First-order inertial algorithms involving dry friction damping

Abstract

In a Hilbert space ${\mathcal H}$, we introduce a new class of first-order algorithms which naturally occur as discrete temporal versions of an inertial differential inclusion jointly involving viscous friction and dry friction. The function $f:{\mathcal H}\rightarrow {\mathbb {R}}$ to be minimized is supposed to be differentiable (not necessarily convex), and enters the algorithm via its gradient. The dry friction damping function $\phi :{\mathcal H}\rightarrow {\mathbb {R}}_+$ is convex with a sharp minimum at the origin, (typically $\phi (x) = r \Vert x\Vert$ with $r >0$). It enters the algorithm via its proximal mapping, which acts as a soft threshold operator on the velocities. As a result, we obtain a new class of splitting algorithms involving separately the proximal and gradient steps. The sequence of iterates has a finite length, and therefore strongly converges towards an approximate critical point $x_{\infty }$ of f (typically $\Vert \nabla f(x_{\infty })\Vert \le r$). Under a geometric property satisfied by the limit point $x_{\infty }$, we obtain geometric and finite rates of convergence. The convergence results tolerate the presence of errors, under the sole assumption of their asymptotic convergence towards zero. By replacing the function f by its Moreau envelope, we extend the results to the case of nonsmooth convex functions. In this case, the algorithm involves the proximal operators of f and $\phi$ separately. Several variants of this algorithm are considered, including the case of the Nesterov accelerated gradient method. We then consider the extension in the case of additive composite optimization, thus leading to new splitting methods. Numerical experiments are given for Lasso-type problems. The performance profiles, as a comparison tool, demonstrate the efficiency of the Nesterov accelerated method with asymptotic vanishing damping combined with dry friction.

Auxiliary results

1.1 Finite-time convergence of the continuous dynamic

Theorem 7

Let $f:{\mathcal H}\rightarrow {\mathbb {R}}$ be a $\mathcal C^1$ function whose gradient is Lipschitz continuous, and let $\phi : {\mathcal H}\rightarrow {\mathbb {R}}$ be a convex continuous function that satisfies (DF). Suppose that the function $\gamma : [t_0, +\infty [ \rightarrow {\mathbb {R}}_+$ belongs to $L^1 ([t_0, T])$ for any $T>t_0$. Then, the following properties hold:

a) For any Cauchy data $(x_0, \dot{x}_0 ) \in {\mathcal H}\times {\mathcal H}$, there exists a unique strong global solution of the Heavy Ball system with Dry Friction

$$\begin{aligned} \mathrm{(HBDF)} \qquad \ddot{x}(t) + \gamma (t)\dot{x}(t) + \partial \phi (\dot{x}(t)) + \nabla f (x(t)) \ni 0, \end{aligned}$$

(65)

satisfying $x(t_0) = x_0$, and $\dot{x}(t_0)=\dot{x}_0$.

b) For any solution trajectory x of $\mathrm{(HBDF)}$ we have:

(i)  $\Vert \dot{x}\Vert \in L^1([t_0,+\infty [,{\mathbb {R}})$, and therefore $x_\infty :=$ $\lim _{t\rightarrow +\infty } x(t)$ exists.

(ii)   The limit point $x_\infty$ is an equilibrium point of $\mathrm{(HBDF)}$, i.e.  

$$\begin{aligned} -\nabla f(x_\infty )\in \partial \phi (0). \end{aligned}$$

(66)

(iii)   If

$$\begin{aligned} -\nabla f(x_\infty )\not \in \text{ boundary }(\partial \phi (0)),\end{aligned}$$

then there exists $t_1\ge 0$ such that $x(t)=x_\infty$ for every $t\ge t_1$.

Proof

An existence proof based on a regularization technique, by using the Moreau-Yosida approximation of $\phi$, was given in [2] in a finite dimensional setting. We present here an original proof of the existence and uniqueness part a) of Theorem 7, in a general Hilbert space, which is based on the study of evolution equations governed by the Lipschitz perturbation of maximally monotone operators (see [21]). It is used in an essential way that $\nabla f$ is Lipschitz continuous over the entire space ${\mathcal H}$.

