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PreprintarXiv licencearXiv

Global Linear Convergence of the Proximal Bundle Method under Unknown Piecewise Smoothness and Quadratic Growth

  • Zhenwei Lin
  • Zhe Zhang
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Abstract

We study why the proximal bundle method (PBM) can perform better in practice when it retains more cutting planes. We consider convex objectives with quadratic growth and an unknown piecewise-smooth structure. Our key observation is that retaining sufficiently many cutting planes allows PBM to exploit the objective's piecewise-smooth structure and behave as if it were optimizing a smooth function. We provide a theoretical explanation for the observed linear convergence of PBM on piecewise-smooth objectives when it retains sufficiently many cutting planes.

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Cite

BibTeXRIS
BibTeX
@misc{lin2026global,
  title = {{Global Linear Convergence of the Proximal Bundle Method under Unknown Piecewise Smoothness and Quadratic Growth}},
  author = {Zhenwei Lin and Zhe Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11480},
  eprint = {2609.11480},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11480},
}
RIS
TY  - UNPB
TI  - Global Linear Convergence of the Proximal Bundle Method under Unknown Piecewise Smoothness and Quadratic Growth
AU  - Zhenwei Lin
AU  - Zhe Zhang
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11480
UR  - https://arxiv.org/abs/2609.11480
AB  - We study why the proximal bundle method (PBM) can perform better in practice when it retains more cutting planes. We consider convex objectives with quadratic growth and an unknown piecewise-smooth structure. Our key observation is that retaining sufficiently many cutting planes allows PBM to exploit the objective's piecewise-smooth structure and behave as if it were optimizing a smooth function. We provide a theoretical explanation for the observed linear convergence of PBM on piecewise-smooth objectives when it retains sufficiently many cutting planes.
ER  -