Gradient Descent Ascent (GDA) methods for min-max optimization problems
typically produce oscillatory behavior that can lead to instability, e.g., in
bilinear settings. To address this problem, we introduce a dissipation term
into the GDA updates to dampen these oscillations. The proposed Dissipative GDA
(DGDA) method can be seen as performing standard GDA on a state-augmented and
regularized saddle function that does not strictly introduce additional
convexity/concavity. We theoretically show the linear convergence of DGDA in
the bilinear and strongly convex-strongly concave settings and assess its
performance by comparing DGDA with other methods such as GDA, Extra-Gradient
(EG), and Optimistic GDA. Our findings demonstrate that DGDA surpasses these
methods, achieving superior convergence rates. We support our claims with two
numerical examples that showcase DGDA's effectiveness in solving saddle point
problems.

梯度下降和上升（GDA）法用于最小 - 最大优化问题时，通常会产生振荡行为，容易导致不稳定性，为解决这一问题，本文提出了一种通过引入耗散项来抑制振荡的方法，称为 Dissipative GDA (DGDA)，DGDA 方法可以看作是在一个状态增广和规则化的鞍点函数上执行标准 GDA，该方法在双线性和强凸 - 强凹设置中具有线性收敛性，并通过与其他方法（如 GDA、Extra-Gradient，以及 Optimistic GDA）的比较来评估其性能，实验结果表明 DGDA 超过了这些方法，实现了更好的收敛速度，通过两个数值例子展示了 DGDA 方法在求解鞍点问题中的有效性。