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Quasi-Variational Inequality Problems over Product Sets with Quasi-monotone Operators

Aussel, Didier and Van, K. Cao and Salas Videla, David Quasi-Variational Inequality Problems over Product Sets with Quasi-monotone Operators. (2019) SIAM Journal on Optimization, 29 (2). 1558-1577. ISSN 1052-6234

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Official URL: https://doi.org/10.1137/18M1191270

Abstract

Quasi-variational inequalities are variational inequalities in which the constraint map depends on the current point. Due to this characteristic, specific proofs have been built to prove adapted existence results. Semicontinuity and generalized monotonicity are assumed and many efforts have been made in the last decades to use the weakest concepts. In the case of quasi-variational inequalities defined on a product of spaces, the existence statements in the literature require pseudomonotonicity of the operator, a hypothesis that is too strong for many applications, in particular in economics. On the other hand, the current minimal hypotheses for existence results for general quasi-variational inequalities are quasi-monotonicity and local upper sign-continuity. But since the product of quasi-monotone (respectively, locally upper sign-continuous) operators is not in general quasi-monotone (respectively, locally upper sign-continuous), it is thus quite difficult to use these general-type existence result in the quasi-variational inequalities defined on a product of spaces. In this work we prove, in an infinite-dimensional setting, several existence results for product-type quasi-variational inequalities by only assuming the quasi-monotonicity and local upper sign-continuity of the component operators. Our technique of proof is strongly based on an innovative stability result and on the new concept of net-lower sign-continuity.

Item Type:Article
HAL Id:hal-02283071
Audience (journal):International peer-reviewed journal
Uncontrolled Keywords:
Institution:French research institutions > Centre National de la Recherche Scientifique - CNRS (FRANCE)
Université de Toulouse > Institut National Polytechnique de Toulouse - INPT (FRANCE)
Université de Toulouse > Université Toulouse III - Paul Sabatier - UPS (FRANCE)
Other partners > Université de Perpignan Via Domitia - UPVD (FRANCE)
Laboratory name:
Funders:
European Funds - Agence Nationale de la Recherche - ANR
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Deposited By: Loetitia MOYA
Deposited On:10 Sep 2019 12:23

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