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Calcul d’un point fixe d’une application prox par la mêthode des approximations successives; conditions de convergence finie. (Computation of a fixed point of a proximal mapping by the successive approximation method; finite convergence conditions). (French) Zbl 0607.65030
Let us consider the proximal mapping $$\Pr ox_{T_ A}=(I+T_ A)^{- 1}$$ associated with the partial inverse $$T_ A$$ (with respect to a subspace A) of the subdifferential T of a polyhedral convex function. In this note we give sufficient conditions which imply the finite convergence of the successive approximation method for computing a fixed point of $$\Pr ox_{T_ A}$$. Using examples we show how these conditions can be realized and we give a case with non finite termination.

##### MSC:
 65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx) 65K05 Numerical mathematical programming methods 90C25 Convex programming 47H05 Monotone operators and generalizations 47H10 Fixed-point theorems 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) 49J40 Variational inequalities