An extension of the second order dynamical system that models Nesterov’s convex gradient method

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Abstract

In this paper we deal with a general second order continuous dynamical system associated to a convex minimization problem with a Frechet differentiable objective function.

We show that inertial algorithms, such as Nesterov’s algorithm, can be obtained via the natural explicit discretization from our dynamical system.

Our dynamical system can be viewed as a perturbed version of the heavy ball method with vanishing damping, however the perturbation is made in the argument of the gradient of the objective function.

This perturbation seems to have a smoothing effect for the energy error and eliminates the oscillations obtained for this error in the case of the heavy ball method with vanishing damping, as some numerical experiments show.

We prove that the value of the objective function in a generated trajectory converges in order $\mathcal{O}(1/t^2)$ to the global minimum of the objective function.

Moreover, we obtain that a trajectory generated by the dynamical system converges to a minimum point of the objective function.

Authors

Cristian Daniel Alecsa
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania

Szilard Csaba Laszlo
Technical University of Cluj-Napoca, Department of Mathematics, Cluj-Napoca, Romania

Titus Pinta
Mathematical Institute University of Oxford, Oxford, England

Keywords

Unconstrained optimization problems; convex optimization; heavy ball method; continuous second order dynamical system; convergence rate; inertial algorithm; Lyapunov energy functions.

Paper coordinates

C.-D. Alecsa, S.C. Laszlo, T. Pinta, An extension of the second order dynamical system that models Nesterov’s convex gradient method, Appl. Math. Optim., 84 (2021), pp. 1687–1716. doi: 10.1007/s00245-020-09692-1

PDF

https://arxiv.org/pdf/1908.02574.pdf

About this paper

Journal

Appl. Math. Optim.

Publisher Name

Springer

DOI

10.1007/s00245-020-09692-1

Print ISSN

0095-4616

Online ISSN

1432-0606

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Paper (preprint) in HTML form

An extension of the second order dynamical system that models Nesterov’s convex gradient method^†^†thanks: This work was supported by a grant of Ministry of Research and Innovation, CNCS - UEFISCDI, project number PN-III-P1-1.1-TE-2016-0266.

Cristian Daniel Alecsa Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania and Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania, e-mail: cristian.alecsa@math.ubbcluj.ro Szilárd Csaba László Technical University of Cluj-Napoca, Department of Mathematics, Str. Memorandumului nr. 28, 400114 Cluj-Napoca, Romania, e-mail: laszlosziszi@yahoo.com Titus Pinţa Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania, e-mail: titus.pinta@gmail.com

Abstract. In this paper we deal with a general second order continuous dynamical system associated to a convex minimization problem with a Frèchet differentiable objective function. We show that inertial algorithms, such as Nesterov’s algorithm, can be obtained via the natural explicit discretization from our dynamical system. Our dynamical system can be viewed as a perturbed version of the heavy ball method with vanishing damping, however the perturbation is made in the argument of the gradient of the objective function. This perturbation seems to have a smoothing effect for the energy error and eliminates the oscillations obtained for this error in the case of the heavy ball method with vanishing damping, as some numerical experiments show. We prove that the value of the objective function in a generated trajectory converges in order $\mathcal{O}(1/t^{2})$ to the global minimum of the objective function. Moreover, we obtain that a trajectory generated by the dynamical system converges to a minimum point of the objective function.

Key Words. convex optimization, heavy ball method, continuous second order dynamical system, convergence rate, inertial algorithm

AMS subject classification. 34G20, 47J25, 90C25, 90C30, 65K10

1 Introduction

Since Su, Boyd and Candès in [33] showed that Nesterov’s accelerated convex gradient method has the exact limit the second order differential equation that governs the heavy ball system with vanishing damping, that is,

\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+{\nabla}g(x(t))=0,\,\,x(t_{0})=u_{0},\,\dot{x}(t_{0})=v_{0},\,t_{0}>0,\,u_{0},v_{0}\in\mathbb{R}^{m},

(1)

with $\alpha=3$ , the latter system has been intensively studied in the literature in connection to the minimization problem $\inf_{x\in\mathbb{R}^{m}}g(x).$ Here $g:\mathbb{R}^{m}\longrightarrow\mathbb{R}$ is a convex Frèchet differentiable function with Lipschitz continuous gradient.