Write $\mathrm{(HBDF)}$ as

$$\begin{aligned} \ddot{x}(t) + \gamma (t)\dot{x}(t)+ \partial \phi (\dot{x}(t)) \ni -\nabla f \left( x_0 + \int _{t_0}^t \dot{x}(\tau )d\tau \right) . \end{aligned}$$

Setting $u(t):= \dot{x}(t)$, this amounts to solving the first-order evolution equation

$$\begin{aligned} \dot{u}(t) + \gamma (t) u(t)+ \partial \phi (u(t)) + \nabla f \left( x_0 + \int _{t_0}^t u(\tau )d\tau \right) \ni 0 \end{aligned}$$

with the Cauchy data $u(t_0)= \dot{x}_0$. Let us introduce the (non-local) operator

$$\begin{aligned} F(u)(t)= \nabla f \left( x_0 + \int _{t_0}^t u(\tau )d\tau \right) . \end{aligned}$$

Thus, we have to solve

$$\begin{aligned} \dot{u}(t) + \gamma (t) u(t) + \partial \phi (u(t)) + F(u)(t) \ni 0. \end{aligned}$$

(67)

For any two trajectories u and v, we have

$$\begin{aligned} \Vert F(u)(t) - F(v)(t) \Vert \le L \int _{t_0}^t \Vert u(\tau )- v(\tau )\Vert d\tau , \end{aligned}$$

where L is the Lipschitz constant of $\nabla f$. Following the approach developed in [21, Proposition 3.12, page 106], we consider the sequence $(u_n)$ defined recursively by

$$\begin{aligned} \dot{u}_{n+1}(t)+ \gamma (t)u_{n+1}(t) + \partial \phi (u_{n+1}(t)) + F(u_n)(t) \ni 0. \end{aligned}$$

(68)

Given $u_n$, the existence and uniqueness of $u_{n+1}$ solution of (68 )with $u_{n+1}(0) = \dot{x}_0$ is ensured by the classical results concerning the evolution equations governed by subdifferentials of convex functions (see [21, Theorem 3.6, page 72], [7, Theorem 17.2.5]). Let’s give $T >t_0$. According to the above Lipschitz continuity property of F, the monotonicity of $\partial \phi$, and $\gamma (t) \ge 0$, we have for all $0 \le t \le T$

$$\begin{aligned} \Vert u_{n+1}(t) - u_{n}(t)\Vert \le L(t-t_0) \int _{t_0}^t \Vert u_{n}(\tau ) - u_{n-1}(\tau )\Vert d\tau , \end{aligned}$$

which gives

$$\begin{aligned} \Vert u_{n+1}(t) - u_{n}(t)\Vert \le \frac{(L (t-t_0)^n}{n!}t^2 \Vert u_1-u_0 \Vert _{L^{\infty }(t_0, T)}. \end{aligned}$$

This implies that $(u_n)$ is a Cauchy sequence for the uniform convergence on $[t_0,T]$. Consequently, it converges uniformly on $[t_0,T]$ to a solution u of (67). So, this uniquely defines $u=\dot{x}$, and at the same time x which is given by to $x(t)= x_0 + \int _{t_0}^t u(\tau )d\tau$.

For part b), we refer to [2, Theorem 3.2 ].

