Proximal Point Algorithm Controlled by a Slowly Vanishing Term: Applications to Hierarchical Minimization

• Authors: Alexandre Cabot
• Venue: SIAM Journal on Optimization
• Year: 2005

Let $\Phi_0:\mathbb{R}^n\to \mathbb{R}\cup \{+\infty\}$ be a closed convex function and $\Phi_1:\mathbb{R}^n\to \mathbb{R}$ be a finite convex function that are bounded from below. Our goal is to build an algorithm which first minimizes the map $\Phi_0$ and secondly the map $\Phi_1$ over the set $S_0:= {\rm argmin}\, \Phi_0$. For that purpose, we define the following proximal-type algorithm: $$ -(x_{n+1}-x_n)/{\lambda_n}\in \partial_{\eta_n} (\Phi_0+\varepsilon_n \Phi_1) (x_{n+1}),\leqno({\mathcal A}_1) $$ where $(\lambda_n)$ is a positive step sequence, $(\eta_n)$ is a summable error sequence, and $(\varepsilon_n)$ is a control sequence tending toward $0$; $\partial_\eta$ denotes the $\eta$-approximate subdifferential. When $(\varepsilon_n)$ is a slow control, i.e., \,$\sum_{n=0}^{+\infty}=\varepsilon_n +\infty$, we prove that, under adequate conditions, the sequence $(x_n)$ defined by $({\mathcal A}_1)$ tends toward an element of $S_1:={\rm argmin\,}_{S_0}\Phi_1$.More generally, given finite convex functions $\Phi_2,\ldots,\Phi_N:\mathbb{R}^n\to \mathbb{R}$, let us define the sets $(S_i)_{i\in\{1,\ldots,N\}}$ by the recursive relation $S_i:={\rm argmin\,}_{S_{i-1}}\Phi_i$. We introduce an extension of algorithm $({\mathcal A}_1)$ to minimize hierarchically each function $\Phi_i$ on the set $S_{i-1}$, for $i\in\{1,\ldots,N\}$.