arXiv subjects

Ya-Ping Fang

Publications and source records attributed to Ya-Ping Fang.

14 recordsLinked to original sources

arXiv2026

A unified continuous-discrete framework for Nesterov acceleration: transitions between convex and strongly convex regimes

Classical Nesterov acceleration employs different choices of damping and inertial parameters in the convex and strongly convex settings, both for continuous-time dynamics and for discrete algorithms. When the strong convexity parameter is small, directly using the strongly convex damping or inertial coefficient may lead to slower early-stage convergence than the corresponding convex choice, despite its favorable asymptotic exponential or linear rate. We develop a unified continuous-discrete framework that encompasses both classical regimes and provides systematic transitions between them. The resulting coefficient families retain the accelerated convex behavior at early stages while attaining the strongly convex asymptotic rate. The continuous-time dynamics arise from a two-state coupling and are analyzed within a unified Lyapunov framework that yields simultaneous $\mathcal{O}(1/t^2)$ and exponential convergence estimates, thereby recovering the classical convex and strongly convex rates. We further derive two classes of accelerated forward-backward algorithms by discretizing the proposed dynamics and establish convergence estimates covering the convex, strongly convex, and intermediate regimes. The framework recovers the classical Nesterov inertial coefficients and generates hyperbolic, exponential, algebraic, and polynomial transition families. Numerical experiments demonstrate the effectiveness of the proposed methods when the strong convexity parameter is small.

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arXiv2026

Fast primal-dual methods for convex-concave bilinear saddle point problems: continuous-time dynamics and discrete algorithms

This paper studies Nesterov accelerated methods for continuously differentiable convex-concave bilinear saddle point problems. For the continuous-time model, we analyze a second-order primal-dual dynamical system with vanishing damping $\alpha/t$, where $\alpha\geq 3$. Under the merely convex-concave setting, we prove convergence of the primal-dual trajectory to a saddle point. In the noncritical regime $\alpha>3$, we further obtain the improved rate $o(1/t^{2})$ for the primal-dual gap and $o(1/t)$ for the velocity, and, under an additional Lipschitz gradient assumption, $o(1/t)$ for the stationarity residual. We then derive a structure-preserving finite-difference discretization, which leads to a fast primal-dual algorithm with Nesterov extrapolation. For a general accelerated parameter sequence ${t_k}$ satisfying $t_{k+1}^2-t_k^2\le \rho t_{k+1}$ with $\rho\in(0,1]$, we prove the $O(1/t_k^{2})$ convergence rate for the primal-dual gap and convergence of the generated sequence. In the noncritical case $\rho<1$, we further establish the improved rate $o(1/t_k^{2})$ for the gap and $o(1/t_k)$ for the stationarity residual. These results provide continuous-discrete acceleration methods for bilinear saddle point problems in the merely convex-concave setting.

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arXiv2026

Convergence of iterates and improved rates for accelerated augmented Lagrangian methods for linearly constrained convex optimization

Motivated by an inertial primal-dual dynamical system with vanishing damping, we propose a class of accelerated augmented Lagrangian methods with Nesterov extrapolation parameters for a linearly constrained convex optimization problem with a differentiable objective function. The framework contains two variants: an implicit-gradient scheme for convex continuously differentiable objectives and a partially explicit scheme for convex smooth objectives. Under suitable parameter conditions, we prove convergence of the primal-dual sequence to a primal-dual solution, together with accelerated estimates for the augmented Lagrangian gap, the feasibility violation, and the objective residual. In the noncritical parameter regime, these estimates are improved from $\mathcal{O}(1/k^2)$ to $o(1/k^2)$. Numerical experiments are also presented to illustrate the theoretical results. To the best of our knowledge, neither $o(1/k^2)$ rates for both feasibility violation and objective residual nor convergence of iterates under the critical parameter condition have been previously established for accelerated augmented Lagrangian-type methods in this setting.

