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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.23843 |
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| _version_ | 1866914224970661888 |
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| author | Lal, Manish Krishan |
| author_facet | Lal, Manish Krishan |
| contents | We study the Reflect-Reflect-Relax (RRR) algorithm in its small-step (flow-limit) regime. In the smooth transversal setting, we show that the transverse dynamics form a hyperbolic sink, yielding exponential decay of a natural gap measure. Under uniform geometric assumptions, we construct a tubular neighborhood of the feasible manifold on which the squared gap defines a strict Lyapunov function, excluding recurrent dynamics and chaotic behavior within this basin.
In the discrete setting, the induced flow is piecewise constant on W-domains and supports Filippov sliding along convergent boundaries, leading to finite-time capture into a solution domain. We prove that small-step RRR is a forward-Euler discretization of this flow, so that solution times measured in rescaled units converge to a finite limit while iteration counts diverge, explaining the emergence of iteration-optimal relaxation parameters. Finally, we introduce a heuristic mesoscopic framework based on percolation and renormalization group to organize performance deterioration near the Douglas-Rachford limit. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23843 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Flow-Limit of Reflect-Reflect-Relax: Existence, Stability, and Discrete-Time Behavior Lal, Manish Krishan Optimization and Control Numerical Analysis Dynamical Systems Primary 37N40, Secondary 49J53, 90C30, 65K10 We study the Reflect-Reflect-Relax (RRR) algorithm in its small-step (flow-limit) regime. In the smooth transversal setting, we show that the transverse dynamics form a hyperbolic sink, yielding exponential decay of a natural gap measure. Under uniform geometric assumptions, we construct a tubular neighborhood of the feasible manifold on which the squared gap defines a strict Lyapunov function, excluding recurrent dynamics and chaotic behavior within this basin. In the discrete setting, the induced flow is piecewise constant on W-domains and supports Filippov sliding along convergent boundaries, leading to finite-time capture into a solution domain. We prove that small-step RRR is a forward-Euler discretization of this flow, so that solution times measured in rescaled units converge to a finite limit while iteration counts diverge, explaining the emergence of iteration-optimal relaxation parameters. Finally, we introduce a heuristic mesoscopic framework based on percolation and renormalization group to organize performance deterioration near the Douglas-Rachford limit. |
| title | The Flow-Limit of Reflect-Reflect-Relax: Existence, Stability, and Discrete-Time Behavior |
| topic | Optimization and Control Numerical Analysis Dynamical Systems Primary 37N40, Secondary 49J53, 90C30, 65K10 |
| url | https://arxiv.org/abs/2512.23843 |