Article Outline
Introduction/Background
Definitions
Formulation
Maximal Pseudomonotonicity
A Generalization of Paramonotone Maps
Pseudoaffine Maps
Pseudomonotone vs. Monotone Maps
Methods/Applications
Conclusions
References
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References
Aussel D, Corvellec JN, Lassonde M (1994) Subdifferential characterization of quasiconvexity and convexity. J Convex Anal 1:195–201
Bianchi M, Hadjisavvas N, Schaible S (2003) On Pseudomonotone Maps T for which \( -T \) is also pseudomonotone. J Convex Anal 10:149–168
Bianchi M, Schaible S (2000) An extension of pseudolinear functions and variational inequality problems. J Optim Theory Appl 104:59–71
Brezis H (1968) Equations et inéquations nonlinéaires dans les espaces vectoriels en dualité. Ann Inst Fourier 18:115–175
Bruck RE Jr (1976) An iterative solution of a variational inequality for certain monotone operators in Hilbert space. Bull Amer Math Soc 81:890–892; Corrigendum, Bull Amer Math Soc 82 (1976) 353
Burachik R, Iusem A (1998) A generalized proximal point algorithm for the variational inequality problem in Hilbert space. SIAM J Optim 8:197–216
Clarke FH (1983) Optimization and nonsmooth Analysis. Wiley Interscience, New York
Crouzeix JP, Marcotte P, Zhu D (2000) Conditions ensuring the applicability of cutting-plane methods for solving variational inequalities. Math Program 88:521–539
Daniilidis A, Hadjisavvas N (1999) On the subdifferentials of pseudoconvex and quasiconvex functions and cyclic monotonicity. J Math Anal Appl 237:30–42
Daniilidis A, Hadjisavvas N (2000) On generalized cyclically monotone operators and proper quasimonotonicity. Optimization 47:123–135
Domokos A, Kolumban J (2000) Comparison of two different types of pseudomonotone mappings, In: Popoviciu E (ed) Seminaire de la théorie de la meilleure approximation, convexité et optimisation. Editura SRIMA, Cluj-Napoca, pp 95–103
Gwinner J (1981) On fixed points and variational inequalities—a circular tour. Nonlinear Anal 5:565–583
Hadjisavvas N (2003) Continuity and maximality properties of pseudomonotone operators. J Convex Anal 10:459–469
Hadjisavvas N (2003) Maximal pseudomonotone operators, In: Crespi GP, Guerraggio A, Miglierina E, Rocca M (eds) Recent advances in Optimization. Datanova Editrice, Milano, pp 87–100
Hadjisavvas N (2006) Translations of quasimonotone maps and monotonicity. Appl Math Lett 19:913–915
Hadjisavvas N, Komlosi S, Schaible S (2005) Handbook of generalized convexity and generalized monotonicity. Springer, New York
Hadjisavvas N, Schaible S (2006) On a generalization of paramonotone maps and its application to solving the Stampacchia variational inequality. Optim 55:593–604
He Y (2004) A relationship between pseudomonotone and monotone mappings. Appl Math Lett 17:459–461
Hu S, Papageorgiou NS (1997) Handbook of Multivalued Analysis, vols I, II. Kluwer, Dordrecht
Isac G, Motreanu D (2004) Pseudomonotonicity and quasimonotonicity by translations versus monotonicity in Hilbert spaces. Austral J Math Anal Appl 1:1–8
Iusem AN (1998) On some properties of paramonotone operators. J Convex Anal 5:269–278
Karamardian S (1976) Complementarity over cones with monotone and pseudomonotone maps. J Optim Theory Appl 18:445–454
Penot J-P, Quang PH (1997) Generalized convexity of functions and generalized monotonicity of set-valued maps. J Optim Theory Appl 92:343–356
Solodov MD, Svaiter BF (2000) An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions. Math Oper Res 25:214–230
Thompson WA, Parke DW (1973) Some properties of generalized concave functions. Oper Res 21:305–313
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Hadjisavvas, N. (2008). Pseudomonotone Maps: Properties and Applications . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_531
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DOI: https://doi.org/10.1007/978-0-387-74759-0_531
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