Iteration Complexity Analysis of Multi-block ADMM for a Family of Convex Minimization Without Strong Convexity

Tianyi Lin, Shiqian Ma, Shuzhong Zhang

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22 Scopus citations

Abstract

The alternating direction method of multipliers (ADMM) is widely used in solving structured convex optimization problems due to its superior practical performance. On the theoretical side however, a counterexample was shown in Chen et al. (Math Program 155(1):57–79, 2016.) indicating that the multi-block ADMM for minimizing the sum of N(N≥ 3) convex functions with N block variables linked by linear constraints may diverge. It is therefore of great interest to investigate further sufficient conditions on the input side which can guarantee convergence for the multi-block ADMM. The existing results typically require the strong convexity on parts of the objective. In this paper, we provide two different ways related to multi-block ADMM that can find an ϵ-optimal solution and do not require strong convexity of the objective function. Specifically, we prove the following two results: (1) the multi-block ADMM returns an ϵ-optimal solution within O(1/ϵ2) iterations by solving an associated perturbation to the original problem; this case can be seen as using multi-block ADMM to solve a modified problem; (2) the multi-block ADMM returns an ϵ-optimal solution within O(1/ϵ) iterations when it is applied to solve a certain sharing problem, under the condition that the augmented Lagrangian function satisfies the Kurdyka–Łojasiewicz property, which essentially covers most convex optimization models except for some pathological cases; this case can be seen as applying multi-block ADMM to solving a special class of problems.

Original languageEnglish (US)
Pages (from-to)52-81
Number of pages30
JournalJournal of Scientific Computing
Volume69
Issue number1
DOIs
StatePublished - Oct 1 2016

Keywords

  • Alternating direction method of multipliers (ADMM)
  • Convergence rate
  • Convex optimization
  • Kurdyka–Łojasiewicz property
  • Regularization

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