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arXiv2026

Trajectory convergence and $o(t^{-2})$ rates for Nesterov accelerated primal-dual dynamics without Lipschitz gradient assumption

We consider the Nesterov accelerated primal-dual dynamical system \[ \begin{cases} \ddot{x}(t)+\dfrac{\alpha}{t}\dot{x}(t) +\nabla f(x(t)) +A^\top\bigl(\lambda(t)+\theta t\dot{\lambda}(t)\bigr)+\beta A^\top(Ax(t)-b)=0,\\[0.6em] \ddot{\lambda}(t)+\dfrac{\alpha}{t}\dot{\lambda}(t) -\bigl(A(x(t)+\theta t\dot{x}(t))-b\bigr)=0, \end{cases} \] which is linked to the linearly constrained optimization problem $ \min_{x\in\mathbb{R}^n} f(x),\ s.t.\ Ax=b, $ where $\alpha\ge 3$ and $f$ is convex and continuously differentiable. In a Hilbert framework, the weak convergence of its trajectory was established by Bo\c{t} and Nguyen (J. Differential Equations, 303:369--406, 2021) under $\alpha>3$ and the Lipschitz continuity assumption on $\nabla f$. In this paper, we prove in finite-dimensional spaces that the trajectory converges to a primal-dual solution for $\alpha\ge3$, without assuming Lipschitz continuity of $\nabla f$. Moreover, when $\alpha>3$, we establish improved $o(t^{-2})$ convergence rates for both the objective residual and the feasibility violation. Our analysis relies on Bregman-distance arguments, instead of the Lipschitz continuity of $\nabla f$. The same strategy can also be extended to time-scaled primal-dual dynamics to obtain analogous convergence results. To the best of our knowledge, this is the first results in this topic without Lipschitz gradient assumption. Our result also present the first work on the convergence of the trajectory of the accelerated primal-dual dynamical system for the critical case $\alpha=3$.

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arXiv2025

Nesterov acceleration for strongly convex-strongly concave bilinear saddle point problems: discrete and continuous-time approaches

In this paper, we study a bilinear saddle point problem of the form $\min_{x}\max_{y} F(x) + \langle Ax, y \rangle - G(y)$, where $F$ and $G$ are $\mu_F$- and $\mu_G$-strongly convex functions, respectively. By incorporating Nesterov acceleration for strongly convex optimization, we first propose an optimal first-order discrete primal-dual gradient algorithm. We show that it achieves the optimal convergence rate $\mathcal{O}\left(\left(1 - \min\left\{\sqrt{\frac{\mu_F}{L_F}}, \sqrt{\frac{\mu_G}{L_G}}\right\}\right)^k\right)$ for both the primal-dual gap and the iterative, where $L_F$ and $L_G$ denote the smoothness constants of $F$ and $G$, respectively. We further develop a continuous-time accelerated primal-dual dynamical system with constant damping. Using the Lyapunov analysis method, we establish the existence and uniqueness of a global solution, as well as the linear convergence rate $\mathcal{O}(e^{-\min\{\sqrt{\mu_F},\sqrt{\mu_G}\}t})$. Notably, when $A = 0$, our methods recover the classical Nesterov accelerated methods for strongly convex unconstrained problems in both discrete and continuous-time. Numerical experiments are presented to support the theoretical convergence rates.

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arXiv2024

Strong convergence of an inertial Tikhonov regularized dynamical system governed by a maximally comonotone operator

In a Hilbert framework, we consider an inertial Tikhonov regularized dynamical system governed by a maximally comonotone operator, where the damping coefficient is proportional to the square root of the Tikhonov regularization parameter. Under an appropriate setting of the parameters, we prove the strong convergence of the trajectory of the proposed system towards the minimum norm element of zeros of the underlying maximally comonotone operator. When the Tikhonov regularization parameter reduces to $\frac{1}{t^q}$ with $0<q<1$, we further establish some convergence rate results of the trajectories. Finally, the validity of the proposed dynamical system is demonstrated by a numerical example.

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arXiv2024

Strong asymptotic convergence of a slowly damped inertial primal-dual dynamical system controlled by a Tikhonov regularization term

We propose a slowly damped inertial primal-dual dynamical system controlled by a Tikhonov regularization term, where the inertial term is introduced only for the primal variable, for the linearly constrained convex optimization problem in a Hilbert space. Under mild conditions on the underlying parameters, by a Lyapunov analysis approach, we prove the strong asymptotic convergence of the trajectory of the proposed dynamic to the minimal norm element of the primal-dual solution set of the problem, along with convergence rate results for the primal-dual gap, the objective residual and the feasibility violation. We perform some numerical experiments to illustrate the theoretical findings.

