arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1408.0974math.OC
收藏

约束变分问题的分裂前后向罚格式

Splitting forward-backward penalty scheme for constrained variational problems

Marc-Olivier Czarnecki, Nahla Noun, Juan Peypouquet

阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

本文研究了求解变分不等式\begin{equation*} A x +\nabla Φ(x)+ N_C (x) \ni 0 \end{equation*}的一种前后分裂算法，其中$H$是实Hilbert空间，$A: H\rightrightarrows H$是极大单调算子，$Φ: H\to\mathbb{R}$是光滑凸函数，$N_C$是闭凸集$C\subset H$的外法锥.约束集$C$表示为两个凸惩罚函数$Ψ_1:H\to\mathbb{R}$和$Ψ_2: H\to\mathbb{R}\cup \{+\infty\}$的最小值集的交集。函数$Ψ_1$是光滑的，函数$Ψ_2$是适当的和下连续的。给定趋于无穷大的惩罚参数序列$(β_n)$和正时间步长序列$(λ_n)$，算法$$ \left\{\begin{array}{rcl} x_1 & \in & H,\\ x_{n+1} & = & (I+λ_n A+λ_nβ_n\partialΨ_2)^{-1}(x_n-λ_n\nablaΦ(x_n)-λ_nβ_n\nablaΨ_1(x_n)),\ n\geq 1. \end{array}\right. $$在平滑部分上执行前向步骤，在其它部分上执行后向步骤。在适当的假设下，我们得到序列$(x_n)$弱遍历收敛到变分不等式的解。当$A$是强单调的或$Φ$是强凸的时，收敛是强的。当$A$是一个真的下连续凸函数的次微分时，我们还得到了整个序列$(x_n)$的弱收敛性。这提供了一个统一的设置几个经典的和最近的结果，在连续和离散的梯度系统的历史研究。

英文摘要：

We study a forward backward splitting algorithm that solves the variational inequality \begin{equation*} A x +\nabla Φ(x)+ N_C (x) \ni 0 \end{equation*} where $H$ is a real Hilbert space, $A: H\rightrightarrows H$ is a maximal monotone operator, $Φ: H\to\mathbb{R}$ is a smooth convex function, and $N_C$ is the outward normal cone to a closed convex set $C\subset H$. The constraint set $C$ is represented as the intersection of the sets of minima of two convex penalization function $Ψ_1:H\to\mathbb{R}$ and $Ψ_2: H\to\mathbb{R}\cup \{+\infty\}$. The function $Ψ_1$ is smooth, the function $Ψ_2$ is proper and lower semicontinuous. Given a sequence $(β_n)$ of penalization parameters which tends to infinity, and a sequence of positive time steps $(λ_n)$, the algorithm $$ \left\{\begin{array}{rcl} x_1 & \in & H,\\ x_{n+1} & = & (I+λ_n A+λ_nβ_n\partialΨ_2)^{-1}(x_n-λ_n\nablaΦ(x_n)-λ_nβ_n\nablaΨ_1(x_n)),\ n\geq 1. \end{array}\right. $$ performs forward steps on the smooth parts and backward steps on the other parts. Under suitable assumptions, we obtain weak ergodic convergence of the sequence $(x_n)$ to a solution of the variational inequality. Convergence is strong when either $A$ is strongly monotone or $Φ$ is strongly convex. We also obtain weak convergence of the whole sequence $(x_n)$ when $A$ is the subdifferential of a proper lower-semicontinuous convex function. This provides a unified setting for several classical and more recent results, in the line of historical research on continuous and discrete gradient-like systems.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