An Oriented Distance Function Application to Gap Functions for Vector Variational Inequalities

Altangerel, L.; Wanka, Gert; Wilfer, Oleg

doi:10.1007/978-1-4614-5131-0_2

Cited by 4 publications

(4 citation statements)

References 17 publications

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“…The following result is Theorem 3.1 in [5] and provides a set-valued gap function for problem (2.1). Proof.…”

Section: Gap Functionsmentioning

confidence: 93%

“…compare Section 3 in [5]. Recall that ∆ K enjoys the representability condition ∆ K (y) < 0 for y ∈ int K; compare Proposition 8.1.…”

Section: Gap Functionsmentioning

confidence: 96%

“…To the best of our knowledge, the introduction of gap functions for vector variational inequalities has been first proposed in [39,54]. Since then, there has been a considerable research effort in the study of gap functions for generalized vector variational inequalities and related problems; see, for example, [5,33,46,48,114,158,159,160] From the computational point of view, such a transformation allows one to apply powerful optimization methods and algorithms for finding solutions of the vector variational inequality. Besides that, gap functions play a fundamental role for developing error bounds for vector variational inequalities.…”

Section: Gap Functionsmentioning

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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“…The following result is Theorem 3.1 in [5] and provides a set-valued gap function for problem (2.1). Proof.…”

Section: Gap Functionsmentioning

confidence: 93%

“…compare Section 3 in [5]. Recall that ∆ K enjoys the representability condition ∆ K (y) < 0 for y ∈ int K; compare Proposition 8.1.…”

Section: Gap Functionsmentioning

confidence: 96%

Section: Gap Functionsmentioning

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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“…The signed distance function has been used in a broad spectrum of applications including binary classification, computer vision, shape analysis, image processing, computational physics, etc. ; see the publications [1,3,9,14,15,14,28] with their bibliographies. We particularly refer the reader to [16] for a thorough and deep study of the class of signed distance functions on finite-dimensional Euclidean spaces with a survey of the previous achievements.…”

Section: Introductionmentioning

confidence: 99%

Convex Analysis of Minimal Time and Signed Minimal Time Functions

Cuong¹,

Mordukhovich²,

Nam³

et al. 2020

Preprint

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In this paper we first consider the class of minimal time functions in the general setting of locally convex topological vector (LCTV) spaces. The results obtained in this framework are based on a novel notion of closedness of target sets with respect to constant dynamics. Then we introduce and investigate a new class of signed minimal time functions, which are generalizations of the signed distance functions. Subdifferential formulas for the signed minimal time and distance functions are obtained under the convexity assumptions on the given data.

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