The gradient and heavy ball with friction dynamical systems: the quasiconvex case
The gradient and heavy ball with friction dynamical systems: the quasiconvex case
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DOI：
10.1007/s10107-007-0109-5
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发表时间：
2009-01-01
影响因子：
2.7
通讯作者：
Munier, J.
中科院分区：
文献类型：
--
作者：
Goudou, X.;Munier, J.
我们考虑梯度系统x(T)+del Phi(x(T))=0和所谓的具有摩擦的重球动力系统x(T)+lambda x(T)+del Phi(x(T))=0，以及它的一个隐式离散(近似)形式，并研究了它们在光滑和拟凸目标函数Phi情况下解的渐近行为。在各种附加假设下，得到了轨迹的极小化性质。最后，我们证明了重球方法的一个不同于梯度方法的极小化性质：至少当Phi是Morse函数时，每个轨迹收敛到Phi的局部极小值的一般性。
We consider the gradient system x(t) + del Phi(x(t)) = 0 and the so-called heavy ball with friction dynamical system " x(t) + lambda x(t) + del Phi(x(t)) = 0, as well as an implicit discrete (proximal) version of it, and study the asymptotic behavior of their solutions in the case of a smooth and quasiconvex objective function Phi. Minimization properties of trajectories are obtained under various additional assumptions. We finally show a minimizing property of the heavy ball method which is not shared by the gradient method: the genericity of the convergence of each trajectory, at least when Phi is a Morse function, towards local minimum of Phi.