Abstract
Very recently, the author gave an upper bound on a decreasing positive sequence. And, he made use of it to improve a classical result of Brézis and Lions concerning the proximal point algorithm for monotone inclusion in an infinite-dimensional Hilbert space. One assumption is the algorithm’s strong convergence. In this paper, we derive a new upper bound on this decreasing positive sequence and thus achieve the same improvement without requiring this assumption.
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References
Dong, Y.D.: The proximal point algorithm revisited. J. Optim. Theory Appl. 161, 478–489 (2014)
Brézis, H., Lions, P.L.: Produits infinis de resolvantes. Isr. J. Math. 29, 329–345 (1978)
Minty, G.J.: Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29, 341–346 (1962)
Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley Interscience, New York (1984)
Martinet, B.: Regularisation d’inéquations variationelles par approximations successives. Rev. Fr. d’Informatique Recherche Opér. 4, 154–158 (1970)
Moreau, J.J.: Proximité et dualité dans un espace Hilbertien. Bull. Soc. Math. Fr. 93, 273–299 (1965)
Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)
Güler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29, 403–419 (1991)
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)
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Dong, Y. Comments on “The Proximal Point Algorithm Revisited”. J Optim Theory Appl 166, 343–349 (2015). https://doi.org/10.1007/s10957-014-0685-5
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DOI: https://doi.org/10.1007/s10957-014-0685-5