Numbers of the connected components of the solution sets of monotone affine vector variational inequalities

Hieu, Vu Trung

doi:10.1007/s10898-018-0678-2

Cited by 2 publications

(3 citation statements)

References 9 publications

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“…Remark 4.6. The set-valued mapping H : R k ⇒ R l , given by relation (4.6), is frequently called base multifunction associated to problem (2.2); see Section 2 in [120].…”

Section: 2mentioning

confidence: 99%

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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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“…Remark 4.6. The set-valued mapping H : R k ⇒ R l , given by relation (4.6), is frequently called base multifunction associated to problem (2.2); see Section 2 in [120].…”

Section: 2mentioning

confidence: 99%

“…Consequences of such results in the field of multi-objective optimization are discussed as well. In [120], Hieu establishes several upper and lower estimates for the maximal number of the connected components of the solution set of a monotone affine vector variational inequality. 3) is a semi-algebraic set.…”

Section: 2mentioning

confidence: 99%

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

2019

Appl. Set-Valued Anal. Optim.

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“…The authors in [8] proved that if the constraint set is bounded then the weak Pareto solution set is connected. Several upper and lower estimates for the maximal number of the connected components of the solution sets of monotone affine vector variational inequalities are established in [4]. The last paper pointed out that the number depends not only on the number of the criteria but also on the number of variables of the vector variational inequality.…”

Section: Introductionmentioning

confidence: 99%

Disconnectedness and unboundedness of the solution sets of monotone vector variational inequalities

Hieu¹

2019

Applicable Analysis

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Numbers of the connected components of the solution sets of monotone affine vector variational inequalities

Cited by 2 publications

References 9 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

Disconnectedness and unboundedness of the solution sets of monotone vector variational inequalities

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