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<p>The Chambolle–Pock method is a versatile three-parameter algorithm designed to solve a broad class of composite convex optimization problems, which encompass two proper, lower semicontinuous, and convex functions, along with a linear operator <i>L</i>. The functions are accessed via their proximal operators, while the linear operator is evaluated in a forward manner. Among the three algorithm parameters <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\tau \)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\sigma \)</EquationSource> </InlineEquation>, and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\theta \)</EquationSource> </InlineEquation>; <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\tau , \sigma &gt;0\)</EquationSource> </InlineEquation> serve as step sizes for the proximal operators, and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\theta \)</EquationSource> </InlineEquation> is an extrapolation step parameter. Previous convergence results have been based on the assumption that <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\theta =1\)</EquationSource> </InlineEquation>. We demonstrate that weak convergence is achievable whenever <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\theta &gt; 1/2\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(\tau \sigma \Vert L\Vert ^2&lt;4/\mathord {\left( 1+2\theta \right) }\)</EquationSource> </InlineEquation>. Moreover, we establish tightness of the step size bound by providing an example that is nonconvergent whenever the second bound is violated.</p>

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The Chambolle–Pock method converges weakly with \(\theta >1/2\) and \(\tau \sigma \Vert L\Vert ^2<4/(1+2\theta )\)

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  • 发表日期：

    2025年10月23日

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  • Sebastian Banert,
  • Manu Upadhyaya,
  • Pontus Giselsson

摘要

The Chambolle–Pock method is a versatile three-parameter algorithm designed to solve a broad class of composite convex optimization problems, which encompass two proper, lower semicontinuous, and convex functions, along with a linear operator L. The functions are accessed via their proximal operators, while the linear operator is evaluated in a forward manner. Among the three algorithm parameters \(\tau \) , \(\sigma \) , and \(\theta \) ; \(\tau , \sigma >0\) serve as step sizes for the proximal operators, and \(\theta \) is an extrapolation step parameter. Previous convergence results have been based on the assumption that \(\theta =1\) . We demonstrate that weak convergence is achievable whenever \(\theta > 1/2\) and \(\tau \sigma \Vert L\Vert ^2<4/\mathord {\left( 1+2\theta \right) }\) . Moreover, we establish tightness of the step size bound by providing an example that is nonconvergent whenever the second bound is violated.