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arXiv2024

Fast convergence rates and trajectory convergence of a Tikhonov regularized inertial primal\mbox{-}dual dynamical system with time scaling and vanishing damping

A Tikhonov regularized inertial primal\mbox{-}dual dynamical system with time scaling and vanishing damping is proposed for solving a linearly constrained convex optimization problem in Hilbert spaces. The system under consideration consists of two coupled second order differential equations and its convergence properties depend upon the decaying speed of the product of the time scaling parameter and the Tikhonov regularization parameter (named the rescaled regularization parameter) to zero. When the rescaled regularization parameter converges rapidly to zero, the system enjoys fast convergence rates of the primal-dual gap, the feasibility violation, the objective residual, and the gradient norm of the objective function along the trajectory, and the weak convergence of the trajectory to a primal-dual solution of the linearly constrained convex optimization problem. When the rescaled regularization parameter converges slowly to zero, the generated primal trajectory converges strongly to the minimal norm solution of the problem under suitable conditions. Finally, numerical experiments are performed to illustrate the theoretical findings.

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arXiv2022

Fast primal-dual algorithm via dynamical system for a linearly constrained convex optimization problem

By time discretization of a second-order primal-dual dynamical system with damping $α/t$ where an inertial construction in the sense of Nesterov is needed only for the primal variable, we propose a fast primal-dual algorithm for a linear equality constrained convex optimization problem. Under a suitable scaling condition, we show that the proposed algorithm enjoys a fast convergence rate for the objective residual and the feasibility violation, and the decay rate can reach $\mathcal{O}(1/k^{α-1})$ at the most. We also study convergence properties of the corresponding primal-dual dynamical system to better understand the acceleration scheme. Finally, we report numerical experiments to demonstrate the effectiveness of the proposed algorithm.

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arXiv2022

"Second-Order Primal'' + "First-Order Dual'' Dynamical Systems with Time Scaling for Linear Equality Constrained Convex Optimization Problems

Second-order dynamical systems are important tools for solving optimization problems, and most of existing works in this field have focused on unconstrained optimization problems. In this paper, we propose an inertial primal-dual dynamical system with constant viscous damping and time scaling for the linear equality constrained convex optimization problem, which consists of a second-order ODE for the primal variable and a first-order ODE for the dual variable. When the scaling satisfies certain conditions, we prove its convergence property without assuming strong convexity. Even the convergence rate can become exponential when the scaling grows exponentially. We also show that the obtained convergence property of the dynamical system is preserved under a small perturbation.

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arXiv2021

Inertial primal-dual methods for linear equality constrained convex optimization problems

In this paper, we propose an inertial accelerated primal-dual method for the linear equality constrained convex optimization problem. When the objective function has a ``nonsmooth + smooth'' composite structure, we further propose an inexact inertial primal-dual method by linearizing the smooth individual function and solving the subproblem inexactly. Assuming merely convexity, we prove that the proposed methods enjoy $\mathcal{O}(1/k^2)$ convergence rate on the objective residual and the feasibility violation in the primal model. Numerical results are reported to demonstrate the validity of the proposed methods.

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arXiv2021

Perturbed primal-dual dynamics with damping and time scaling coefficients for affine constrained convex optimization problems

In Hilbert space, we propose a family of primal-dual dynamical system for affine constrained convex optimization problem. Several damping coefficients, time scaling coefficients, and perturbation terms are thus considered. By constructing the energy functions, we investigate the convergence rates with different choices of the damping coefficients and time scaling coefficients. Our results extend the inertial dynamical approaches for unconstrained convex optimization problems to affine constrained convex optimization problems.

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arXiv2020

Asymptotic behavior of a nonautonomous evolution equation governed by a quasi-nonexpansive operator

We study the asymptotic behavior of the trajectory of a nonautonomous evolution equation governed by a quasi-nonexpansive operator in Hilbert spaces. We prove the weak convergence of the trajectory to a fixed point of the operator by relying on Lyapunov analysis. Under a metric subregularity condition, we further derive a flexible global exponential-type rate for the distance of the trajectory to the set of fixed points. The results obtained are applied to analyze the asymptotic behavior of the trajectory of an adaptive Douglas-Rachford dynamical system, which is applied for finding a zero of the sum of two operators, one of which is strongly monotone while the other one is weakly monotone.

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arXiv2020

Convergence Rates of Inertial Primal-Dual Dynamical Methods for Separable Convex Optimization Problems

In this paper, we propose a second-order continuous primal-dual dynamical system with time-dependent positive damping terms for a separable convex optimization problem with linear equality constraints. By the Lyapunov function approach, we investigate asymptotic properties of the proposed dynamical system as the time $t\to+\infty$. The convergence rates are derived for different choices of the damping coefficients. We also show that the obtained results are robust under external perturbations.

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