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

Lefebvre, Odile; Michelot, Christian

C. R. Acad. Sci., Paris, Sér. I 303, 905-908 (1986).

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
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

Keywords:

maximal monotone operator; proximal mapping; partial inverse; subdifferential; polyhedral convex function; convergence; successive approximation; fixed point