Proximal Decomposition on the Graph of a Maximal Monotone Operator

Abstract : We present an algorithm to solve: Find $(x, y) \in A\times A^\bot$ such that $y\in Tx$, where $A$ is a subspace and $T$ is a maximal monotone operator. The algorithm is based on the proximal decomposition on the graph of a monotone operator and we show how to recover Spingarn's decomposition method. We give a proof of convergence that does not use the concept of partial inverse and show how to choose a scaling factor to accelerate the convergence in the strongly monotone case. Numerical results performed on quadratic problems confirm the robust behaviour of the algorithm.
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Philippe Mahey, Said Oualibouch, Tao Pham Dinh. Proximal Decomposition on the Graph of a Maximal Monotone Operator. SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 1995, 5 (2), pp.454-466. ⟨10.1137/0805023⟩. ⟨hal-01644645⟩

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