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A Tuning-Free Primal-Dual Splitting Algorithm for Large-Scale Semidefinite Programming

Yinjun Wang, Haixiang Lan, Yinyu Ye

TL;DR

A tuning-free variant of Primal-Dual Hybrid Gradient that provides a unified approach to analyze the convergence of generic adaptive PDHG, including the proposed tuning-free algorithm and various existing ones is proposed and analyzed.

Abstract

This paper proposes and analyzes a tuning-free variant of Primal-Dual Hybrid Gradient (PDHG), and investigates its effectiveness for solving large-scale semidefinite programming (SDP). The core idea is based on the combination of two seemingly unrelated results: (1) the equivalence of PDHG and Douglas-Rachford splitting (DRS); (2) the asymptotic convergence of non-stationary DRS. This combination provides a unified approach to analyze the convergence of generic adaptive PDHG, including the proposed tuning-free algorithm and various existing ones. Numerical experiments are conducted to show the performance of our algorithm, highlighting its superior convergence speed and robustness in the context of SDP.

A Tuning-Free Primal-Dual Splitting Algorithm for Large-Scale Semidefinite Programming

TL;DR

A tuning-free variant of Primal-Dual Hybrid Gradient that provides a unified approach to analyze the convergence of generic adaptive PDHG, including the proposed tuning-free algorithm and various existing ones is proposed and analyzed.

Abstract

This paper proposes and analyzes a tuning-free variant of Primal-Dual Hybrid Gradient (PDHG), and investigates its effectiveness for solving large-scale semidefinite programming (SDP). The core idea is based on the combination of two seemingly unrelated results: (1) the equivalence of PDHG and Douglas-Rachford splitting (DRS); (2) the asymptotic convergence of non-stationary DRS. This combination provides a unified approach to analyze the convergence of generic adaptive PDHG, including the proposed tuning-free algorithm and various existing ones. Numerical experiments are conducted to show the performance of our algorithm, highlighting its superior convergence speed and robustness in the context of SDP.
Paper Structure (11 sections, 8 theorems, 33 equations, 1 figure, 1 table, 5 algorithms)

This paper contains 11 sections, 8 theorems, 33 equations, 1 figure, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Algorithms and Convergence
  3. Notation
  4. Tuning-Free PDHG for SDP
  5. Numerical Experiments
  6. Settings and parameter configurations
  7. Results and discussion
  8. Appendix
  9. Proof of Theorem \ref{['theorem_pdhg4sdp']}
  10. Proof of Theorem \ref{['theorem_pdhg4sdp_balancing']}
  11. Proof of Theorem \ref{['theorem_pdhg4sdp_drs']}

Key Result

Theorem 2.1

If the adjustment of $\left\{(\alpha_k,\theta_k,\beta_k)\right\}_k$ in Algorithm algm:pdhg_sdp follows that where $0<\alpha_{\min}\leq\alpha_{\max}<\infty$ and $R< \frac{1}{\lambda_{\max}(\mathcal{A}^T\mathcal{A})}$. Then Algorithm algm:pdhg_sdp weakly converges to $(X^*, y^*)$ such that $0 \in C + \partial_{X}\mathbb{I}_{\mathbb{S}_{+}^{n\times n}} (X^*) + \partial_{X}\mathbb{I}_{=b} (\mathcal

Figures (1)

  • Figure 1: The rows from top to bottom are the results of RG, MC, and SNL. The columns from left to right are the results of $\texttt{rng(1)}$, $\texttt{rng(2)}$, $\texttt{rng(3)}$, and $\texttt{rng(4)}$.

Theorems & Definitions (13)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem A.1
  • proof
  • Lemma A.1
  • proof
  • Theorem A.1
  • proof
  • Theorem A.1
  • ...and 3 more