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Fast Rates and Strong Convergence of Tikhonov-Regularized Mixed-Order Primal-Dual Dynamics for Linearly Constrained Optimization Without Eventual Ball Conditions — arXiv:2609.06420

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.06420
License (as archived): CC0-1.0
Archived: 2026-10-16T14:24:03.546Z
SHA-256 of the archived text: 2a8bd5a80926ee1c46aeb41ef500386d40e5555f6a8340c221eb357e8f4f5a60
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How to cite

Fast Rates and Strong Convergence of Tikhonov-Regularized Mixed-Order Primal-Dual Dynamics for Linearly Constrained Optimization Without Eventual Ball Conditions — arXiv:2609.06420 — arXiv (en), https://arxiv.org/abs/2609.06420 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_cea331bf7459 (Metraipolis archive dc_cea331bf7459, sha256 2a8bd5a80926ee1c46aeb41ef500386d40e5555f6a8340c221eb357e8f4f5a60)

BibTeX

@misc{dccea331bf7459,
  title = {Fast Rates and Strong Convergence of Tikhonov-Regularized Mixed-Order Primal-Dual Dynamics for Linearly Constrained Optimization Without Eventual Ball Conditions — arXiv:2609.06420},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.06420},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_cea331bf7459; sha256 2a8bd5a80926ee1c46aeb41ef500386d40e5555f6a8340c221eb357e8f4f5a60},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.06420. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In this paper, we study a Tikhonov-regularized mixed-order primal--dual dynamical system with implicit Hessian damping for linearly constrained convex optimization problems in finite-dimensional Euclidean spaces, where the primal equation is second order and incorporates the viscous damping term \(\delta\sqrt{\varepsilon(t)}\,\dot x(t)\), whereas the multiplier equation remains first order. By constructing a new class of energy functions, for a general Tikhonov regularization coefficient \(\varepsilon(t)\), we prove the strong convergence of the primal trajectory and derive fast convergence rates under the same parameter assumptions, without imposing any eventual inside/outside-ball condition. More precisely, the primal trajectory converges to the minimum-norm solution, and the multiplier converges to a compatible KKT multiplier, while the convergence rates of the Lagrangian gap, feasibility violation, and objective residual are \(o(\varepsilon(t))\), and the convergence rate of the velocity norm is \(o(\sqrt{\varepsilon(t)})\). For the critical case \(\varepsilon(t)=c/t^2\), in which the damping coefficient \(\delta\sqrt{\varepsilon(t)}\) reduces to \(\delta\sqrt{c}/t\), we establish the sharper convergence rates \(o(t^{-2})\) for the Lagrangian gap, feasibility violation, and objective residual, together with \(o(t^{-1})\) for the velocity norm, which improve the corresponding \(O(t^{-2})\) and \(O(t^{-1})\) decay estimates obtained in the related literature. Most importantly, when the proposed dynamical system is specialized to the finite-dimensional unconstrained setting, our analysis answers the open question on strong convergence in this critical regime posed by Attouch and László [Math. Methods Oper. Res., 99 (2024), pp.~307--347].

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-09 in math.OC).
Categorie arXiv: Optimization and Control (math.OC).
Pagina del lavoro: https://arxiv.org/abs/2609.06420 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T14:09:21.670Z dalla pagina https://arxiv.org/catchup/math.OC/2026-09-09?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).