Painlevé–Kuratowski convergences of the solution sets for generalized vector quasi-equilibrium problems

Anh, Lâm Quốc; Bantaojai, Thanatporn; Hùng, Nguyễn Văn; Tâm, Võ Minh; Wangkeeree, Rabian

doi:10.1007/s40314-017-0548-4

Cited by 14 publications

(14 citation statements)

References 42 publications

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“…On the other hand, from a practical point of view, one cannot expect that the data of the vector variational inequality (2.2), here the constraining set C ⊆ R l and the mapping F : R l → Mat k×l (R), are known. Due to this incompleteness, many authors have studied problem (2.2) with respect to contaminated/perturbed data, depending on some parameters signifying the level of error in a certain sense; see [6,49,52,58,85,89,124,162] and the references therein. To be precise, let U and V be Banach spaces, the so-called parameter spaces, let E : U ⇒ R l be a set-valued mapping with non-empty, closed and convex values, and let F : V × R l → Mat k×l (R) be a mapping.…”

Section: Stability and Sensitivity Analysismentioning

confidence: 99%

“…Note that an initial concept of well-posedness for scalar optimization problems has been introduced in [183] for the first time, known as Tykhonov well-posedness. In addition to the previous research areas related to stability and sensitivity analysis, the Painlevé-Kuratowski upper and lower convergence of perturbed vector variational inequalities has been investigated in [6,87].…”

Section: Stability and Sensitivity Analysismentioning

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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Section: Stability and Sensitivity Analysismentioning

confidence: 99%

Section: Stability and Sensitivity Analysismentioning

confidence: 99%

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

2019

Appl. Set-Valued Anal. Optim.

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“…Such error estimates have played a vital role in convergence analysis of iterative algorithms for solving variational inequalities. In recent years, there have been many studies on gap functions for different models on different topics such as iterative algorithms [23], the Painlevé-Kuratowski convergence [2], stability of solutions [3,[20][21][22] and error bounds [6,13,[24][25][26]. We also refer the reader to [1,5,7,11,14,[27][28][29] and the references therein for a more detailed discussion of interesting topic.…”

Section: Introductionmentioning

confidence: 99%

Gap Functions and Error Bounds for Variational–Hemivariational Inequalities

et al. 2020

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In this paper we investigate the gap functions and regularized gap functions for a class of variational-hemivariational inequalities of elliptic type. First, based on regularized gap functions introduced by Yamashita and Fukushima, we establish some regularized gap functions for the variational-hemivariational inequalities. Then, the global error bounds for such inequalities in terms of regularized gap functions are derived by using the properties of the Clarke generalized gradient. Finally, an application to a stationary nonsmooth semipermeability problem is given to illustrate our main results.

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“…It was first introduced by Auslender to transform the variational inequality problem into an equivalent optimization problem. The gap function in Auslender for the scalar variational inequality problem (with

H_{1} = {double-struckR}^{n}

) is defined by

ω (x) = \sup_{y \in P} ⟨ f (x), x - y ⟩ .

In recent years, there have been many authors studying gap functions for different models on different topics such as iterative algorithms, the Painlevé‐Kuratowski convergence, and stability of solutions . On the other hand, in 1992, Fukushima extended to the concept of regularized gap function for the scalar variational inequality problem based on the gap function of Auslender as follows:

ω_{θ} (x) = \sup_{y \in P} \{⟨ f (x), x - y ⟩ - \frac{1}{2 θ} ‖ x - y ‖^{2}\},

where

θ > 0

is a regularized parameter.…”

Section: Introductionmentioning

confidence: 99%

Regularized gap functions and error bounds for split mixed vector quasivariational inequality problems

2020

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In this paper, we consider a class of split mixed vector quasivariational inequality problems in real Hilbert spaces and establish new gap functions by using the method of the nonlinear scalarization function. Further, we obtain some error bounds for the underlying split mixed vector quasivariational inequality problems in terms of regularized gap functions. Finally, we give some examples to illustrate our results. The results obtained in this paper are new.

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Painlevé–Kuratowski convergences of the solution sets for generalized vector quasi-equilibrium problems

Cited by 14 publications

References 42 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

Gap Functions and Error Bounds for Variational–Hemivariational Inequalities

Regularized gap functions and error bounds for split mixed vector quasivariational inequality problems

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