{"ID":2824499,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.23843","arxiv_id":"2512.23843","title":"The Flow-Limit of Reflect-Reflect-Relax: Existence, Stability, and Discrete-Time Behavior","abstract":"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.","short_abstract":"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 f...","url_abs":"https://arxiv.org/abs/2512.23843","url_pdf":"https://arxiv.org/pdf/2512.23843v1","authors":"[\"Manish Krishan Lal\"]","published":"2025-12-29T20:17:58Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"math.DS\",\"math.NA\"]","methods":"[\"Generative Adversarial Network\"]","has_code":false}