Another proof and a generalization of a theorem of H. H. Bauschke on monotone operators

Abstract

ABSTRACT Using the concept of partial-inverse of monotone operators due to Spingarn, we present a new and simple proof of a result – Theorem 2 in Bauschke [A note on the paper by Eckstein and Svaiter on “General projective splitting methods for sums of maximal monotone operators”. SIAM J Control Optim. 2009;48(4):2513–2515] – of Heinz H. Bauschke. Our proof is based on the maximal monotonicity of the partial-inverse and on the (asymptotic) closedness principle on the graph of maximal monotone operators in the topology. We also present a generalization of Bauschke's theorem to the more general setting of ε–enlargements of monotone maps.