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Vector Variational Inequalities Involving Set-valued Mappings via Scalarization with Applications to Error Bounds for Gap Functions

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Abstract

In this paper, by using the scalarization approach of Konnov, several kinds of strong and weak scalar variational inequalities (SVI and WVI) are introduced for studying strong and weak vector variational inequalities (SVVI and WVVI) with set-valued mappings, and their gap functions are suggested. The equivalence among SVVI, WVVI, SVI, WVI is then established under suitable conditions and the relations among their gap functions are analyzed. These results are finally applied to the error bounds for gap functions. Some existence theorems of global error bounds for gap functions are obtained under strong monotonicity and several characterizations of global (respectively local) error bounds for the gap functions are derived.

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Authors and Affiliations

  1. School of Mathematics and Information, China West Normal University, Nanchong, Sichuan, 637002, P.R. China

    J. Li

  2. Department of Mathematics, University of Pisa, Largo B. Pontecorvo 5, 56127, Pisa, Italy

    G. Mastroeni

Authors
  1. J. Li
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  2. G. Mastroeni
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Corresponding author

Correspondence to J. Li.

Additional information

Communicated by F. Giannessi.

This work was supported by the National Natural Science Foundation of China (60804065), the Applied Research Project of Sichuan Province, the National Natural Science Foundation of Sichuan Province (07ZA123) and the Fundation of China West Normal University (08B075).

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Li, J., Mastroeni, G. Vector Variational Inequalities Involving Set-valued Mappings via Scalarization with Applications to Error Bounds for Gap Functions. J Optim Theory Appl 145, 355–372 (2010). https://doi.org/10.1007/s10957-009-9625-1

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