In [33] the authors proved that

g(x(t))-\min g=\mathcal{O}\left(\frac{1}{t^{2}}\right)

for every $\alpha\geq 3,$ however they did not show the convergence of a generated trajectory to a minimum of the objective function $g.$

In [10], Attouch, Chbani, Peypouquet and Redont considered the case $\alpha>3$ in (1), and showed that the generated trajectory $x(t)$ converges to a minimizer of $g$ as $t\longrightarrow+\infty$ . Actually in [10] the authors considered the perturbed version of the heavy ball system with vanishing damping, that is,

\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+{\nabla}g(x(t))=h(t),\,\,x(t_{0})=u_{0},\,\dot{x}(t_{0})=v_{0},\,t_{0}>0,\,u_{0},v_{0}\in\mathbb{R}^{m},

(2)

where $h:[t_{0},+\infty)\longrightarrow\mathbb{R}$ is a small perturbation therm that satisfies $\int_{t_{0}}^{+\infty}t\|h(t)\|dt<+\infty.$ Beside the convergence of a generated trajectory $x(t)$ to a minimizer of $g$ , they showed that also in this case the convergence rate of the objective function along the trajectory, that is $g(x(t))-\min g,$ is of order $\mathcal{O}\left(\frac{1}{t^{2}}\right)$ .

Another perturbed version of (1) was studied by Attouch, Peypouquet and Redont in [12]. They assumed that the objective $g$ is twice continuously differentiable and the perturbation of their system is made at the damping therm. More precisely, they studied the dynamical system with Hessian driven damping

\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+\beta{\nabla}^{2}g(x(t))\dot{x}(t)+{\nabla}g(x(t))=0,\,\,x(t_{0})=u_{0},\,\dot{x}(t_{0})=v_{0},\,t_{0}>0,\,u_{0},v_{0}\in\mathbb{R}^{m},

(3)

where $\alpha>0$ and $\beta>0.$ In case $\alpha>3,\,\beta>0$ they showed the convergence of a generated trajectory to a minimizer of $g$ . Moreover, they obtained that in this case the convergence rate of the objective function along the trajectory, that is $g(x(t))-\min g,$ is of order ${o}\left(\frac{1}{t^{2}}\right)$ .

Further, Attouch, Chbani and Riahi in [8] studied the subcritical case $\alpha\leq 3$ and they proved that in this case the convergence rates of the objective function $g$ along the trajectory generated by (1), i.e $g(x(t))-\min g,$ is of order $\mathcal{O}\left(\frac{1}{t^{\frac{2}{3}\alpha}}\right)$ .

Another approach is due to Aujol, Dossal and Rondepierre [2], who assumed that beside convexity, the objective $g$ in (1) satisfies some geometrical conditions, such as the Łojasiewicz property. The importance of their results obtained in [2] is underlined by the fact that applying the classical Nesterov scheme on a convex objective function without studying its geometrical properties may lead to sub-optimal algorithms.

It is worth mentioning the work of Aujol and Dossal [1], who did not assumed the convexity of $g,$ but the convexity of the function $(g(x(t))-g(x^{*}))^{\beta}$ , where $\beta$ is strongly related to the damping parameter $\alpha$ and $x^{*}$ is a global minimizer of $g.$ Under these assumptions, they obtained some general convergence rates and also the convergence of the generated trajectories of (1). In case $\beta=1$ they results reduce to the results obtained in [10, 12, 8].

However, the convergence of the trajectories generated by the continuous heavy ball system with vanishing damping (1), in the general case when $g$ is nonconvex is still an open question. Some important steps in this direction have been made in [20] (see also [18]), where convergence of the trajectories of a perturbed system, have been obtained in a nonconvex setting. More precisely in [20] is considered the system

\ddot{x}(t)+\left(\gamma+\frac{\alpha}{t}\right)\dot{x}(t)+{\nabla}g(x(t))=0,\,x(t_{0})=u_{0},\,\dot{x}(t_{0})=v_{0},

(4)

where $t_{0}>0,u_{0},v_{0}\in\mathbb{R}^{m},\,\gamma>0,\,\alpha\in\mathbb{R}.$ Note that here $\alpha$ can take nonpositive values. For $\alpha=0$ we recover the dynamical system studied in [16]. According to [20], the trajectory generated by the dynamical system (4) converges to a critical point of $g,$ if a regularization of $g$ satisfies the Kurdyka-Łojasiewicz property.

