Title: On the Arrow-Hurwicz differential system for linearly constrained convex minimization Show Chinese title

Title: 关于求解具有线性约束的凸极小化的阿罗-赫维茨微分系统

Abstract: In a real Hilbert space setting, we reconsider the classical Arrow-Hurwicz differential system in view of solving linearly constrained convex minimization problems. We investigate the asymptotic properties of the differential system and provide conditions for which its solutions converge towards a saddle point of the Lagrangian associated with the convex minimization problem. Our convergence analysis mainly relies on a `Lagrangian identity' which naturally extends on the well-known descent property of the classical continuous steepest descent method. In addition, we present asymptotic estimates on the decay of the solutions and the primal-dual gap function measured in terms of the Lagrangian. These estimates are further refined to the ones of the classical damped harmonic oscillator provided that second-order information on the objective function of the convex minimization problem is available. Finally, we show that our results directly translate to the case of solving structured convex minimization problems. Numerical experiments further illustrate our theoretical findings. Show Chinese abstract

Abstract: 在实Hilbert空间框架下，我们重新考虑经典的Arrow-Hurwicz微分系统，以解决具有线性约束的凸极小化问题。 我们研究了微分系统的渐近性质，并给出了其解收敛至与凸极小化问题相关的Lagrange函数鞍点的条件。 我们的收敛性分析主要依赖于一个“Lagrange恒等式”，该恒等式自然地扩展了经典连续最速下降法的已知下降性质。 此外，我们提供了关于解和基于Lagrange函数的原始-对偶间隙函数衰减的渐近估计。 这些估计进一步细化为经典的阻尼谐振子的估计，前提是凸极小化问题的目标函数具有二阶信息。 最后，我们证明了我们的结果可以直接推广到求解结构化凸极小化问题的情况。 数值实验进一步验证了我们的理论发现。