快速收敛速率与无最终球条件的线性约束优化中Tikhonov正则化混合阶原始-对偶动力学的强收敛性
Fast Rates and Strong Convergence of Tikhonov-Regularized Mixed-Order Primal-Dual Dynamics for Linearly Constrained Optimization Without Eventual Ball Conditions
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中文总结 AI 辅助
本文提出带隐式Hessian阻尼的Tikhonov正则化混合阶原始-对偶动力学，在无最终球条件下证明强收敛并给出快速速率，改进临界情形的衰减估计并解答开放问题。
中文摘要 AI 辅助
本文研究有限维欧几里得空间中线性约束凸优化问题的带隐式Hessian阻尼的Tikhonov正则化混合阶原始-对偶动力系统，其中原始方程是二阶的并包含粘性阻尼项\\(\delta\sqrt{\varepsilon(t)}\\,\dot x(t)\\)，而对偶乘子方程保持一阶。通过构造一类新的能量函数，对于一般的Tikhonov正则化系数\\(\varepsilon(t)\\)，我们在相同的参数假设下证明了原始轨迹的强收敛性并推导出快速收敛速率，无需施加任何最终球内/球外条件。更精确地，原始轨迹收敛到最小范数解，乘子收敛到相容的KKT乘子，而拉格朗日间隙、可行性违背和目标残差的收敛速率为\\(o(\varepsilon(t))\\)，速度范数的收敛速率为\\(o(\sqrt{\varepsilon(t)})\\)。对于临界情形\\(\varepsilon(t)=c/t^2\\)，其中阻尼系数\\(\delta\sqrt{\varepsilon(t)}\\)退化为\\(\delta\sqrt{c}/t\\)，我们建立了拉格朗日间隙、可行性违背和目标残差的更精细收敛速率\\(o(t^{-2})\\)，以及速度范数的\\(o(t^{-1})\\)，这改进了相关文献中获得的相应\\(O(t^{-2})\\)和\\(O(t^{-1})\\)衰减估计。最重要的是，当所提出的动力系统特化为有限维无约束情形时，我们的分析回答了Attouch和László [Math. Methods Oper. Res., 99 (2024), pp. 307–347]提出的关于该临界状态下强收敛性的开放问题。
英文摘要
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 \(δ\sqrt{\varepsilon(t)}\,\dot x(t)\), 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 \(δ\sqrt{\varepsilon(t)}\) reduces to \(δ\sqrt{c}/t\), 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].
发表机构
- School of Mathematical Sciences, University of Electronic Science and Technology of China（电子科技大学数学科学学院）
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