AI 中文总结
该研究针对带双线性耦合的光滑鞍点问题提出两种含Hessian驱动阻尼的惯性原始对偶动力系统，推导了不同凸性条件下的收敛速率，并将其应用于仿射约束凸优化以补充现有结果。
AI 中文摘要
针对具有双线性耦合的光滑鞍点问题，本文提出了两种具有Hessian驱动阻尼特性的惯性原始对偶动力系统。对于凸-凹函数，我们建立了原始对偶间隙的$\u27e8mathcal{O}\left( \frac{1}{t^2} \right)\u27e9收敛速率；对于强凸-强凹函数，在无需知晓强凸参数的情况下，我们得到了$\u27e8mathcal{O}\left( \frac{1}{t^{α-1}} \right)\u27e9的渐近速率（$\u27e8α\ge 3\u27e9为阻尼参数），而在已知强凸参数时可获得加速线性收敛速率。作为所提出惯性系统的应用，我们还考虑了仿射约束凸优化问题，构建了一个带Hessian驱动阻尼的惯性系统，补充了现有研究结果。
英文摘要
Featuring Hessian-driven damping, two inertial primal dual dynamical systems are proposed for solving smooth saddle point problems with bilinear coupling. For convex-concave functions, we establish a convergence rate $\mathcal{O}\left( \frac{1}{t^2} \right)$ for the primal dual gap; for strongly convex-strongly concave functions, we obtain an asymptotic rate $\mathcal{O}\left( \frac{1}{t^{α-1}} \right)$ ($α\ge 3$ is the damping parameter) without knowledge of the strong convexity parameters, and an accelerated linear convergence rate when the strong convexity parameters are known. As an application of the proposed inertial systems, we also consider the affinely constrained convex optimization problem, and develop an inertial system with Hessian-driven damping, which complements existing results.
