Two-Step Fixed-Point Proximity Algorithms for Multi-block Separable Convex Problems

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• 作者：Qia Li ; Yuesheng Xu ; Na Zhang
• 关键词：Multi ; block separable convex problems ; Fixed ; point proximity algorithms ; Two ; step algorithms
• 刊名：Journal of Scientific Computing
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：70
• 期：3
• 页码：1204-1228
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Algorithms; Computational Mathematics and Numerical Analysis; Appl.Mathematics/Computational Methods of Engineering; Theoretical, Mathematical and Computational Physics;
• 出版者：Springer US
• ISSN：1573-7691
• 卷排序：70

文摘

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 \(O\left( \frac{1}{k}\right) \) 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.