Convergence of a Hybrid Projection-Proximal Point Algorithm Coupled with Approximation Methods in Convex Optimization

Authors:
Felipe Alvarez;Miguel Carrasco;Karine Pichard
Affiliations:
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (CNRS UMI 2807), Universidad de Chile, Casilla 170/3, Correo 3, Santiago, Chile;Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170/3, Correo 3, Santiago, Chile;Centro de Modelamiento Matemático (CNRS UMI 2807), Universidad de Chile, Casilla 170/3, Correo 3, Santiago, Chile
Venue:
Mathematics of Operations Research
Year:
2005

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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 general hybrid iterative 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, focusing on 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.