Type Inequality and Optimization Problems

Jong Kyu Kim,Salahuddin Salahuddin

doi:10.3390/math8101844

Jong Kyu Kim, Salahuddin Salahuddin

Open Access

https://doi.org/10.3390/math8101844

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Journal: Mathematics	Publication Date: Oct 20, 2020
Citations: 2	License type: CC BY 4.0

Affiliation: Kyungnam University, Jazan University

Abstract

In this paper, our goal was to establish the relationship between solutions of local sharp vector variational type inequality and sharp efficient solutions of vector optimization problems, also Minty local sharp vector variational type inequality and sharp efficient solutions of vector optimization problems, under generalized approximate η-convexity conditions for locally Lipschitzian functions.

Highlights

The research of variational inequality problems is a part of development in the theory of optimization since optimization problems can often be specialized to the solution of variational inequality problems
In 1984, Loridan [13] studied the concept of e-efficient solutions for vector minimization problems where the function to be optimized has its values in the Rn space, which is a generalization of the classical problem for Pareto solution
(2) reduces to weak local sharp vector variational inequalities (WLSVVI) for finding x0 ∈ X, there exists a δ-neighborhood of x0 and for any τ > 0, such that x ∈ B( x0, δ) ∩ X and max max 1≤i ≤ p x0∗ ∈∂ f i ( x0 )

Summary

Introduction

The research of variational inequality problems is a part of development in the theory of optimization since optimization problems can often be specialized to the solution of variational inequality problems. Later in 1986, White [14] extended e-optimality for scalar problems to vector maximization problems, or efficiency problems, with m objective functions defined on a subset of Rn. In 1993, Burke et al [15] studied the concept of weak sharp minima for scalar optimization problem which was motivated by the application in convex and convex composite mathematical programming. In 2016, Zhu [16] suggested the necessary optimal conditions for the weak local sharp efficient solution of a constrained multi-objective optimization problem by using the generalized. Fermat formula, the Mordukhovich subdifferential for maximum functions, the fuzzy sum rule for Fréchet subdifferentials, and the sum rule for Mordukhovich subdifferentials, and got the some sufficient optimal conditions respectively for the local and global weak sharp efficient solutions of such a multi-objective optimization problem, by applying the approximate projection method, and some appropriate convexity and affineness conditions.

Preliminaries

Local Sharp Vector Variational Type Inequalities

Minty Local Sharp Vector Variational Type Inequalities

Conclusions