Operator Splitting with Hamilton-Jacobi-based Proximals
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910249287417856 |
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| author | Di, Nicholas Chi, Eric C. Fung, Samy Wu |
| author_facet | Di, Nicholas Chi, Eric C. Fung, Samy Wu |
| contents | Operator splitting algorithms are a cornerstone of modern first-order optimization, decomposing complex problems into simpler subproblems solved via proximal operators. However, most functions lack closed-form proximal operators, which has long restricted these methods to a narrow set of problems. Hamilton-Jacobi-based proximal operator (HJ-Prox) is a recent derivative-free Monte Carlo technique based on Hamilton-Jacobi PDE theory, that approximates proximal operators numerically. In this work, we introduce a unified framework for operator splitting via HJ-Prox, which allows for deployment of operator splitting even when functions are not proximable. We prove that replacing exact proximal steps with HJ-Prox in algorithms such as proximal point, proximal gradient descent, Douglas-Rachford splitting, Davis-Yin splitting, and primal-dual hybrid gradient preserves convergence guarantees under mild assumptions. Numerical experiments demonstrate HJ-Prox is competitive and effective on a wide variety of statistical learning tasks. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_22370 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Operator Splitting with Hamilton-Jacobi-based Proximals Di, Nicholas Chi, Eric C. Fung, Samy Wu Optimization and Control 65K10 Operator splitting algorithms are a cornerstone of modern first-order optimization, decomposing complex problems into simpler subproblems solved via proximal operators. However, most functions lack closed-form proximal operators, which has long restricted these methods to a narrow set of problems. Hamilton-Jacobi-based proximal operator (HJ-Prox) is a recent derivative-free Monte Carlo technique based on Hamilton-Jacobi PDE theory, that approximates proximal operators numerically. In this work, we introduce a unified framework for operator splitting via HJ-Prox, which allows for deployment of operator splitting even when functions are not proximable. We prove that replacing exact proximal steps with HJ-Prox in algorithms such as proximal point, proximal gradient descent, Douglas-Rachford splitting, Davis-Yin splitting, and primal-dual hybrid gradient preserves convergence guarantees under mild assumptions. Numerical experiments demonstrate HJ-Prox is competitive and effective on a wide variety of statistical learning tasks. |
| title | Operator Splitting with Hamilton-Jacobi-based Proximals |
| topic | Optimization and Control 65K10 |
| url | https://arxiv.org/abs/2601.22370 |