Solving monotone inclusions via compositions of nonexpansive averaged operators

By Patrick L. Combettes. 2004

487 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/2f3e9d3ba10ad55abfe4db780c2e5a60901b149b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Forward-Douglas–Rachford splitting and forward-partial inverse method for solving monotone inclusions (2012, methodology): “The proposed method generalizes the partial inverse method proposed in [9] and the forward–backward splitting.[30]”
Fixed-Point Algorithms for Inverse Problems in Science and Engineering (2011, methodology): “In 1979, Lions and Mercier [56] extended the DouglasRachford algorithm to an operator splitting method for finding a zero of the sum of two maximal monotone operators (see [31] for an historical account and theoretical details).”
A Primer on Monotone Operator Methods (2015, background): “When an iteration is a strict contraction, we get strong convergence [3, 7, 33,34,36,44,103].”
Proximal Algorithms (2013, background): “Building on Rockafellar’s work, there is a substantial literature studying the proximal point algorithm in the monotone operator setting; this is closely connected to the literature on set-valued mappings, fixed point theory, nonexpansive mappings, and variational inequalities [202, 37, 103, 175, 81”
A splitting algorithm for dual monotone inclusions involving cocoercive operators (2011, background): “29) has the structure of the forward-backward splitting algorithm [13].”
The Split Common Null Point Problem (2011, methodology): “the class of splitting methods. References on splitting methods and their applications can be found in Eckstein’s Ph.D. thesis [32], in Tseng’s work [49, 50, 51] and more recently in Combettes et al. [28, 29, 30]. One splitting method of interest is the following forward-backward algorithm: xk+1 = J”
Fast alternating linearization methods for minimizing the sum of two convex functions (2009, unknown): “…( ) is a subgradient of φ ( ) and γ φ ( p ψ ( v )) satisﬁes the ﬁrst-order optimality conditions for (2.4), i.e., Therefore, from (2.7), (2.10) and (2.11) it follows that (2.12) , ∀ k, (2.13) where x ∗ is an optimal solution of (1.1) and k n is the number of iterations until the k -th for which F…”
A strong convergence result involving an inertial forward–backward algorithm for monotone inclusions (2017, background): “There have been many works concerning the problem of finding zero points of the sum of two monotone operators (in Hilbert spaces) and accretive operators (in Banach spaces); see [25,26,34,36,48,53,65,66].”
A proximal decomposition method for solving convex variational inverse problems (2008, methodology): “The algorithm adopted in [25] is based on the Douglas–Rachford splitting method [22, 38] and operates via the updating rule ⌊ yn+ (1)2 = proxγf2 yn + an yn+1 = yn + λn ( proxγf1(2yn+ (1)2 − yn) + bn − yn+ (1)2 ) , (1.”
Hybrid methods with regularization for minimization problems and asymptotically strict pseudocontractive mappings in the intermediate sense (2009, background): “3 (see [20,21]) Let S, T, V : H → H be given operators.”
Signal Recovery by Proximal Forward-Backward Splitting (2005, methodology): “The following theorem is an adaption of some results from [25], which provides a more general iteration in which the coeﬃcient γ is made iteration-dependent, errors are allowed in the evaluation of the operators prox γf 1 and ∇ f 2 , and a relaxation sequence ( λ n ) n ∈ N is introduced.”
Variable metric forward–backward splitting with applications to monotone inclusions in duality (2012, unknown): “Keywords: cocoercive operator, composite operator, demiregularity, duality, forward-backward splitting algorithm, monotone inclusion, monotone operator, primal-dual algorithm, quasi-Fejér sequence, variable metric.”
Firmly Nonexpansive Mappings and Maximally Monotone Operators: Correspondence and Duality (2011, methodology): “We shall conclude this paper with some comments regarding prototypical applications of the above results to splitting methods (see also [9] and [17] for further information and various variants).”