zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On the convergence of the proximal point algorithm for convex minmization. (English) Zbl 0737.90047

This paper studies convergence properties of the proximal point algorithm (PPA) for the convex minimization problem min xH f(x), where f:HR{} is a proper, lower-semicontinuous (lsc) function in a Hilbert space H.

In the literature, the convergence properties of the PPA are studied only in the case where f has a minimizer, and the convergence rate of the algorithm is given only in the case where f is strongly convex. In this paper, the authors give convergence of the PPA under the weakest conditions, even in cases where f has no minimizer, or is unbounded from below. Convergence rate results are given in terms of the residual f(x k )-f(u) where u is an arbitrary point in H rather than in terms of the closeness of x k to a minimizer of f. Under the minimal condition that f is proper, lower-semicontinuous, it is proved that the PPA, with positive parameters {λ k } k=1 , converges in general if and only if σ n = k=1 n λ k . An open question of Rockafellar is settled by giving an example of a PPA for which x n converges weakly but not strongly to a minimizer of f.

90C25Convex programming
65K05Mathematical programming (numerical methods)
49M37Methods of nonlinear programming type in calculus of variations
90-08Computational methods (optimization)