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.

摘要

使用由 Spingarn 引起的单调算子的部分逆概念，我们提出了一个新的和简单的结果证明——Bauschke 中的定理 2 [Eckstein 和 Svaiter 在论文中关于“最大单调和的一般投影分裂方法”的注释运营商”。SIAM J 控制优化。2009;48(4):2513–2515] – Heinz H. Bauschke。我们的证明是基于部分逆的最大单调性和最大单调算子图上的（渐近）封闭性原则。$\text{虚弱的}\times \text{强的}$拓扑。我们还将 Bauschke 定理推广到更一般的ε - 单调映射放大设置。