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A proximal point algorithm converging strongly for general errors. (English) Zbl 1202.90271
Summary: A proximal point algorithm (PPA) for maximal monotone operators with appropriate regularization parameters is considered. A strong convergence result for the PPA is stated and proved under the general condition that the error sequence tends to zero in norm. Note that R. T. Rockafellar [SIAM J. Control Optimization 14, 877–898 (1976; Zbl 0358.90053)] assumed summability for the error sequence to derive weak convergence of the PPA in its initial form, and this restrictive condition on the errors has been extensively used so far for different versions of the PPA. Thus, this Note provides a solution to a long standing open problem and in particular offers new possibilities towards the approximation of the minimum points of convex functionals.

MSC:
90C48Programming in abstract spaces
90C56Derivative-free methods; methods using generalized derivatives
References:
[1]Boikanyo O.A., Moroşanu G.: Modified Rockafellar’s algorithms. Math. Sci. Res. J. 13(5), 101–122 (2009)
[2]Lehdili N., Moudafi A.: Combining the proximal point for convex optimization. Optimization 37, 239–252 (1996) · Zbl 0863.49018 · doi:10.1080/02331939608844217
[3]Moroşanu G.: Nonlinear Evolution Equations and Applications. D Reidel, Dordrecht (1988)
[4]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
[5]Song Y., Yang C.: A note on the paper ”A regularization method for the proximal point algorithm”. J. Glob. Optim. 43, 171–174 (2009) · Zbl 1165.49012 · doi:10.1007/s10898-008-9279-9
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[7]Xu H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66(2), 240–256 (2002) · Zbl 1013.47032 · doi:10.1112/S0024610702003332