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Convergence of a hybrid projection-proximal point algorithm coupled with approximation methods in convex optimization

Felipe Alvarez (falvarez***at***dim.uchile.cl)
Miguel Carrasco (migucarr***at***dim.uchile.cl)
Karine Pichard (kpichard***at***dim.uchile.cl)

Abstract: In order to minimize a closed convex function that is approximated by a sequence of better behaved functions, we investigate the global convergence of a generic diagonal hybrid algorithm, which consists of an inexact relaxed proximal point step followed by a suitable orthogonal projection onto a hyperplane. The latter permits to consider a fixed relative error criterion for the proximal step. We provide various sets of conditions ensuring the global convergence of this algorithm. The analysis is valid for nonsmooth data in infinite-dimensional Hilbert spaces. Some examples are presented, in particular some penalty/barrier methods in convex programming. We also show that some results can be adapted to the zero-finding problem for a maximal monotone operator.

Keywords: Parametric approximation; diagonal iteration; proximal point; hybrid method ; global convergence

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Global Optimization (Other )

Citation: Mathematics of Operations Research Vol. 30, No. 4, November 2005, pp. 966-984

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Entry Submitted: 11/03/2004
Entry Accepted: 11/03/2004
Entry Last Modified: 02/28/2006

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