An Extragradient-Based Alternating Direction Method for Convex Minimization

Tianyi Lin, Shiqian Ma, Shuzhong Zhang

Research output: Contribution to journalArticlepeer-review

20 Scopus citations

Abstract

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 ϵ-optimal solution within O(1 / ϵ) 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.

Original languageEnglish (US)
Pages (from-to)35-59
Number of pages25
JournalFoundations of Computational Mathematics
Volume17
Issue number1
DOIs
StatePublished - Feb 1 2017

Bibliographical note

Funding Information:
Shiqian Ma: Research of this author was supported in part by a Direct Grant of The Chinese University of Hong Kong (Project ID: 4055016) and the Hong Kong Research Grants Council General Research Funds Early Career Scheme (Project ID: CUHK 439513).

Funding Information:
Shuzhong Zhang: Research of this author was supported in part by the NSF Grant CMMI-1161242.

Publisher Copyright:
© 2015, SFoCM.

Keywords

  • Alternating direction method
  • Extragradient
  • Fused logistic regression
  • Iteration complexity
  • Lasso

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