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Pseudomonotone Operators: A Survey of the Theory and Its Applications

Author

Listed:
  • N. Hadjisavvas

    (University of the Aegean)

  • S. Schaible

    (Chung Yuan Christian University)

  • N.-C. Wong

    (National Sun Yat-Sen University)

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.

Suggested Citation

  • N. Hadjisavvas & S. Schaible & N.-C. Wong, 2012. "Pseudomonotone Operators: A Survey of the Theory and Its Applications," Journal of Optimization Theory and Applications, Springer, vol. 152(1), pages 1-20, January.
    • Handle: RePEc:spr:joptap:v:152:y:2012:i:1:d:10.1007_s10957-011-9912-5
    DOI: 10.1007/s10957-011-9912-5
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    References listed on IDEAS

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    1. Varian, Hal R., 1988. "Revealed preference with a subset of goods," Journal of Economic Theory, Elsevier, vol. 46(1), pages 179-185, October.
    2. I. V. Konnov, 1998. "On Quasimonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 99(1), pages 165-181, October.
    3. Asen L. Dontchev & William W. Hager, 1994. "Implicit Functions, Lipschitz Maps, and Stability in Optimization," Mathematics of Operations Research, INFORMS, vol. 19(3), pages 753-768, August.
    4. J.-P. Penot & P. H. Quang, 1997. "Generalized Convexity of Functions and Generalized Monotonicity of Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 92(2), pages 343-356, February.
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    Cited by:

    1. Boţ, R.I. & Csetnek, E.R. & Vuong, P.T., 2020. "The forward–backward–forward method from continuous and discrete perspective for pseudo-monotone variational inequalities in Hilbert spaces," European Journal of Operational Research, Elsevier, vol. 287(1), pages 49-60.
    2. Alfredo Iusem & Felipe Lara, 2019. "Existence Results for Noncoercive Mixed Variational Inequalities in Finite Dimensional Spaces," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 122-138, October.

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