论文标题
与渐近消失阻尼的原始双重动力学系统的时间重新缩放
Time rescaling of a primal-dual dynamical system with asymptotically vanishing damping
论文作者
论文摘要
在这项工作中，我们通过二阶动力系统在线性相等性约束下最小化连续可区分的凸函数，该系统具有渐近消失的阻尼项。正在考虑的系统是文献中先前发现的另一个系统的时间版本。我们显示了沿生成的轨迹的原始偶发性间隙，可行性度量和目标函数值的快速收敛性。这些收敛速率现在取决于重新缩放参数，因此可以通过适当选择上述参数来改善。当目标函数具有Lipschitz的连续梯度时，我们表明，原始的偶型轨迹渐近地收敛到原始的二重式最佳解决方案，以解决潜在的最小化问题。我们还表现出沿着原始轨迹和沿双轨迹的相应线性算子的伴随的梯度收敛速率的提高。即使在不受约束的情况下，某些轨迹收敛结果似乎是新的。我们通过数值实验说明了理论结果。
In this work, we approach the minimization of a continuously differentiable convex function under linear equality constraints by a second-order dynamical system with an asymptotically vanishing damping term. The system under consideration is a time rescaled version of another system previously found in the literature. We show fast convergence of the primal-dual gap, the feasibility measure, and the objective function value along the generated trajectories. These convergence rates now depend on the rescaling parameter, and thus can be improved by choosing said parameter appropriately. When the objective function has a Lipschitz continuous gradient, we show that the primal-dual trajectory asymptotically converges weakly to a primal-dual optimal solution to the underlying minimization problem. We also exhibit improved rates of convergence of the gradient along the primal trajectories and of the adjoint of the corresponding linear operator along the dual trajectories. Even in the unconstrained case, some trajectory convergence result seems to be new. We illustrate the theoretical outcomes through numerical experiments.
