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Pseudomonotone\({_{\ast}}\) maps and the cutting plane property

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Abstract

Pseudomonotone\({_{\ast}}\) maps are a generalization of paramonotone maps which is very closely related to the cutting plane property in variational inequality problems (VIP). In this paper, we first generalize the so-called minimum principle sufficiency and the maximum principle sufficiency for VIP with multivalued maps. Then we show that pseudomonotonicity\({_{\ast}}\) of the map implies the “maximum principle sufficiency” and, in fact, is equivalent to it in a sense. We then present two applications of pseudomonotone\({_{\ast}}\) maps. First we show that pseudomonotone\({_{\ast}}\) maps can be used instead of the much more restricted class of pseudomonotone+ maps in a cutting plane method. Finally, an application to a proximal point method is given.

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Authors and Affiliations

  1. Department of Product and Systems Design Engineering, University of the Aegean, 84100, Hermoupolis, Syros, Greece

    Nicolas Hadjisavvas

  2. A.G. Anderson Graduate School of Management, University of California, Riverside, CA, 92512, USA

    Siegfried Schaible

Authors
  1. Nicolas Hadjisavvas
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  2. Siegfried Schaible
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Correspondence to Nicolas Hadjisavvas.

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Hadjisavvas, N., Schaible, S. Pseudomonotone\({_{\ast}}\) maps and the cutting plane property. J Glob Optim 43, 565–575 (2009). https://doi.org/10.1007/s10898-008-9335-5

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  • DOI: https://doi.org/10.1007/s10898-008-9335-5

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