An Extragradient-Based Alternating Direction Method for Convex Minimization

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• 作者：Tianyi Lin ; Shiqian Ma ; Shuzhong Zhang
• 关键词：Alternating direction method ; Extragradient ; Iteration complexity ; Lasso ; Fused logistic regression
• 刊名：Foundations of Computational Mathematics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：17
• 期：1
• 页码：35-59
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Numerical Analysis; Economics, general; Applications of Mathematics; Linear and Multilinear Algebras, Matrix Theory; Math Applications in Computer Science; Computer Science, general;
• 出版者：Springer US
• ISSN：1615-3383
• 卷排序：17

文摘

In this paper, we consider the problem of minimizing the sum of two convex functions subject to linear linking constraints. The classical alternating direction type methods usually assume that the two convex functions have relatively easy proximal mappings. However, many problems arising from statistics, image processing and other fields have the structure that while one of the two functions has an easy proximal mapping, the other function is smoothly convex but does not have an easy proximal mapping. Therefore, the classical alternating direction methods cannot be applied. To deal with the difficulty, we propose in this paper an alternating direction method based on extragradients. Under the assumption that the smooth function has a Lipschitz continuous gradient, we prove that the proposed method returns an \(\epsilon \)-optimal solution within \(O(1/\epsilon )\) iterations. We apply the proposed method to solve a new statistical model called fused logistic regression. Our numerical experiments show that the proposed method performs very well when solving the test problems. We also test the performance of the proposed method through solving the lasso problem arising from statistics and compare the result with several existing efficient solvers for this problem; the results are very encouraging.