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Peaceman-Rachford Splitting Method Converges Ergodically for Solving Convex Optimization Problems

Kaihuang Chen, Defeng Sun, Yancheng Yuan, Guojun Zhang, Xinyuan Zhao

TL;DR

The paper studies ergodic convergence for the Peaceman-Rachford splitting method with semi-proximal terms applied to convex optimization problems of the form $\min_{y,z} f_1(y)+f_2(z)$ subject to $B_1y+B_2z=c$. It grounds the analysis by recasting the preconditioned ADMM (pADMM) as a degenerate proximal point algorithm (dPPA) and proves global ergodic convergence with an $O(1/k)$ rate for the ergodic sequences, under mild Lipschitz assumptions on the resolvent. By establishing an equivalence between pADMM and dPPA, the authors extend these ergodic guarantees to the PR splitting method (a special case with $\rho=2$) for solving COPs, while noting that pointwise convergence may fail in general. Numerical experiments on LP benchmarks show that restarting the ergodic PR sequence yields substantial practical gains, outperforming DR-based variants in many instances. An analytical example illustrates the limits of pointwise convergence and underscores the value of ergodic guarantees for PR in large-scale COPs.

Abstract

In this paper, we prove that the ergodic sequence generated by the Peaceman-Rachford (PR) splitting method with semi-proximal terms converges for convex optimization problems (COPs). Numerical experiments on the linear programming benchmark dataset further demonstrate that, with a restart strategy, the ergodic sequence of the PR splitting method with semi-proximal terms consistently outperforms both the point-wise and ergodic sequences of the Douglas-Rachford (DR) splitting method. These findings indicate that the restarted ergodic PR splitting method is a more effective choice for tackling large-scale COPs compared to its DR counterparts.

Peaceman-Rachford Splitting Method Converges Ergodically for Solving Convex Optimization Problems

TL;DR

The paper studies ergodic convergence for the Peaceman-Rachford splitting method with semi-proximal terms applied to convex optimization problems of the form subject to . It grounds the analysis by recasting the preconditioned ADMM (pADMM) as a degenerate proximal point algorithm (dPPA) and proves global ergodic convergence with an rate for the ergodic sequences, under mild Lipschitz assumptions on the resolvent. By establishing an equivalence between pADMM and dPPA, the authors extend these ergodic guarantees to the PR splitting method (a special case with ) for solving COPs, while noting that pointwise convergence may fail in general. Numerical experiments on LP benchmarks show that restarting the ergodic PR sequence yields substantial practical gains, outperforming DR-based variants in many instances. An analytical example illustrates the limits of pointwise convergence and underscores the value of ergodic guarantees for PR in large-scale COPs.

Abstract

In this paper, we prove that the ergodic sequence generated by the Peaceman-Rachford (PR) splitting method with semi-proximal terms converges for convex optimization problems (COPs). Numerical experiments on the linear programming benchmark dataset further demonstrate that, with a restart strategy, the ergodic sequence of the PR splitting method with semi-proximal terms consistently outperforms both the point-wise and ergodic sequences of the Douglas-Rachford (DR) splitting method. These findings indicate that the restarted ergodic PR splitting method is a more effective choice for tackling large-scale COPs compared to its DR counterparts.
Paper Structure (7 sections, 4 theorems, 36 equations, 2 figures, 1 algorithm)

This paper contains 7 sections, 4 theorems, 36 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation.
  3. Ergodic convergence of the dPPA
  4. Ergodic convergence of the PR splitting method
  5. Numerical experiment
  6. Conclusion
  7. An analytical example of the ergodic convergence of the PR splitting method

Key Result

proposition thmcounterproposition

Let $\mathcal{T} : \mathbb{W} \rightarrow 2^{\mathbb{W}}$ be a maximal monotone operator with $\mathcal{T}^{-1}(0) \neq \emptyset$, and let $\mathcal{M}$ be an admissible preconditioner. Then the ergodic sequences $\{{\bar{w}}_a^k\}$ and $\{w_a^k\}$, generated by the dPPA alg:dPPA with $\rho \in (0,

Figures (2)

  • Figure 1: Performance comparison of algorithms ($\sigma=1$) on the “ex10” instance from Mittelmann’s LP benchmark
  • Figure 2: Performance profiles of solving times for tested algorithms ($\sigma=1$) on 49 instances from Mittelmann’s LP benchmark set

Theorems & Definitions (7)

  • proposition thmcounterproposition
  • proof
  • theorem 1
  • proof
  • proposition thmcounterproposition: sun2024accelerating
  • corollary thmcountercorollary
  • remark thmcounterremark