Two-Step Fixed-Point Proximity Algorithms for Multi-block Separable Convex Problems
Qia Li,Yuesheng Xu,Na Zhang +2 more
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TL;DR: In this paper, a class of two-step fixed-point iterative schemes for solving multi-block separable convex problems based on the characterization of the solutions of the optimization problems by proximity operators of the convex functions involved in their objective functions is proposed.
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Abstract: Multi-block separable convex problems recently received considerable attention. Optimization problems of this type minimize separable convex objective functions with linear constraints. Challenges encountered in algorithmic development applying the classic alternating direction method of multipliers (ADMM) come from the fact that ADMM for the optimization problems of this type is not necessarily convergent. However, it is observed that ADMM applying to problems of this type outperforms numerically many of its variants with guaranteed theoretical convergence. The goal of this paper is to develop convergent and computationally efficient algorithms for solving multi-block separable convex problems. We first characterize the solutions of the optimization problems by proximity operators of the convex functions involved in their objective functions. We then design a class of two-step fixed-point iterative schemes for solving these problems based on the characterization. We further prove convergence of the iterative schemes and show that they have $$O\left( \frac{1}{k}\right) $$O1k of convergence rate in the ergodic sense and the sense of the partial primal-dual gap, where k denotes the iteration number. Moreover, we derive specific two-step fixed-point proximity algorithms (2SFPPA) from the proposed iterative schemes and establish their global convergence. Numerical experiments for solving the sparse MRI problem demonstrate the numerical efficiency of the proposed 2SFPPA.
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Citations
A New Insight on Augmented Lagrangian Method with Applications in Machine Learning
Jianchao Bai,Linyuan Jia,Zheng Peng +2 more
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Efficient dual ADMMs for sparse compressive sensing MRI reconstruction
TL;DR: Wang et al. as mentioned in this paper focused on the algorithmic improvement for the sparse reconstruction of CS-MRI, especially considering a non-smooth convex minimization problem which is composed of the sum of a total variation regularization term and a $$\ell _1$$ -norm term of the wavelet transformation.
Implicit Fixed-point Proximity Framework for Optimization Problems and Its Applications
Xiaoxia Liu
- 01 Jan 2018
TL;DR: Theoretical analysis is provided for the convergence of implicit algorithms proposed under the framework, and numerical experiments on image reconstruction models demonstrate that the proposed implicit fixed-point proximity algorithms work well in comparison with existing explicit fixed- point proximity algorithms in terms of the consumed computational time and accuracy of the solutions.
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Efficient Dual ADMMs for Sparse Compressive Sensing MRI Reconstruction
TL;DR: Wang et al. as discussed by the authors focused on the algorithmic improvement for the sparse reconstruction of CS-MRI, especially considering a non-smooth convex minimization problem which is composed of the sum of a total variation regularization term and a $\ell_1$-norm term of the wavelet transformation.
Improved proximal ADMM with partially parallel splitting for multi-block separable convex programming
Min Sun,Hongchun Sun +1 more
TL;DR: For a type of multi-block separable convex programming raised in machine learning and statistical inference, a proximal alternating direction method of multiplier with partially parallel splitting is proposed, which has the following nice properties.
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