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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 ${min}_{x\in H}f\left(x\right)$, where $f:H↦R\cup \left\{\infty \right\}$ 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 $f\left({x}_{k}\right)-f\left(u\right)$ 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 ${\left\{{\lambda }_{k}\right\}}_{k=1}^{\infty }$, converges in general if and only if ${\sigma }_{n}={\sum }_{k=1}^{n}{\lambda }_{k}\to \infty$. 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 $f$.

##### MSC:
 90C25 Convex programming 65K05 Mathematical programming (numerical methods) 49M37 Methods of nonlinear programming type in calculus of variations 90-08 Computational methods (optimization)