Asymptotic behavior of compositions of under-relaxed nonexpansive operators

Jean-Bernard Baillon (1) , Patrick Louis Combettes (2) , Roberto Cominetti (3)

1 SAMM - Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne)
2 LJLL - Laboratoire Jacques-Louis Lions
3 DII - Departamento de Ingenieria Industrial [Santiago]

In general there exists no relationship between the fixed point sets of the composition and of the average of a family of nonexpansive operators in Hilbert spaces. In this paper, we establish an asymptotic principle connecting the cycles generated by under-relaxed compositions of nonexpansive operators to the fixed points of the average of these operators. In the special case when the operators are projectors onto closed convex sets, we prove a conjecture by De Pierro which has so far been established only for projections onto affine subspaces.

Mots clés: Cyclic projections; De Pierro's conjecture; fixed point; nonexpansive operator; projection operator; under-relaxed cycles.

Domaines: Mathématiques [math]; Optimisation et contrôle [math.OC]

Jean-Bernard Baillon, Patrick Louis Combettes, Roberto Cominetti. Asymptotic behavior of compositions of under-relaxed nonexpansive operators. Journal of Dynamics and Games (JDG), 2014, JDG, 1 (3), pp.331-346. ⟨hal-01159001⟩