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Pseudomonotone Operators: A Survey of the Theory and Its Applications

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Abstract

The notion of pseudomonotone operator in the sense of Karamardian has been studied for 35 years and has found many applications in variational inequalities and economics. The purpose of this survey paper is to present the most fundamental results in this field, starting from the earliest developments and reaching the latest results and some open questions. The exposition includes: the relation of (generally multivalued) pseudomonotone operators to pseudoconvex functions; first-order characterizations of single-valued, differentiable pseudomonotone operators; application to variational inequalities; the notion of equivalence of pseudomonotone operators and its application to maximality; a generalization of paramonotonicity and its relation to the cutting-plane method; and the relation to the revealed preference problem of mathematical economics.

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Authors and Affiliations

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

    N. Hadjisavvas

  2. Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li, Taiwan

    S. Schaible

  3. Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, 804, Taiwan

    N.-C. Wong

Authors
  1. N. Hadjisavvas
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  2. S. Schaible
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  3. N.-C. Wong
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Corresponding author

Correspondence to N. Hadjisavvas.

Additional information

This paper was completed while the author N. Hadjisavvas was visiting the Department of Applied Mathematics, National Sun Yat-Sen University, Taiwan. The author wishes to thank the Department for its hospitality.

N.-C. Wong partially supported by Taiwan NSC grant no. 99-2115-M-110-007-MY3.

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Hadjisavvas, N., Schaible, S. & Wong, NC. Pseudomonotone Operators: A Survey of the Theory and Its Applications. J Optim Theory Appl 152, 1–20 (2012). https://doi.org/10.1007/s10957-011-9912-5

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