Fast Rates and Strong Convergence of Tikhonov-Regularized Mixed-Order Primal-Dual Dynamics for Linearly Constrained Optimization Without Eventual Ball Conditions
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 L\'aszl\'o [Math. Methods Oper. Res., 99 (2024), pp.~307--347].