Variable Metric Three Operator Splitting

Abstract

To minimize the sum f + g + h of three convex functions when f is differentiable with a Lipschitz-continuous gradient, we present a proximal algorithm, variable-metric three operator splitting (VMTOS), a generalization of the three operator splitting (TOS) scheme of Davis and Yin. We establish convergence guarantees by controlling changes in the metric. The smallest saddle-point residual among the first k primal-dual iterates is Õ(k^−1/2). When f and the convex conjugate h^* are strongly convex and the accepted metrics and their inverses are uniformly bounded, the rate improves to Õ(β^k) for some β∈ (0,1). We develop a practical Barzilai--Borwein update for diagonal metrics that satisfies our bound on metric changes. Under the same strong convexity assumptions, this update guarantees the required uniform metric bounds, yielding R-linear convergence. Numerical experiments on benchmark optimization problems show that VMTOS is competitive with other proximal algorithms.

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Cite

BibTeX RIS

BibTeX

@misc{suh2026variable,
  title = {{Variable Metric Three Operator Splitting}},
  author = {Hansol Suh and Tobin Isaac},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38935},
  eprint = {2610.38935},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38935},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - Variable Metric Three Operator Splitting
AU  - Hansol Suh
AU  - Tobin Isaac
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38935
UR  - https://arxiv.org/abs/2610.38935
AB  - To minimize the sum f + g + h of three convex functions when f is differentiable with a Lipschitz-continuous gradient, we present a proximal algorithm, variable-metric three operator splitting (VMTOS), a generalization of the three operator splitting (TOS) scheme of Davis and Yin. We establish convergence guarantees by controlling changes in the metric. The smallest saddle-point residual among the first k primal-dual iterates is Õ(k^−1/2). When f and the convex conjugate h^* are strongly convex and the accepted metrics and their inverses are uniformly bounded, the rate improves to Õ(β^k) for some β∈ (0,1). We develop a practical Barzilai--Borwein update for diagonal metrics that satisfies our bound on metric changes. Under the same strong convexity assumptions, this update guarantees the required uniform metric bounds, yielding R-linear convergence. Numerical experiments on benchmark optimization problems show that VMTOS is competitive with other proximal algorithms.
ER  -

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