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On convergence criteria of generalized proximal point algorithms. (English) Zbl 1147.65049
The authors analyze some generalized proximal point algorithms for finding a zero of maximal monotone operators, which include the previously known proximal point algorithms as special cases. Weak and strong convergence of the proposed proximal point algorithms are proved under some mild conditions. No numerical tests are presented.

65K10Optimization techniques (numerical methods)
49J40Variational methods including variational inequalities
[1]Noor, M. Aslam: Some developments in general variational inequalities, Appl. math. Comput. 152, 199-277 (2004) · Zbl 1134.49304 · doi:10.1016/S0096-3003(03)00558-7
[2]Noor, M. Aslam: Projection-proximate methods for general variational inequalities, J. math. Anal. appl. 318, 53-62 (2006) · Zbl 1086.49005 · doi:10.1016/j.jmaa.2005.05.024
[3]V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leiden, 1976.
[4]Bnouhachem, A.; Noor, M. Aslam: Inexact proximal point method for general variational inequalities, J. math. Anal. appl. 324, 1195-1212 (2006) · Zbl 1101.49026 · doi:10.1016/j.jmaa.2006.01.014
[5]Chidume, C. E.; Chidume, C. O.: Iterative approximation of fixed points of nonexpansive mappings, J. math. Anal. appl. 318, 288-295 (2006) · Zbl 1095.47034 · doi:10.1016/j.jmaa.2005.05.023
[6]Eckstein, J.; Bertsckas, D. P.: On the Douglas – Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. programming 55, 293-318 (1992) · Zbl 0765.90073 · doi:10.1007/BF01581204
[7]Gol’shtein, E. G.; Tret’yakov, N. V.: Modified Lagrangians in convex programming and their generalizations, Math. programming stud. 10, 86-97 (1979) · Zbl 0404.90069
[8]Güler, O.: On the convergence of the proximal point algorithm for convex optimization, SIAM J. Control optim. 29, 403-419 (1991) · Zbl 0737.90047 · doi:10.1137/0329022
[9]Halpern, B.: Fixed points of nonexpanding maps, Bull. amer. Math. soc. 73, 957-961 (1967) · Zbl 0177.19101 · doi:10.1090/S0002-9904-1967-11864-0
[10]Han, D.; He, B. S.: A new accuracy criterion for approximate proximal point algorithms, J. math. Anal. appl. 263, 343-354 (2001) · Zbl 0995.65062 · doi:10.1006/jmaa.2001.7535
[11]He, B. S.: Inexact implicit methods for monotone general variational inequalities, Math. programming 86, 113-123 (1999) · Zbl 0979.49006 · doi:10.1007/s101070050086
[12]Marino, G.; Xu, H. K.: Convergence of generalized proximal point algorithm, Comm. pure appl. Anal. 3, 791-808 (2004) · Zbl 1095.90115 · doi:10.3934/cpaa.2004.3.791
[13]Rockafellar, R. T.: Monotone operators and the proximal point algorithm, SIAM J. Control optim. 14, 877-898 (1976) · Zbl 0358.90053 · doi:10.1137/0314056
[14]Solodov, M. V.; Svaiter, B. F.: Forcing strong convergence of proximal point iterations in a Hilbert space, Math. programming 87, 189-202 (2000) · Zbl 0971.90062 · doi:10.1007/s101079900113
[15]Solodov, M. V.; Svaiter, B. F.: A unified framework for some inexact proximal point algorithms, Numer. funct. Anal. optim. 22, 1013-1035 (2001) · Zbl 1052.49013 · doi:10.1081/NFA-100108320
[16]Suzuki, T.: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. math. Anal. appl. 305, 227-239 (2005) · Zbl 1068.47085 · doi:10.1016/j.jmaa.2004.11.017
[17]Suzuki, T.: A sufficient and necessary condition for halpern-type strong convergence to fixed points of nonexpansive mappings, Proc. amer. Math. soc. 135, 99-106 (2007) · Zbl 1117.47041 · doi:10.1090/S0002-9939-06-08435-8
[18]Xu, H. K.: Iterative algorithms for nonlinear operators, J. London math. Soc. 66, 240-256 (2002)