A splitting primal-dual proximity algorithm for solving composite optimization problems

Tang, Yuchao; Zhu, Chuanxi; Wen, Meng; Peng, Jigen

doi:10.1007/s10114-016-5625-x

Cited by 18 publications

(18 citation statements)

References 21 publications

(37 reference statements)

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“…We set γ = 1.9/( A 2 + ρ 1 D 1 2 + ρ 2 D 2 2 ) and ρ 1 = ρ 2 = 1 for ADMM. For SPDP and Pre-SPDP, the parameters are set to be the same as those in [7]. We select Algorithm 4. as the total variation.…”

Section: Methodsmentioning

confidence: 99%

“…In this section, we will employ the proposed iterative algorithms to solve an optimization model that is suitable for reconstructing computed tomography (CT) images from a set of undersampled and potentially noisy projection measurements. The proposed iterative algorithms are compared with several representative algorithms, including ADMM [36], splitting primal-dual proximity algorithm (SPDP) [7], and preconditioned SPDP (Pre-SPDP) [7]. All the experiments are accomplished by Matlab and on a standard Lenovo laptop with Intel(R) Core(TM) i7-4712MQ 2.3GHz CPU and 4GB RAM.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…. Therefore, the splitting primal-dual proximity algorithm and the preconditioned splitting primal-dual proximity algorithm developed in [7] can be used to solve the regularized general PICCS model (5.3). In addition, the ADMM [36] provides a very general framework for solving various convex optimization problems including the regularized general PICCS model (5.3).…”

Section: Computed Tomography Image Reconstruction Problemmentioning

confidence: 99%

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A First-Order Splitting Method for Solving a Large-Scale Composite Convex Optimization Problem

Tang¹

2019

JCM

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The forward-backward operator splitting algorithm is one of the most important methods for solving the optimization problem of the sum of two convex functions, where one is differentiable with a Lipschitz continuous gradient and the other is possibly nonsmooth but proximable. It is convenient to solve some optimization problems in the form of dual or primal-dual problems. Both methods are mature in theory. In this paper, we construct several efficient first-order splitting algorithms for solving a multi-block composite convex optimization problem. The objective function includes a smooth function with a Lipschitz continuous gradient, a proximable convex function that may be nonsmooth, and a finite sum of a composition of a proximable function and a bounded linear operator. To solve such an optimization problem, we transform it into the sum of three convex functions by defining an appropriate inner product space. On the basis of the dual forward-backward splitting algorithm and the primal-dual forward-backward splitting algorithm, we develop several iterative algorithms that involve only computing the gradient of the differentiable function and proximity operators of related convex functions. These iterative algorithms are matrix-inversion-free and completely splitting algorithms. Finally, we employ the proposed iterative algorithms to solve a regularized general prior image constrained compressed sensing (PICCS) model that is derived from computed tomography (CT) image reconstruction under sparse sampling of projection measurements. Numerical results show that the proposed iterative algorithms outperform other algorithms.

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Section: Methodsmentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Computed Tomography Image Reconstruction Problemmentioning

confidence: 99%

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A First-Order Splitting Method for Solving a Large-Scale Composite Convex Optimization Problem

Tang¹

2019

JCM

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“…We mention in passing that similar techniques are also adopted in [8,4,12,14]. Compared to the schemes developed in [8,12,14], if a scheme is established based on PDFP with f 1 nonzero in (1.4), it's more convenient for us to choose parameters in applications, as shown in [5]. However, if a scheme is constructed based on PDFP by viewing f 1 equal to 0, it requires to compute an additional symmetric step.…”

Section: C)mentioning

confidence: 99%

“…Then the PDFP algorithm can be applied and formulated in parallel form due to the separability of f 2 on its variables. Similar technique has also been used in [8,4,12,14] and we present the details here for completeness.…”

Section: Algorithm and Its Deductionmentioning

confidence: 99%

A Primal-Dual Fixed Point Algorithm for Multi-Block Convex Minimization

Chen¹

2016

JCM

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We extend a primal-dual fixed point algorithm (PDFP) proposed in [5] to solve two kinds of separable multi-block minimization problems, arising in signal processing and imaging science. This work shows the flexibility of applying PDFP algorithm to multi-block problems and illustrate how practical and fully decoupled schemes can be derived, especially for parallel implementation of large scale problems. The connections and comparisons to the alternating direction method of multiplier (ADMM) are also present. We demonstrate how different algorithms can be obtained by splitting the problems in different ways through the classic example of sparsity regularized least square model with constraint. In particular, for a class of linearly constrained problems, which are of great interest in the context of multi-block ADMM, can be solved by PDFP with a guarantee of convergence. Finally, some experiments are provided to illustrate the performance of several schemes derived by the PDFP algorithm.

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An inertial primal‐dual fixed point algorithm for composite optimization problems

Wen¹,

Tang²,

Peng³

2022

Math Methods in App Sciences

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In this paper, we consider an inertial primal-dual fixed point algorithm (IPDFP) to compute the minimizations of the sum of a non-smooth convex function and a finite family of composite non-smooth convex functions, each one of which is composed of a non-smooth convex function and a bounded linear operator. This is a full splitting approach, in the sense that non-smooth functions are processed individually via their proximity operators. The convergence of the IPDFP is obtained by reformulating the problem to the sum of three non-smooth convex functions. Furthermore, we propose a preconditioning technique for the IPDFP. The key idea of the preconditioning technique is that the constant iterative parameters are updated self-adaptively in the iteration process. What's more, we also give a simple and easy way to choose the diagonal preconditioners while the convergence of the iterative algorithms is maintained. This work brings together and notably extends several classical splitting schemes, like the primal-dual method proposed by Chambolle and Pock, and the recent proximity algorithms of Charles et al. designed for the L 1 /TV image denoising model. The iterative algorithm is used for solving non-differentiable convex optimization problems arising in image processing.

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A splitting primal-dual proximity algorithm for solving composite optimization problems

Cited by 18 publications

References 21 publications

A First-Order Splitting Method for Solving a Large-Scale Composite Convex Optimization Problem

A First-Order Splitting Method for Solving a Large-Scale Composite Convex Optimization Problem

A Primal-Dual Fixed Point Algorithm for Multi-Block Convex Minimization

An inertial primal‐dual fixed point algorithm for composite optimization problems

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