Splitting Algorithms for the Sum of Two Nonlinear Operators

By Pierre-Louis Lions, B. Mercier. 1979

Subfields: Mathematical Optimization, Numerical Methods & Scientific Computing

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Citations in the corpus

Stochastic Gradient Descent-Ascent and Consensus Optimization for Smooth Games: Convergence Analysis under Expected Co-coercivity (2021, methodology): “Note that in our definition we consider the inverse of the co-coercive constant from Lions and Mercier [1979] which is the constant ` such that 〈ξ(x)− ξ(y), x− y〉 ≥ `‖ξ(x)− ξ(y)‖(2).”
Distributed Fixed Point Methods with Compressed Iterates (2019, methodology): “The Douglas–Rachford splitting (or ADMM) [31] is an algorithm allowing to minimize G + H , where G and H are convex nonsmooth functions.”
GADMM: Fast and Communication Efficient Framework for Distributed Machine Learning (2019, background): “First, it worth mentioning that the convergence rate of Gauss-Seidel ADMM for sum of two separable convex functions has been studied in (Lions and Mercier, 1979; He and Yuan, 2012, 2015).”
Optimization with Sparsity-Inducing Penalties (2011, background): “The proximal operator itself dates back to [95] and proximal methods themselves date back to [82, 92].”
On the O(1=k) convergence of asynchronous distributed alternating Direction Method of Multipliers (2013, background): “imal updates at each iteration, and studies convergence in the context of ﬁnding zeros of the sum of two maximal monotone operators (mo re speciﬁcally, the Douglas-Rachford operator), see [10], [13], [28]. The recent contribution [20] considered solving problem (1) (with C= 2) with ADMM and showed t”
Computational Optimal Transport (2018, methodology): “The DR algorithm [Lions and Mercier, 1979] is specifically tailored to solve nonsmooth structured problems of the form min x∈H F (x) +G(x), (7.”
An efficient augmented Lagrangian method with applications to total variation minimization (2013, methodology): “In the early 1980s, Glowinski et al. studied this type of problems in depth using the ALM and operator-splitting methods [14, 16], which also have close ties to earlier works such as [27].”
An introduction to Total Variation for Image Analysis (2009, background): “This is an example of “splitting” (introduced first by Douglas and Rachford, see for instance [48] for a general form): one solves successively one step of the gradient descent of F (in an explicit way), and one step of the gradient descent of G (in an implicit way), in order to obtain a “full” grad”
Parallel Multi-Block ADMM with o(1 / k) Convergence (2013, unknown): “Its convergence rate has been actively studied; see [21, 19, 18, 13, 8, 20] and the references therein.”
A Stochastic forward-backward splitting method for solving monotone inclusions in Hilbert spaces (2014, methodology): “Since the seminal works [34, 43], forward-backward splitting methods have been considerably developed to be more flexible, to achieve better convergence properties, and to allow for numerical errors [5, 17, 42, 6, 51, 16].”
A Three-Operator Splitting Scheme and its Optimization Applications (2015, background): “We are now ready to show that FDRS can converge arbitrarily slowly.”