Further results concerning the heavy ball method and its extensions can be found in [6, 7, 11, 14, 22, 23].

1.1 An extension of the heavy ball method and the Nesterov type algorithms obtained via explicit discretization

What one can notice concerning the heavy ball system and its variants is, that despite of the result of Su et al. [33], this system will never give through the natural implicit/explicit discretization the Nesterov algorithm. This is due to the fact that the gradient of $g$ is evaluated in $x(t),$ and this via discretization will become $g(x_{n})$ , (or $g(x_{n+1})$ ) and never of the form $g(y_{n}),\,y_{n}=x_{n}+\alpha_{n}(x_{n}-x_{n-1})$ as Nesterov’s gradient method requires. Another observation is that using the same approach as in [33], one can show, see [26], that (1), (and also (4)), models beside Nesterov’s algorithm other algorithms too. In this paper we overcome the deficiency emphasized above, by introducing a dynamical system that via explicit discretization leads to inertial algorithms of gradient type. To this end, let us consider the optimization problem

(P)\ \inf_{x\in\mathbb{R}^{m}}g(x)

(5)

where $g:\mathbb{R}^{m}\longrightarrow\mathbb{R}$ is a convex Fréchet differentiable, function with $L_{g}$ -Lipschitz continuous gradient, i.e. there exists $L_{g}\geq 0$ such that $\|{\nabla}g(x)-{\nabla}g(y)\|\leq L_{g}\|x-y\|$ for all $x,y\in\mathbb{R}^{n}.$

We associate to (5) the following second order dynamical system:

\left\{\begin{array}[]{ll}\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+\nabla g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)=0\\ x(t_{0})=u_{0},\,\dot{x}(t_{0})=v_{0},\end{array}\right.

(6)

where $u_{0},v_{0}\in\mathbb{R}^{m},$ $t_{0}\geq 0$ and $\alpha>0,\,\beta\in\mathbb{R},\,\gamma\geq 0.$

Remark 1

The connection of (6) with the heavy ball system with vanishing damping (1) is obvious, the latter one can be obtained from (6) for $\gamma=\beta=0.$ The study of the dynamical system (6) in connection to the optimization problem (5) is motivated by the following facts:

1. The dynamical system (6) leads via explicit discretization to inertial algorithms. In particular Nesterov’s algorithm can be obtained via this natural discretization.

2. A generated trajectory and the objective function value in this trajectory in general have a better convergence behaviour than a trajectory generated by the heavy ball system with vanishing damping (1), as some numerical examples shows.

3. The same numerical experiments reveal that the perturbation term $\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)$ in the argument of the gradient of the objective function $g$ has a smoothing effect and annihilates the oscillations obtained in case of the dynamical system (1) for the errors $g(x(t)-\min g$ and $\|x(t)-x^{*}\|,$ where $x^{*}$ is a minimizer of $g.$

4. A trajectory $x(t)$ generated by the dynamical system (6) ensures the convergence rate of order $\mathcal{O}\left(\frac{1}{t^{2}}\right)$ for the decay $g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)-\min g$ , provided it holds that $\alpha>3,\,\gamma>0,\,\beta\in\mathbb{R}$ or $\alpha\geq 3,\,g=0,\,\beta\geq 0.$

5. A trajectory $x(t)$ generated by the dynamical system (6) ensures the same convergence rate of order $\mathcal{O}\left(\frac{1}{t^{2}}\right)$ for the decay $g(x(t))-\min g$ as the heavy ball method with vanishing damping, for the cases $\alpha>3,\,\gamma>0,\,\beta\in\mathbb{R}$ and $\alpha>3,\,\gamma=0,\,\beta\geq 0.$

6. The convergence of a generated trajectory $x(t)$ to a minimizer of the objective function $g$ can be obtained in case $\alpha>3,\,\gamma>0$ and $\beta\in\mathbb{R}$ and also in the case $\alpha>3,\,\gamma=0$ and $\beta\geq 0.$

Remark 2

Nevertheless, in case $\gamma=0$ and $\beta<0$ the dynamical system (6) can generate periodical solutions, hence the convergence of a generated trajectory to a minimizer of the objective is hopeless. To illustrate this fact, for $\beta<0,\,\alpha>0,\,\gamma=0$ consider the strongly convex objective function $g:\mathbb{R}\longrightarrow\mathbb{R},$ $g(x)=\frac{\alpha}{-2\beta}x^{2}$ . Then, taking into account that $\gamma=0$ the dynamical system (6) becomes

\left\{\begin{array}[]{ll}\ddot{x}(t)+\frac{\alpha}{-\beta}x(t)=0,\\ x(0)=0,\,\dot{x}(0)=\sqrt{\frac{\alpha}{-\beta}}.\end{array}\right.

