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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Karamardian, S.: Complementarity problems over cones with monotone or pseudomonotone maps. J. Optim. Theory Appl. 18, 445–454 (1976)
Papageorgiou, N.S., Kyritsi, S.Th.: Handbook of Applied Analysis. Springer, Dordrecht, Heidelberg, London, New York (2009)
Aussel, D., Corvellec, J.-N., Lassonde, M.: Subdifferential characterization of quasiconvexity and convexity. J. Convex Anal. 1, 195–201 (1994)
Hadjisavvas, N.: Translations of quasimonotone maps and monotonicity. Appl. Math. Lett. 19, 913–915 (2006)
Penot, J.P., Quang, P.H.: Generalized convexity of functions and generalized monotonicity of set-valued maps. J. Optim. Theory Appl. 92, 343–356 (1997)
Rockafellar, T.: Generalized directional derivatives and subgradients of nonconvex functions. Can. J. Math. 32, 257–280 (1980)
Correa, R., Jofré, A., Thibault, L.: Subdifferential monotonicity as characterization of convex functions. Numer. Funct. Anal. Optim. 15, 531–535 (1994)
Hadjisavvas, N.: Generalized convexity, generalized monotonicity and nonsmooth analysis. In: Hadjisavvas, N., Komlosi, S., Schaible, S. (eds.) Handbook of Generalized Convexity and Generalized Monotonicity, pp. 465–500. Springer, New York (2005)
Daniilidis, A., Hadjisavvas, N., Martinez Legaz, J.-E.: An appropriate subdifferential for quasiconvex functions. SIAM J. Optim. 12, 407–420 (2001)
Brighi, L., John, R.: Characterizations of pseudomonotone maps and economic equilibrium. J. Stat. Manag. Syst. 5, 253–273 (2002)
Karamardian, S., Schaible, S., Crouzeix, J.-P.: Characterizations of generalized monotone maps. J. Optim. Theory Appl. 76, 399–413 (1993)
Daniilidis, A., Hadjisavvas, N.: Coercivity conditions and variational inequalities. Math. Program. 86, 433–438 (1999)
Cottle, R.W., Yao, J.-C.: Pseudo-monotone complementarity problems in Hilbert space. J. Optim. Theory Appl. 75, 281–295 (1992)
Yao, J.-C.: Multi-valued variational inequalities with K-pseudomonotone operators. J. Optim. Theory Appl. 83, 391–403 (1994)
Aussel, D., Hadjisavvas, N.: On quasimonotone variational inequalities. J. Optim. Theory Appl. 121, 445–450 (2002)
Konnov, I.V.: On quasimonotone variational inequalities. J. Optim. Theory Appl. 99, 165–181 (1998)
Hadjisavvas, N.: Continuity and maximality properties for pseudomonotone operators. J. Convex Anal. 10, 459–469 (2003)
Dontchev, A.L., Hager, W.: Implicit functions, Lipschitz maps, and stability in optimization. Math. Oper. Res. 19, 753–768 (1994)
Hadjisavvas, N., Schaible, S.: Pseudomonotone∗ maps and the cutting plane property. J. Glob. Optim. 43, 565–575 (2009)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley Interscience, New York (1983)
Levy, A.B., Poliquin, R.A.: Characterizing the single-valuedness of multifunctions. Set-Valued Anal. 5, 351–364 (1997)
Bruck, R.E. Jr.: An iterative solution of a variational inequality for certain monotone operators in Hilbert space, Research announcement. Bull. Am. Math. Soc. 81, 890–892 (1975)
Bruck, R.E. Jr.: Corrigendum. Bull. Am. Math. Soc. 82, 353 (1976)
Crouzeix, J.P., Marcotte, P., Zhu, D.: Conditions ensuring the applicability of cutting-plane methods for solving variational inequalities. Math. Program. 88, 521–539 (2000)
Castellani, M., Giuli, M.: A characterization of the solution set of pseudoconvex extremum problems. J. Convex Anal. 19(1) (2012, to appear)
Hadjisavvas, N., Schaible, S.: On a generalization of paramonotone maps and its application to solving the Stampacchia variational inequality. Optimization 55, 593–604 (2006)
Burke, J.V., Ferris, M.C.: Characterization of solution sets of convex programs. Oper. Res. Lett. 10, 57–60 (1991)
Varian, H.R.: Revealed preference with a subset of goods. J. Econ. Theory 46, 179–185 (1988)
Crouzeix, J.P., Eberhard, A., Ralph, D.: A geometrical insight on pseudoconvexity and pseudomonotonicity. Math. Program., Ser. B 123, 61–83 (2010)
Crouzeix, J.P., Keraghel, A., Rahmani, N.: Integration of pseudomonotone maps and the revealed preference problem. Optimization (2011, to appear). doi:10.1080/02331934.2010.531135
Author information
Authors and Affiliations
Corresponding author
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.
Rights and permissions
About this article
Cite this article
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
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-011-9912-5