Solving Composite Monotone Inclusions in Reflexive Banach Spaces by Constructing Best Bregman Approximations from Their Kuhn-Tucker Set
Journal of convex analysis, Tome 23 (2016) no. 2, pp. 481-51
We introduce the first operator splitting method for composite monotone inclusions outside of Hilbert spaces. The proposed primal-dual method constructs iteratively the best Bregman approximation to an arbitrary point from the Kuhn-Tucker set of a composite monotone inclusion. Strong convergence is established in reflexive Banach spaces without requiring additional restrictions on the monotone operators or knowledge of the norms of the linear operators involved in the model. The monotone operators are activated via Bregman distance-based resolvent operators. The method is novel even in Euclidean spaces, where it provides an alternative to the usual proximal methods based on the standard distance.
Mots-clés :
Best approximation, Banach space, Bregman distance, duality, Legendre function, monotone operator, operator splitting, primal-dual algorithm
@article{JCA_2016_23_2_JCA_2016_23_2_a6,
author = {P. L. Combettes and Q. V. Nguyen},
title = {Solving {Composite} {Monotone} {Inclusions} in {Reflexive} {Banach} {Spaces} by {Constructing} {Best} {Bregman} {Approximations} from {Their} {Kuhn-Tucker} {Set}},
journal = {Journal of convex analysis},
pages = {481--51},
year = {2016},
volume = {23},
number = {2},
url = {http://geodesic.mathdoc.fr/item/JCA_2016_23_2_JCA_2016_23_2_a6/}
}
TY - JOUR AU - P. L. Combettes AU - Q. V. Nguyen TI - Solving Composite Monotone Inclusions in Reflexive Banach Spaces by Constructing Best Bregman Approximations from Their Kuhn-Tucker Set JO - Journal of convex analysis PY - 2016 SP - 481 EP - 51 VL - 23 IS - 2 UR - http://geodesic.mathdoc.fr/item/JCA_2016_23_2_JCA_2016_23_2_a6/ ID - JCA_2016_23_2_JCA_2016_23_2_a6 ER -
%0 Journal Article %A P. L. Combettes %A Q. V. Nguyen %T Solving Composite Monotone Inclusions in Reflexive Banach Spaces by Constructing Best Bregman Approximations from Their Kuhn-Tucker Set %J Journal of convex analysis %D 2016 %P 481-51 %V 23 %N 2 %U http://geodesic.mathdoc.fr/item/JCA_2016_23_2_JCA_2016_23_2_a6/ %F JCA_2016_23_2_JCA_2016_23_2_a6
P. L. Combettes; Q. V. Nguyen. Solving Composite Monotone Inclusions in Reflexive Banach Spaces by Constructing Best Bregman Approximations from Their Kuhn-Tucker Set. Journal of convex analysis, Tome 23 (2016) no. 2, pp. 481-51. http://geodesic.mathdoc.fr/item/JCA_2016_23_2_JCA_2016_23_2_a6/