A method for nonlinear constraints in minimization problems

By M. Powell. 1969

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Citations in the corpus

Regularizing Text Categorization with Clusters of Words (2016, background): “…λglas and λlas 1: while update in weights not small do 2: θ = argmin θ Ωlas(θ)+L(θ)+ ρ2 V∑ i=1 Ni(θi−µi)2 3: for g = 1 to G do 4: vg = proxΩglas,λgρ (zg) 5: end for 6: u = u+ ρ(v−Mθ) 7: end while Their method uses the alternating directions method of multipliers (Hestenes, 1969; Powell, 1969).”
An efficient augmented Lagrangian method with applications to total variation minimization (2013, methodology): “For solving this class of problems, we will pro-pose and study an algorithm in the framework of the classic augmented Lagrangian multiplier (ALM) method, ﬁrst suggested by Hestenes [21] and Powell [31].”
Prox-PDA: The Proximal Primal-Dual Algorithm for Fast Distributed Nonconvex Optimization and Learning Over Networks (2017, methodology): “The proposed algorithm builds upon the classical augmented Lagrangian (AL) method (Bertsekas, 1982; Powell, 1969).”
Fast Image Recovery Using Variable Splitting and Constrained Optimization (2009, unknown): “The so-calledaugmented Lagrangian method (ALM) [37], also known as themethod of multipliers(MM) [28], [38], consists in minimizingLA(z, λ, μ) with respect toz, keeping λ fixed, then updatingλ, and repeating these two steps until some convergence criterion is satisfied.”
Linear Convergence Rate of a Class of Distributed Augmented Lagrangian Algorithms (2013, methodology): “The method of multipliers based on the augmented Lagrangian has been proposed in the late 1960s [34], [35].”
A note on the inertial proximal point method (2015, methodology): “It turns out that the PPM is a very powerful algorithmic tool and contains many well known algorithms as special cases, including the classical augmented Lagrangian method [20, 29], the Douglas-Rachford splitting method [15] and the alternating direction method of multipliers [18, 17].”
Making the Most of Bag of Words: Sentence Regularization with Alternating Direction Method of Multipliers (2014, methodology): “We propose instead to apply the alternating directions method of multipliers (ADMM; Hestenes, 1969; Powell, 1969).”
Convergence results of two-step inertial proximal point algorithm (2022, methodology): “The PPA (2) is a very powerful algorithmic tool and contains many well known algorithms as special cases, including the classical augmented Lagrangian method [33,49], the Douglas-Rachford splitting method [27, 37] and the alternating direction method of multipliers [30,31].”
BM3D Frames and Variational Image Deblurring (2011, methodology): “This AL technique, introduced independently by Hestenes [13] and Powell [14] is now widely used for minimization of convex functionals under linear equality constraints.”
An immersogeometric variational framework for fluid–structure interaction: application to bioprosthetic heart valves (2014, methodology): “We consider an iterative approximation of the multiplier, based on the work of Hestenes [128] and Powell [129].”
Linearized augmented Lagrangian and alternating direction methods for nuclear norm minimization (2012, methodology): “…with the eﬀort of reducing the computation of SVD. Recently, it has been shown in the literature that the augmented Lagrangian method (ALM) [25, 39] and the alternating direction method (ADM) [17] are very eﬃcient for some convex programming problems arising from various applications, provided…”
An efficient algorithm for total variation regularization with applications to the single pixel camera and compressive sensing (2010, methodology): “In 1969, Hestenes [70] and Powell [71] independently proposed the augmented Lagrangian method which successfully avoided this inherent problem by introducing explicit Lagrangian multiplier estimates at each iteration into the objective function.”
Sparse and low-rank matrix decomposition via alternating direction method (2013, background): “Let C = A * + B * be the available data, where A * and B * are, respectively, the original sparse and low-rank matrices that we wish to recover.”
Separable approximations and decomposition methods for the augmented Lagrangian (2013, unknown): “The augmented Lagrangian idea was first introduced independently by Hestenes [13] and Powell [26] and convergence of the associated augmented Lagrangian method was established later by Tyrell Rockafellar [32,33].”
A topology optimization method based on the level set method incorporating a fictitious interface energy (2010, methodology): “In addition, the volume constraint is handled using the augmented Lagrangian method [86–88].”
Robust Matrix Completion via Joint Schatten p-Norm and lp-Norm Minimization (2012, methodology): “We use the Augmented Lagrangian Method [15], [16], [17] to solve this problem, and focus on the solutions to the related subproblems.”
Interior Methods for Nonlinear Optimization (2002, background): “Slack variables pervade linear programming, and the idea of shifted constraints underlies the motivation in [86] of augmented Lagrangian methods for nonlinearly constrained optimization.”