Abstract
In this paper, we consider the proximal point algorithm for the problem of finding zeros of any given maximal monotone operator in an infinite-dimensional Hilbert space. For the usual distance between the origin and the operator’s value at each iterate, we put forth a new idea to achieve a new result on the speed at which the distance sequence tends to zero globally, provided that the problem’s solution set is nonempty and the sequence of squares of the regularization parameters is nonsummable. We show that it is comparable to a classical result of Brézis and Lions in general and becomes better whenever the proximal point algorithm does converge strongly. Furthermore, we also reveal its similarity to Güler’s classical results in the context of convex minimization in the sense of strictly convex quadratic functions, and we discuss an application to an ϵ-approximation solution of the problem above.
Similar content being viewed by others
References
Martinet, B.: Regularisation d’inéquations variationelles par approximations successives. Rev. Fr. Inform. Rech. Oper. 4, 154–158 (1970)
Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)
Brézis, H., Lions, P.L.: Produits infinis de resolvantes. Isr. J. Math. 29, 329–345 (1978)
Güler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29, 403–419 (1991)
Zaslavski, A.J.: Proximal point algorithm for finding a common zero of a finite family of maximal monotone operators in the presence of computational errors. Nonlinear Anal. 75(16), 6071–6087 (2012)
Minty, G.J.: Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29, 341–346 (1962)
Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis, Interscience Publications. Wiley, New York (1984)
Moreau, J.J.: Proximité et dualité dans un espace Hilbertien. Bull. Soc. Math. Fr. 93, 273–299 (1965)
Zaslavski, A.J.: Maximal monotone operators and the proximal point algorithm in the presence of computational errors. J. Optim. Theory Appl. 150, 20–32 (2011)
Luque, F.J.: Asymptotic convergence analysis of the proximal point algorithm. SIAM J. Control Optim. 22, 277–293 (1984)
Pennanen, T.: Local convergence of the proximal point algorithm and multiplier methods without monotonicity. Math. Oper. Res. 27, 193–202 (2002)
Dong, Y.D.: A new relative error criterion for the proximal point algorithm. Nonlinear Anal. 64, 2143–2148 (2006)
Hardy, G.H.: Divergent Series. Clarendon Press, Oxford (1949)
Nemirovski, A., Onn, S., Rothblum, U.: Accuracy certificates for computational problems with convex structure. Math. Oper. Res. 35(11), 52–78 (2010)
Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)
Lawrence, J., Spingarn, J.E.: On fixed points of non-expansive piecewise isometric mappings. Proc. Lond. Math. Soc. 55, 605–624 (1987)
Eckstein, J., Bertsekas, D.P.: On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55(3), 293–318 (1992)
Combettes, P.L.: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 53, 475–504 (2004)
Dong, Y.D., Fischer, A.: A family of operator splitting methods revisited. Nonlinear Anal. 72, 4307–4315 (2010)
Acknowledgements
The author is very grateful to the associated editor and the referees for their valuable suggestions and helpful comments which improve the presentation of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Qamrul Hasan Ansari.
Rights and permissions
About this article
Cite this article
Dong, Y. The Proximal Point Algorithm Revisited. J Optim Theory Appl 161, 478–489 (2014). https://doi.org/10.1007/s10957-013-0351-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-013-0351-3