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<p>In this paper, we are concerned with a class of convex-concave saddle point problems, where one of the objective parts is assumed to be a convex and smooth function with Lipschitz continuous gradient. By exploiting the bilinear structure of the objective, we first propose a practical accelerated Proximal Primal-Dual algorithm (PPD+), which possesses an <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(O(1/N^2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <msup> <mi>N</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> convergence rate measured by the residual between two successive iterates, where <i>N</i> represents the iteration counter. In some cases, considering that the underlying subproblems of PPD+ cannot be easily solved exactly or up to a high precision, we further propose two inexact versions of the PPD+ under absolute and relative error criteria. Finally, we employ a restarting technique to enhance our algorithms for the purpose of making them more robust and efficient. A series of numerical experiments demonstrate that our algorithms perform well in practice.</p>

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Practical proximal primal-dual algorithms for structured saddle point problems

  • 期刊论文

  • 发表日期：

    2025年10月21日

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  • Yunfei Qu,
  • Hongjin He,
  • Tao Zhang,
  • Deren Han

摘要

In this paper, we are concerned with a class of convex-concave saddle point problems, where one of the objective parts is assumed to be a convex and smooth function with Lipschitz continuous gradient. By exploiting the bilinear structure of the objective, we first propose a practical accelerated Proximal Primal-Dual algorithm (PPD+), which possesses an \(O(1/N^2)\) O ( 1 / N 2 ) convergence rate measured by the residual between two successive iterates, where N represents the iteration counter. In some cases, considering that the underlying subproblems of PPD+ cannot be easily solved exactly or up to a high precision, we further propose two inexact versions of the PPD+ under absolute and relative error criteria. Finally, we employ a restarting technique to enhance our algorithms for the purpose of making them more robust and efficient. A series of numerical experiments demonstrate that our algorithms perform well in practice.