(7)

Now, the periodical function $x(t)=\sin\sqrt{\frac{\alpha}{-\beta}}t$ is a solution of (7), consequently do not exist the limit $\lim_{t\longrightarrow+\infty}x(t).$

We emphasize, that despite the novelty of the dynamical system (6), its formulation is natural since by explicit discretization leads to inertial gradient methods, in particular the famous Polyak and Nesterov numerical schemes can be obtained from (6). For other inertial algorithms of gradient type we refer to [17, 19, 26].

Indeed, explicit discretization of (6), with the constant stepsize $h$ , $t_{n}=nh,\,\,x_{n}=x(t_{n})$ leads to

\frac{x_{n+1}-2x_{n}+x_{n-1}}{h^{2}}+\frac{\alpha}{nh^{2}}(x_{n}-x_{n-1})+{\nabla}g\left(x_{n}+\left(\frac{\gamma}{h}+\frac{\beta}{nh^{2}}\right)(x_{n}-x_{n-1})\right)=0.

Equivalently, the latter equation can be written as

x_{n+1}=x_{n}+\left(1-\frac{\alpha}{n}\right)(x_{n}-x_{n-1})-h^{2}{\nabla}g\left(x_{n}+\left(\frac{\gamma}{h}+\frac{\beta}{nh^{2}}\right)(x_{n}-x_{n-1})\right).

(8)

Now, setting $h^{2}=s$ and denoting the constants $\frac{\gamma}{h}$ and $\frac{\beta}{h^{2}}$ still with $\gamma$ and $\beta$ , we get the following general inertial algorithm:

Let $x_{0},x_{-1}\in\mathbb{R}^{m}$ and for all $n\in\mathbb{N}$ consider the sequences

\left\{\begin{array}[]{lll}y_{n}=x_{n}+\left(1-\frac{\alpha}{n}\right)(x_{n}-x_{n-1})\\ z_{n}=x_{n}+\left({\gamma}+\frac{\beta}{n}\right)(x_{n}-x_{n-1})\\ x_{n+1}=y_{n}-s{\nabla}g\left(z_{n}\right).\end{array}\right.

(9)

However, from a practical point of view it is more convenient to work with the following equivalent formulation: Let $x_{0},x_{-1}\in\mathbb{R}^{m}$ and for all $n\in\mathbb{N}$ consider the sequences

\left\{\begin{array}[]{lll}y_{n}=x_{n}+\frac{n}{n+\alpha}(x_{n}-x_{n-1})\\ \\ z_{n}=x_{n}+\frac{\gamma n+\beta}{n+\alpha}(x_{n}-x_{n-1})\\ \\ x_{n+1}=y_{n}-s{\nabla}g\left(z_{n}\right),\end{array}\right.

(10)

where $\alpha>0,\,\beta\in\mathbb{R}$ and $\gamma\geq 0.$

Remark 3

Notice that for $\gamma=\beta=0$ we obtain a variant of Polyak’s algorithm [31] and for $\gamma=1$ and $\beta=0$ we obtain Nesterov’s algorithm [29]. An interesting fact is that Algorithm (10) allows different inertial steps and this approach seems to be new in the literature.

Remark 4

Independently to us, very recently, a system similar to (6) was studied by Muehlebach and Jordan in [27] and they show that Nesterov’s accelerated gradient method can be obtained from their system via a semi-implicit Euler discretization scheme. They considered a constant damping instead of $\frac{\alpha}{t}$ and also took $\beta=0.$ Further they also treated the ODE

\ddot{x}(t)+\frac{3}{t+2}\dot{x}(t)+s\nabla g\left(x(t)+\frac{t-1}{t+2}\dot{x}(t)\right)=0

which for $s=1$ is obviously equivalent to the particular case of the governing ODE from (6), obtained for $\alpha=3,\,\gamma=1,\,\beta=-3.$ However, the freedom of controlling the parameters $\beta$ and $\gamma$ in (6) is essential as the next numerical experiments show.

