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A regularization method for the proximal point algorithm. (English) Zbl 1131.90062
Summary: A regularization method for the proximal point algorithm of finding a zero for a maximal monotone operator in a Hilbert space is proposed. Strong convergence of this algorithm is proved.

MSC:
90C30Nonlinear programming
49J40Variational methods including variational inequalities
47J20Inequalities involving nonlinear operators
65J15Equations with nonlinear operators (numerical methods)
References:
[1]Brezis H. (1973). Operateurs Maximaux Monotones et Semi-Groups de Contractions dans les Espaces de Hilbert. North-Holland, Amsterdam
[2]Goebel, K. and Kirk, W.A. (1990), Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, Vol. 28, Cambridge University Press.
[3]Güler O. (1991) On the convergence of the proximal point algorithm for convex optimization. SIAM Journal of Control Optimization 29:403–419 · Zbl 0737.90047 · doi:10.1137/0329022
[4]Kamimura S. and Takahashi W. (2003), Strong convergence of a proximal-type algorithm in a Banach space. SIAM Journal of Optimization 13(3):938–945 · Zbl 1101.90083 · doi:10.1137/S105262340139611X
[5]Lehdili N. and Moudafi A. (1996). Combining the proximal algorithm and Tikhonov regularization. Optimization 37:239–252 · Zbl 0863.49018 · doi:10.1080/02331939608844217
[6]Rockafellar R.T. (1976). Monotone operators and the proximal point algorithm. SIAM Journal of Control Optimization 14: 877–898 · Zbl 0358.90053 · doi:10.1137/0314056
[7]Solodov M.V. and Svaiter B.F. (2000), Forcing strong convergence of proximal point iterations in a Hilbert space. Mathematical Programming, Series A 87:189–202
[8]Xu H.K. (2002). Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society 66:240–256 · Zbl 1013.47032 · doi:10.1112/S0024610702003332