Remark 8

With the condition $-\nabla f(x_\infty )\not \in \text{ boundary }(\partial \phi (0)),$ the finite-time convergence of the trajectory to a stationary point of the dynamic (HBDF) is ensured, i.e. there exists $t_1\ge 0$ such that $x(t)=x_\infty$ for every $t\ge t_1$. In addition, an estimate of the final time could be given. In fact, we can show, by integrating the differential inequality satisfied by $\alpha (t)= \Vert \dot{x}(t)\Vert ^2$

$$\begin{aligned} \dot{\alpha }(t) + 2\epsilon h \sqrt{\alpha (t)}\le 0,\;t\in [0,+\infty [ \end{aligned}$$

(69)

that

$$\begin{aligned} t_1\le t_0 +\frac{2\Vert \dot{x}(t_0)\Vert }{\mathrm{dist}\Big (-\nabla f(x_\infty ), \mathrm{boundary}(\partial \phi (0) \Big )},\end{aligned}$$

where $t_0$ is the first time instant such that

$$\begin{aligned}&\nabla f(x(t))\in \nabla f(x_\infty )+B(0,\varepsilon ),\hbox { for all } t\ge t_0,\hbox { with }\varepsilon \\&\quad =\frac{1}{2} \mathrm{dist}\Big (-\nabla f(x_\infty ), \mathrm{boundary}(\partial \phi (0) \Big ). \end{aligned}$$

We refer to [3] for Lagrangian systems.

Remark 9

The conclusions of Theorem 7 are valid under the key assumption $-\nabla f(x_\infty ) \not \in \text{ boundary }(\partial \phi (0))$. Since the boundary of the convex set $\partial \phi (0)$ has an empty interior, it is reasonable to think that the circumstances leading to the relation $-\nabla f(x_\infty )\in \text{ boundary }(\partial \phi (0))$ are “exceptional”. More precisely, we conjecture that generically with respect to the initial data $(x_0,\dot{x}_0)\in {\mathbb {R}}^n \times {\mathbb {R}}^n$, the point $x_\infty =\lim _{t\rightarrow +\infty }x(t)$ satisfies the condition $-\nabla f(x_\infty ) \not \in \text{ boundary }(\partial \phi (0))$. Consequently, this would give a generic finite-time stabilization result in the case of dry friction.

Let us give a counter-example to convergence in finite-time when the condition $-\nabla f(x_\infty ) \not \in \text{ boundary }(\partial \phi (0))$ is not satisfied, i.e.  $-\nabla f(x_\infty ) \in \text{ boundary }(\partial \phi (0))$. For that purpose, take ${\mathcal H}={\mathbb {R}}$, $\phi :=|\,.\,|$ (so that $\partial \phi (0)=[-1,1]$), $\gamma =2$ and $f:=|\,.\,|^2/2$. The differential inclusion $\mathrm{(HBDF)}$ then reads

$$\begin{aligned} \ddot{x}(t)+ \text{ sign }(\dot{x}(t))+2\, \dot{x}(t)+x(t)\ni 0. \end{aligned}$$

Let us choose as initial conditions $x(0)=-2$ and $\dot{x}(0)=1$. The unique solution of $\mathrm{(HBDF)}$ is given by $x(t)=-1-e^{-t}$, $t\ge 0$. The trajectory tends toward the value $x_\infty =-1$, which satisfies $-f'(x_\infty )=1 \in \text{ boundary }(\partial \phi (0))$. However the convergence does not hold in a finite-time.

Remark 10

It is natural to know if convergence in finite-time is specific to the dry friction situation $0\in \mathrm{int}(\partial \phi (0))$. To answer this question Amann-Diaz [5] and Diaz-Linan [24] considered the damped oscillator in ${\mathcal H}= {\mathbb {R}}$

$$\begin{aligned} \ddot{x}(t)+ |\dot{x}(t)|^{\alpha -1} \dot{x}(t)) +x(t)= 0, \end{aligned}$$

where $\alpha \in ]0,1[$. This corresponds to a sub-linear friction, the case of dry friction corresponds to the limiting case $\alpha =0$. They have shown the existence of two curves in the phase space such that, for the solution trajectories with initial data $(x_0, \dot{x}_0)$ belonging to these two curves, there is finite-time stabilization at the origin. Using both energetic and geometrical arguments, they showed that for many other initial data, the solution tends to zero in infinite time, at the rate $t^{-\frac{\alpha }{1-\alpha }}$.