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<p>The primal-dual hybrid gradient (PDHG) algorithm is widely utilized for solving a broad class of composite convex optimization problems due to its low computational complexity and high efficiency. In this paper, we first analyze a generic PDHG that employs dynamic primal and dual stepsizes and relaxes the classical stepsize condition to the maximal extent, beyond which further improvement is impossible. We establish pointwise convergence and sublinear convergence rate results by connecting the generic PDHG with an indefinite proximal alternating direction method of multipliers. Furthermore, an adaptive PDHG is proposed, which adjusts stepsizes based on the primal and dual residuals while adhering to the convergence condition of the generic PDHG. Extensive numerical experiments demonstrate the superior performance of the proposed adaptive PDHG.</p>

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An Adaptive PDHG Algorithm with Relaxed Stepsize Condition for Composite Convex Optimization

  • 期刊论文

  • 发表日期：

    2025年4月15日

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  • Zengyun Shan,
  • Haiwen Xu,
  • Junfeng Yang

摘要

The primal-dual hybrid gradient (PDHG) algorithm is widely utilized for solving a broad class of composite convex optimization problems due to its low computational complexity and high efficiency. In this paper, we first analyze a generic PDHG that employs dynamic primal and dual stepsizes and relaxes the classical stepsize condition to the maximal extent, beyond which further improvement is impossible. We establish pointwise convergence and sublinear convergence rate results by connecting the generic PDHG with an indefinite proximal alternating direction method of multipliers. Furthermore, an adaptive PDHG is proposed, which adjusts stepsizes based on the primal and dual residuals while adhering to the convergence condition of the generic PDHG. Extensive numerical experiments demonstrate the superior performance of the proposed adaptive PDHG.