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The proximal algorithm. (English) Zbl 0692.90079
New methods in optimization and their industrial uses, Proc. Symp., Pau/Fr. and Paris/Fr. 1987, ISNM, Int. Ser. Numer. Math. 87, 73-87 (1989).
[For the entire collection see Zbl 0679.00019.]
The proximal algorithm or, more exactly, “the proximal point algorithm”, according to Rockafellar’s terminology, is mainly the successive approximations method for finding fixed points of some nonexpansive mappings in Hilbert spaces. This method, which is therefore not new but was used before [M. A. Krasnosel’skij, Usp. Mat. Nauk 10, No.1(63), 123-127 (1955; Zbl 0064.120); Z. Opial, Bull. Am. Math. Soc. 73, 591-597 (1967; Zbl 0179.199)] for regularizing linear equations [M. A. Krasnosel’skij, Usp. Mat. Nauk 15, No.3(93), 161- 165 (1960; Zbl 0095.315); A. V. Krjanev, Sov. Math., Dokl. 14, 673- 676 (1973; Zbl 0294.47008)] seems to have been applied for the first time to convex optimization by B. Martinet [Rev. Française Inform. Rech. Oper. 4, No.R-3, 154-158 (1970; Zbl 0215.211); “Algorithmes pour la résolution de problèmes d’optimisation et de minimax”, Thèse d’Etat, Univ. Grenoble (1972)]. The first important results (approximate version, linear and finite convergence) in the general framework of maximal monotone inclusions are due to R. T. Rockafellar [Math. Oper. Res. 1, 97-116 (1976; Zbl 0402.90076)]. More recently, a lot of work has been devoted to this algorithm and nowadays it is still the object of intensive investigation.
In this survey we present the main properties and applications of the proximal algorithm and some recent developments in convex programming, especially its association with other approximation methods, for instance, epiconvergence.
Reviewer: B.Lemaire

90C30 Nonlinear programming
90-02 Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming
65K05 Numerical mathematical programming methods
49M30 Other numerical methods in calculus of variations (MSC2010)