Generalized vector variational-like inequalities and vector optimization

Ansari, Qamrul Hasan; Rezaie, Mahboubeh; Zafarani, J.

doi:10.1007/s10898-011-9686-1

Cited by 26 publications

(11 citation statements)

References 18 publications

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“…Within the last decades, several existence results for vector variational inequalities of type (2.1) and extensions of it have been published in the literature. However, a careful study of all results shows that only a handful of different techniques are used; compare, for example, [7,8,12,13,31,40,53,81,82,101,111,115,133,136,138,140,141,146,163,165,176,186,193,194] and the references therein. We will therefore focus on the basic vector variational inequality (2.1) only since all results and methods for generalized vector variational inequalities are based on the results for problem (2.1).…”

Section: Existence Resultsmentioning

confidence: 99%

Vector variational inequalities and related topics: A survey of theory and applications

2019

Appl. Set-Valued Anal. Optim.

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In this survey paper, we give a detailed introduction to some of the recent developments in the field of vector variational inequalities and related problems. By giving several examples and presenting the necessary mathematical background and theories, the survey attempts to draw a broad audience and is accessible to students in mathematics and engineering. In doing this, we will study several scalarization methods for vector variational inequalities, which are necessary, sufficient, or both, for the original problem. We further analyze topological and algebraical properties of the solution set. In particular, the existence of solutions is discussed by presenting several existence results. For this purpose, coercivity conditions ensuring, in a certain sense, the boundedness of the data of the vector variational inequality are required. We will further give a precise overview about existence results and techniques, which are known in the literature. Besides that, we analyze a regularization method for non-coercive vector variational inequalities, which consists of approximating solutions of non-coercive problems by a family of regularized vector variational inequalities. After that, we will study relations between vector variational inequalities and multi-objective optimization problems. Motivated by the duality principle in optimization, we also investigate two inverse vector variational inequalities. Furthermore, we consider gap functions for vector variational inequalities, which enable us to study equivalent optimization problems instead. A completely different approach consists of replacing the vector variational inequality by a parametric system or intersection problem. The idea of image space analysis is to study the vector problem in the image space, using one of the previous reformulations. Since vector variational inequalities are ill-posed in general, in the sense that they may either have no solution or multiple solutions, we study stability and sensitivity analysis results. Especially continuity properties of the corresponding solution mapping are investigated. Finally, we give a brief analysis of stochastic vector variational inequalities, generalized problems and numerical methods.

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Section: Existence Resultsmentioning

confidence: 99%

Vector variational inequalities and related topics: A survey of theory and applications

2019

Appl. Set-Valued Anal. Optim.

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“…where K is a nonempty subset of X, T : X 2 L(X,Y) , which has been studied by Ansari et al [4]. (III) If, for each pair of parameters (l, μ) ∧ × ∨, C(x) = C, h(y, x) = y -x and j(y, x) = 0 for all x, y K (l), where C is a proper closed convex cone, then the problem (2.1) is reduced to the following vector quasi-variational inequality problem:…”

Section: Preliminariesmentioning

confidence: 99%

“…Since Giannessi, vector variational inequalities were investigated by many authors in abstract spaces and widely applied to transportation, finance, and economics, mathematical physics, engineering sciences and many others (see, for instance, [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and the reference therein).…”

Section: Introductionmentioning

confidence: 99%

Stability analysis for parametric generalized vector quasi-variational-like inequality problems

et al. 2012

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In this article, we consider a class of parametric generalized vector quasi-variationallike inequality problem (for short, (PGVQVLIP)) in Hausdorff topological vector spaces, where the constraint set K and a set-valued mapping T are perturbed by different parameters, and establish the nonemptiness and upper semicontinuity of the solution mapping S for (PGVQVLIP) under some suitable conditions. By virtue of the gap function, sufficient conditions for the H-continuity and B-continuity of the solution mapping S of (PGVQVLIP) are also derived. Moreover, examples are provided for illustrating the presented results. 2000 Mathematics Subject Classification: 49J40; 90C33.

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“…The aim of this paper is to derive the existence results of GMVVLI-1. It is well known that GMVVLI-1 and GMVVLI-2 are extensions of the vector variational inequality, which is a generalized form of a variational inequality, having applications in different areas of optimization, optimal control, operations research, economics equilibrium and free boundary value problems, see, for instance, [1][2][3] and the references therein. Now we give some definitions and lemma needed for the proof of the existence results.…”

Section: Introductionmentioning

confidence: 99%

Existence of Solution to Generalized Multivalued Vector Variational-like Inequalities

Wu¹,

Shen²,

Zhao³

2016

Proceedings of the 2016 International Conference on Intelligent Control and Computer Application

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Generalized vector variational-like inequalities and vector optimization

Cited by 26 publications

References 18 publications

Vector variational inequalities and related topics: A survey of theory and applications

Vector variational inequalities and related topics: A survey of theory and applications

Stability analysis for parametric generalized vector quasi-variational-like inequality problems

Existence of Solution to Generalized Multivalued Vector Variational-like Inequalities

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