A Primal–Dual Splitting Method for Convex Optimization Involving Lipschitzian, Proximable and Linear Composite Terms

By Laurent Condat. 2012

1,002 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Numerical Analysis

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Citations in the corpus

Distributed Fixed Point Methods with Compressed Iterates (2019, methodology): “Vu-Condat splitting Vu condat splitting [11, 30] is an optimization algorithm to minimize a sum of three convex functions F (x) +G(x) +H(Ax) where A is a matrix.”
A Direct Algorithm for 1-D Total Variation Denoising (2013, methodology): “Recent iterative methods based on operator splitting, which exploit both the primal and dual formulations of the problems and use variable stepsize strategies or Nesterov-style accelerations, are quite efficient when applied to TV-based problems [19]–[21].”
Stochastic Approximations and Perturbations in Forward-Backward Splitting for Monotone Operators (2015, background): “1 [20, 53] (see also [23]) and therefore solved via Theorem 4.”
Warped proximal iterations for monotone inclusions (2019, background): “7) reduces to the variable metric forward-backward algorithm studied in [27], which itself captures the frameworks of [15, 28, 37, 41, 52, 54].”
A Coordinate Descent Primal-Dual Algorithm and Application to Distributed Asynchronous Optimization (2014, methodology): “(5) The ADMM+ is an instance of the primal dual algorithm recently proposed by Vũ [1] and Condat [2], see also [31]: Theorem 2 ([1], [2]) Given a Euclidean spaceE , consider the minimization probleminfy∈E f̄(y) + ḡ(y) + h(y) where ḡ, h ∈ Γ0(E) and wheref̄ is convex and differentiable on E with an L−”
Dualize, Split, Randomize: Fast Nonsmooth Optimization Algorithms (2020, methodology): “Moreover, the condition γ 2 ξ of Lemma 2 is equivalent to ν/ 2 < 1 τ − γ k L 2 which is exactly the condition (i) of Theorem 3.1 in [13].”
Decentralized Consensus Optimization With Asynchrony and Delays (2016, methodology): “Problem (8) can be solved iteratively by the primal-dual algorithm that is adapted from [38], [39]: { Xk+1 = proxαr [ X − α∇s(Xk ) − V (2Y k+1 − Y k ) ] ,”
Sparse Modeling for Image and Vision Processing (2014, methodology): “For example, proximal splitting techniques [Combettes and Pesquet, 2011, Chambolle and Pock, 2011, Condat, 2013, Raguet et al., 2013], such as the alternating direction method of multipliers [see Boyd et al., 2011], are often used in signal processing.”
Distributed Linearized Alternating Direction Method of Multipliers for Composite Convex Consensus Optimization (2015, methodology): “2) We establish that when the gradient is deterministic PG-ADMM is equivalent to primal-dual algorithms for saddle-point problems proposed in [12] and [13]—hence, SPG-ADMM extends these algorithms to noisy gradient setting.”
A Convex Formulation for Hyperspectral Image Superresolution via Subspace-Based Regularization (2014, background): “Different approaches can be followed to factorize matrix Z , and two of them will be brieﬂy mentioned here.”
ARock: an Algorithmic Framework for Asynchronous Parallel Coordinate Updates (2015, background): “, the metric in the Condat-Vũ primal-dual splitting [16, 57]), then we have”
Playing with Duality: An overview of recent primal?dual approaches for solving large-scale optimization problems (2014, background): “Convergence guarantees were established in [44], as well as for more general version of this algorithm in [45]:”
A New Primal–Dual Algorithm for Minimizing the Sum of Three Functions with a Linear Operator (2016, methodology): “Condat-Vu : The Condat-Vu algorithm [12,34] is a generalization of the Chambolle-Pock algorithm for problem (1).”
An introduction to continuous optimization for imaging (2016, methodology): “L. Condat (2013a), ‘A direct algorithm for 1D total variation denoising’, IEEE Signal Proc.”
Distributed Proximal Splitting Algorithms with Rates and Acceleration (2020, methodology): “On the other hand, the latter encompasses the Condat–Vũ algorithm [22,23], which is not the case of our approach.”
A primal-dual algorithm with optimal stepsizes and its application in decentralized consensus optimization (2017, background): “λ ≤ 1/λ max(AA⊤) and γ < 2/L, where L is the Lipschitz constant of ∇f. An optimal primal-dual algorithm 3 There are many other algorithms for solving (2) and its extensions. For example, Condat-Vu [4,7,21] solves a more general problem than (2) with an additional non-diﬀerential function. However”