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On the Convergence of the Proximal Point Algorithm for Convex Minimization

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Abstract

The proximal point algorithm (PPA) for the convex minimization problem min x member of H$/f(x), where f:H → R union {∞} is a proper, lower semicontinuous (lsc) function in a Hilbert space H is considered. Under this minimal assumption on f, it is proved that the PPA, with positive parameters {λ k} k=1∞, converges in general if and only if σ n = Σ k=1n λ k → ∞. Global convergence rate estimates for the residual f$x n) - f(u), where x n is the nth iterate of the PPA and u member of H is arbitrary are given. 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.