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Summary: Convergence of a hybrid projection-proximal point algorithm coupled with
approximation methods in convex optimization
Felipe Alvarez
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (CNRS UMI 2807),
Universidad de Chile, Casilla 170/3, Correo 3, Santiago, Chile.
email: falvarez@dim.uchile.cl
Miguel Carrasco
Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170/3, Correo 3, Santiago, Chile.
email: migucarr@dim.uchile.cl
Karine Pichard
Centro de Modelamiento Matemático (CNRS UMI 2807), Universidad de Chile, Casilla 170/3, Correo 3,
Santiago, Chile.
email: kpichard@dim.uchile.cl
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 xed 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 innite-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-nding problem for a maximal monotone operator.
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