Forward-Douglas–Rachford splitting and forward-partial inverse method for solving monotone inclusions

By Luis M. Briceño-Arias. 2012

111 citations recorded by the source corpus.

We provide two weakly convergent algorithms for finding a zero of the sum of a maximally monotone operator, a cocoercive operator, and the normal cone to a closed vector subspace of a real Hilbert space. The methods exploit the intrinsic structure of the problem by activating explicitly the cocoercive operator in the first step, and taking advantage of a vector space decomposition in the second step. The second step of the first method is a Douglas–Rachford iteration involving the maximally monotone operator and the normal cone. In the second method, it is a proximal step involving the partial inverse of the maximally monotone operator with respect to the vector subspace. Connections between the proposed methods and other methods in the literature are provided. Applications to monotone inclusions with finitely many maximally monotone operators and optimization problems are examined.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/12c43f36fe73d6da0f40329e42b9e0dc76b62554

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Citations in the corpus

A New Primal–Dual Algorithm for Minimizing the Sum of Three Functions with a Linear Operator (2016, unknown): “of many existing two-operator splitting schemes such as forward-backward splitting [19], backward-forward splitting [21,2], Peaceman-Rachford splitting [11,20], and forward-Douglas-Rachford splitting [3]. The proposed algorithm PD3O has the following iteration ((z;s) !(z+;s+)): x 2(I+ @g) 1 (z(1.2a)”
Proximal Splitting Algorithms: A Tour of Recent Advances, with New Twists. (2020, methodology): “The Davis–Yin algorithm generalizes the forward–Douglas– Rachford algorithm and the Generalized forward–backward algorithm [93–95].”
A Three-Operator Splitting Scheme and its Optimization Applications (2015, background): “Consider the forward-Douglas-Rachford splitting (FDRS) operator [9], TFDRS := IH−PV +JγA ◦ (2PV − IH − γPV ◦ C ′ ◦ PV ), for solving 0 ∈ Ax + C ′x + NV x where A is maximal monotone, C ′ is cocoercive, V is a closed vector space, NV is the normal cone operator of V , and PV denote the projection to ”
A Dynamical System Associated with the Fixed Points Set of a Nonexpansive Operator (2014, unknown): “The same can be done for other iterative schemes which have their origins in the discrete Krasnosel’skĭı–Mann algorithm, like are the generalized forwardbackward splitting algorithm in [19] and the forward-Douglas-Rachford splitting algorithm in [12].”
SURVEY: SIXTY YEARS OF DOUGLAS–RACHFORD (2018, methodology): “They provided weak convergence results for a method which employs a DR step applied to the normal cone operator and the maximal monotone operator [58].”