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<p>In this paper, we consider a broad class of nonconvex and nonsmooth optimization problems, where one objective component is a nonsmooth weakly convex function composed with a linear operator. By integrating variable smoothing techniques with first-order methods, we propose a variable smoothing alternating proximal gradient algorithm that features flexible parameter choices for step sizes and smoothing levels. Under mild assumptions, we establish that the iteration complexity to reach an <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varepsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>-approximate stationary point is <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {O}(\varepsilon ^{-3})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>ε</mi> <mrow> <mo>-</mo> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The proposed algorithm is evaluated on sparse signal recovery and image denoising problems. Numerical experiments demonstrate its effectiveness and superiority over existing algorithms.</p>

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Variable Smoothing Alternating Proximal Gradient Algorithm for Coupled Composite Optimization

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月26日

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  • Xian-Jun Long,
  • Kang Zeng,
  • Gao-Xi Li,
  • Minh N. Dao,
  • Zai-Yun Peng

摘要

In this paper, we consider a broad class of nonconvex and nonsmooth optimization problems, where one objective component is a nonsmooth weakly convex function composed with a linear operator. By integrating variable smoothing techniques with first-order methods, we propose a variable smoothing alternating proximal gradient algorithm that features flexible parameter choices for step sizes and smoothing levels. Under mild assumptions, we establish that the iteration complexity to reach an \(\varepsilon \) ε -approximate stationary point is \(\mathcal {O}(\varepsilon ^{-3})\) O ( ε - 3 ) . The proposed algorithm is evaluated on sparse signal recovery and image denoising problems. Numerical experiments demonstrate its effectiveness and superiority over existing algorithms.