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Zbl 0737.90047
Güler, Osman
On the convergence of the proximal point algorithm for convex minmization.
(English)
[J] SIAM J. Control Optimization 29, No.2, 403-419 (1991). ISSN 0363-0129; ISSN 1095-7138/e

This paper studies convergence properties of the proximal point algorithm (PPA) for the convex minimization problem $\min\sb{x\in H}f(x)$, where $f: H\mapsto R\cup\{\infty\}$ is a proper, lower-semicontinuous ($lsc$) function in a Hilbert space $H$.\par 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 $f(x\sb k)-f(u)$ where $u$ is an arbitrary point in $H$ rather than in terms of the closeness of $x\sb k$ to a minimizer of $f$. Under the minimal condition that $f$ is proper, lower-semicontinuous, it is proved that the PPA, with positive parameters $\{\lambda\sb k\}\sb{k=1}\sp \infty$, converges in general if and only if $\sigma\sb n=\sum\sb{k=1}\sp n \lambda\sb k\to \infty$. An open question of Rockafellar is settled by giving an example of a PPA for which $x\sb n$ converges weakly but not strongly to a minimizer of $f$.
[Xue Guoliang (Minneapolis)]
MSC 2000:
*90C25 Convex programming
65K05 Mathematical programming (numerical methods)
49M37 Methods of nonlinear programming type
90-08 Computational methods (optimization)

Keywords: proximal point algorithm; convex minimization; convergence rate

Cited in: Zbl 1071.65082 Zbl 0971.90062

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