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On cyclic and \(n\)-cyclic monotonicity of bifunctions. (English) Zbl 1329.47050

The authors generalize some well-known connections between the maximal monotone bifunctions and their Fitzpatrick functions to the maximal cyclically monotone bifunctions and the corresponding Fitzpatrick transforms.

MSC:

47H05 Monotone operators and generalizations
47S30 Constructive operator theory
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
90C48 Programming in abstract spaces
Full Text: DOI

References:

[1] Ait Mansour, M., Chbani, Z., Riahi, H.: Recession bifunction and solvability of noncoercive equilibrium problems. Commun. Appl. Anal. 7, 369-377 (2003) · Zbl 1085.49501
[2] Alizadeh, M.H.: Monotone and generalized monotone bifunctions and their application to operator theory. PhD thesis, Department of Product and Systems Design Engineering, University of the Aegean (2012) · Zbl 1242.90288
[3] Alizadeh, M.H., Hadjisavvas, N.: On the Fitzpatrick transform of a monotone bifunction. Optimization 62, 693-701 (2013) · Zbl 1285.47059 · doi:10.1080/02331934.2011.653975
[4] Alizadeh, M.H., Hadjisavvas, N.: Local boundedness of monotone bifunctions. J. Glob. Optim. 53, 231-241 (2012) · Zbl 1274.90489 · doi:10.1007/s10898-011-9677-2
[5] Bartz, S., Bauschke, H.H., Borwein, J., Reich, S., Wang, X.: Fitzpatrick functions, cyclic monotonicity and Rockafellar’s antiderivative. Nonlinear Anal. 66, 1198-1223 (2007) · Zbl 1119.47050 · doi:10.1016/j.na.2006.01.013
[6] Bartz, S., Reich, S.: Minimal antiderivatives and monotonicity. Nonlinear Anal. 74, 59-66 (2011) · Zbl 1251.47046 · doi:10.1016/j.na.2010.08.015
[7] Bartz, S., Reich, S.: Abstract convex optimal antiderivatives. Ann. Inst. H. Poincaré Anal. Non Linéaire 29, 435-454 (2012) · Zbl 1259.47064 · doi:10.1016/j.anihpc.2012.01.004
[8] Bauschke, H.H., Wang, X.: A convex-analytical approach to extension results for n-cyclically monotone operators. Set-Valued Anal. 15, 297-306 (2007) · Zbl 1133.47037 · doi:10.1007/s11228-006-0029-1
[9] Bauschke, H.H., Wang, X.: An explicit example of a maximal 3-cyclically monotone operator with bizarre properties. Nonlinear Anal. 69, 2875-2891 (2008) · Zbl 1170.47033 · doi:10.1016/j.na.2007.08.059
[10] Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123-145 (1994) · Zbl 0888.49007
[11] Borwein, J.M.: Maximal monotonicity via convex analysis. J. Convex Anal. 13, 561-586 (2006) · Zbl 1111.47042
[12] Boţ, R.I., Grad, S.-M.: Approaching the maximal monotonicity of bifunctions via representative functions. J. Convex Anal. 19, 713-724 (2012) · Zbl 1267.47079
[13] Boţ, R.I., Csetnek, E.R.: On extension results for n-cyclically monotone operators in reflexive Banach spaces. J. Math. Anal. Appl. 367, 693-698 (2010) · Zbl 1187.47038 · doi:10.1016/j.jmaa.2010.02.036
[14] Burachik, R.S., Fitzpatrick, S.: On a family of convex functions associated to subdifferentials. J. Nonlinear Convex Anal. 6, 165-171 (2005) · Zbl 1076.46056
[15] Hadjisavvas, N., Khatibzadeh, H.: Maximal monotonicity of bifunctions. Optimization 59, 147-160 (2010) · Zbl 1250.47050 · doi:10.1080/02331930801951116
[16] Iusem, A.N.: On the maximal monotonicity of diagonal subdifferential operators. J. Convex Anal. 18, 489-503 (2011) · Zbl 1223.90076
[17] Iusem, A.N., Svaiter, B.F.: On diagonal subdifferential operators in nonreflexive Banach spaces. Set-Valued Anal. 20, 1-14 (2012) · Zbl 1242.90288 · doi:10.1007/s11228-011-0189-5
[18] Kassay, G., Reich, S., Sabach, S.: Iterative methods for solving systems of variational inequalities in reflexive Banach spaces. SIAM J. Optim. 21, 1319-1344 (2011) · Zbl 1250.47064 · doi:10.1137/110820002
[19] Penot, J.-P.: The relevance of convex analysis for the study of monotonicity. Nonlinear Anal. 58, 855-871 (2004) · Zbl 1078.47008 · doi:10.1016/j.na.2004.05.018
[20] Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33, 209-216 (1970) · Zbl 0199.47101 · doi:10.2140/pjm.1970.33.209
[21] Zeidler, E.: Nonlinear Functional Analysis and its Applications, Vol. II/B, Nonlinear Monotone Operators. Springer, Berlin (1990) · Zbl 0684.47029 · doi:10.1007/978-1-4612-0981-2
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