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An Introduction to Vector Variational Inequalities and Some New Results

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Abstract

This survey paper discusses some new results on vector variational inequalities. It can serve as an elementary introduction to vector variational inequalities and vector optimization problems. The focus point is made on the results about connectedness structure of the solution sets, which are obtained by a scalarization method and properties of semi-algebraic sets. The first major theorem says that both Pareto solution set and weak Pareto solution set of a vector variational inequality, where the constraint set is polyhedral convex and the basic operators are given by polynomial functions, have finitely many connected components. The second major theorem asserts that both proper Pareto solution set and weak Pareto solution set of a vector variational inequality, where the convex constraint set is given by polynomial functions and all the components of the basic operators are polynomial functions, have finitely many connected components, provided that the Mangasarian-Fromovitz Constraint Qualification is satisfied at every point of the constraint set. In addition, it has been established that if the proper Pareto solution set is dense in the Pareto solution set, then the latter also has finitely many connected components. Consequences of the results for vector optimization problems are shown.

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Acknowledgments

The author was supported by Project VAST.HTQT.PHAP.02/14-15. He thanks Prof. Jen-Chih Yao and Dr. Nguyen Thi Thu Huong for the successful research collaboration, and Mr. Vu Xuan Truong for useful discussions on connectedness structure of semi-algebraic sets. Proposition 1 in this paper is due to Mr. Vu Trung Hieu.

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  1. Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, 10307, Vietnam

    Nguyen Dong Yen

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  1. Nguyen Dong Yen
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Correspondence to Nguyen Dong Yen.

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Dedicated to Professor Franco Giannessi on the occasion of his 80th birthday

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Yen, N.D. An Introduction to Vector Variational Inequalities and Some New Results. Acta Math Vietnam 41, 505–529 (2016). https://doi.org/10.1007/s40306-015-0168-2

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