FAST CONVERGENCE OF INERTIAL DYNAMICS AND ALGORITHMS WITH ASYMPTOTIC VANISHING DAMPING
FAST CONVERGENCE OF INERTIAL DYNAMICS AND ALGORITHMS WITH ASYMPTOTIC VANISHING DAMPING
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惯性动力学和渐近消失阻尼算法的快速收敛
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
P. Redont
中科院分区：
文献类型：
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作者：
H. Attouch;Z. Chbani;J. Peypouquet;P. Redont
在Hilbert空间H中，研究了二阶微分方程<$（t）+ α t <$（t）+<$Φ（x（t））= g（t）的轨迹在t → +∞时的快速收敛性，其中<$Φ是凸连续可微函数Φ：H → R的梯度，α是正参数，g：[t0，+∞[→ H]是小扰动项.在这个惯性系中，粘性阻尼系数αt渐近地消失，但不会太快。对于α ≥ 3，且<$+∞ t0 t <$g（t）<$dt < +∞，仅设argminΦ <$=<$，我们证明了上述系统的任何轨迹满足快速收敛性质Φ（x（t））−min H Φ ≤ Ct 2.此外，当α > 3时，我们证明了任何轨线都弱收敛于Φ的极小值。强收敛是在各种实际情况下建立的。这些结果补充了Su，Boyd和Candès在未扰动情况g = 0下的O（t−2）收敛速度。这个系统的时间离散化，以及它的一些变体，提供了新的快速收敛算法，扩展了Nesterov介绍的结构化凸最小化的快速方法的领域，并由Beck和Teboulle进一步开发了FISTA。这项研究也补充了Chambolle和Dossal的最新进展。
In a Hilbert space setting H, we study the fast convergence properties as t → +∞ of the trajectories of the second-order differential equation ẍ(t) + α t ẋ(t) +∇Φ(x(t)) = g(t), where ∇Φ is the gradient of a convex continuously differentiable function Φ : H → R, α is a positive parameter, and g : [t0,+∞[→ H is a small perturbation term. In this inertial system, the viscous damping coefficient αt vanishes asymptotically, but not too rapidly. For α ≥ 3, and ∫+∞ t0 t∥g(t)∥dt < +∞, just assuming that argminΦ ̸= ∅, we show that any trajectory of the above system satisfies the fast convergence property Φ(x(t))−min H Φ ≤ C t2 . Moreover, for α > 3, we show that any trajectory converges weakly to a minimizer of Φ. The strong convergence is established in various practical situations. These results complement the O(t−2) rate of convergence for the values obtained by Su, Boyd and Candès in the unperturbed case g = 0. Time discretization of this system, and some of its variants, provides new fast converging algorithms, expanding the field of rapid methods for structured convex minimization introduced by Nesterov, and further developed by Beck and Teboulle with FISTA. This study also complements recent advances due to Chambolle and Dossal.