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

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 $\text{weak}\times \text{strong}$ topology. We also present a generalization of Bauschke's theorem to the more general setting of ε–enlargements of monotone maps.

Keywords:

• Maximal monotone operators
• partial-inverse
• closedness principle
• operator splitting
• ε–enlargements

2000 Mathematics Subject Classifications:

• 47H05
• 90C25
• 90C30

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work of this author was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) grants no. 405214/2016-2 and 304692/2017-4.