论文标题
关于复合优化的半齿牛顿方法的局部收敛
On the local convergence of the semismooth Newton method for composite optimization
论文作者
论文摘要
在本文中，我们考虑了从一阶类型方法得出的一类大型非线性方程，以解决复合优化问题。建立半齿牛顿型方法的超线性收敛速率用于求解非线性方程的方法通常会假定B-雅各布的非语言或方程的平滑度。我们研究了这两种情况的可行性。对于非语言条件，我们介绍了广泛的一般性表征，并说明它们是一些示例的易于检查的标准。对于平滑性条件，我们证明它在本地固定，用于从复合优化问题中得出的大量剩余映射。此外，我们研究了平滑度条件的轻松版本 - 平滑度仅限于某些主动歧管。我们提出了一种利用这种结构的概念算法，并证明其具有超线性收敛速率。
In this paper, we consider a large class of nonlinear equations derived from first-order type methods for solving composite optimization problems. Traditional approaches to establishing superlinear convergence rates of semismooth Newton-type methods for solving nonlinear equations usually postulate either nonsingularity of the B-Jacobian or smoothness of the equation. We investigate the feasibility of both conditions. For the nonsingularity condition, we present equivalent characterizations in broad generality, and illustrate that they are easy-to-check criteria for some examples. For the smoothness condition, we show that it holds locally for a large class of residual mappings derived from composite optimization problems. Furthermore, we investigate a relaxed version of the smoothness condition - smoothness restricted to certain active manifolds. We present a conceptual algorithm utilizing such structures and prove that it has a superlinear convergence rate.