1.2 Some numerical experiments

In this section we consider two numerical experiments for the trajectories generated by the dynamical system (6) for a strongly convex and for a convex but not strongly convex objective function.

Everywhere in the following numerical experiments we consider the continuous time dynamical system (6), solved numerically with a Runge Kutta 4-5 (ode45) adaptive method in MATLAB. We solved the dynamical system with ode45 on the interval $[1,100]$ and the plot in Fig. 1 - Fig. 2 show the energy error $|g(x(t))-g(x^{*})|$ on the left, and the iterate error $\|x(t)-x^{*}\|$ on the right.

We show the evolution of the two errors with respect to different values for $\alpha$ , $\beta$ and $\gamma$ , including the case that yields the Heavy Ball with Friction. One can observe that the best choice is not $\gamma=\beta=0$ which is the case of heavy ball system with vanishing damping (1).

1. Consider the function $g:\mathbb{R}^{2}\longrightarrow\mathbb{R},$

g(x,y)=2x^{2}+5y^{2}-4x+10y+7.

Then, $g$ is a strongly convex function, ${\nabla}g(x,y)=(4x-4,10y+10)$ , further $x^{*}=(1,-1)$ is the unique minimizer of $g$ and $g^{*}=g(1,-1)=0.$

We compare the convergence behaviour of the generated trajectories of (6) by taking into account the following instances.

$\alpha$	$\beta$	$\gamma$	Color
3	0	0	blue
3.1	1	0	red
3.1	0	0.5	yellow
3.1	1	1	purple
3.1	-1	1	green

The result are depicted in Figure.1 for the starting points $u_{0}=v_{0}=(-5,30)$ and $u_{0}=v_{0}=(5,-30)$ , respectively.

Refer to caption — (a) $u_{0}=v_{0}=(-5,30).$

2. In the next experiment we consider the convex, but not strongly convex function $g:\mathbb{R}^{2}\longrightarrow\mathbb{R},$

g(x,y)=x^{4}+5y^{2}-4x-10y+8.

Then, ${\nabla}g(x,y)=(4x^{3}-4,10y-10)$ , further $x^{*}=(1,1)$ is the unique minimizer of $g$ and $g^{*}=g(1,1)=0.$

We compare the convergence behaviour of the generated trajectories of (6) by taking into account the following instances.

$\alpha$	$\beta$	$\gamma$	Color
3.1	0	0	blue
3.1	2	0	red
3.1	0	1	yellow
3.1	0.5	0.5	purple
3.1	-0.5	1	green

The result are depicted in Figure.2 for the starting points $u_{0}=v_{0}=(-1,5)$ and $u_{0}=v_{0}=(2,-2)$ , respectively.

Remark 5

Observe that in all cases the trajectories generated by dynamical system (6) have a better behaviour than the trajectories generated by the heavy ball system with vanishing damping (1). As we have emphasized before, it seems that the perturbation $\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)$ in the argument of the gradient of the objective function has a smoothing effect. The choice of the parameters $\gamma$ and $\beta$ will be validated by the theoretical results from Section 3, where we show that in case $\alpha>3,\,\gamma>0,\,\beta\in\mathbb{R}$ and also in the case $\alpha>3,\,\gamma=0,\,\beta\geq 0$ the energy error $g(x(t)-\min g$ is of order $\mathcal{O}\left(\frac{1}{t^{2}}\right)$ just as the case of heavy ball method. Further, for the values $\alpha>3,\,\gamma>0,\,\beta\in\mathbb{R}$ and for $\alpha>0,\,\gamma=0,\,\beta\geq 0$ we are able to show that a generated trajectory $x(t)$ converges to a minimum of the objective function $g.$

1.3 The organization of the paper

The outline of the paper is the following. In the next section we show the existence and uniqueness of the trajectories generated by the system (6). Further we show that the third order derivative exists almost for every $t\geq t_{0}$ and we give some estimates of the third order derivative in terms of velocity and acceleration. We do not assume the convexity of the objective function $g$ in these results. However, it seems that the assumption that $g$ has Lipschitz continuous gradient is essential in obtaining existence and uniqueness of the generated trajectories of the dynamical system (6). In Section 3 we deal with the convergence analysis of the generated trajectories. We introduce a general energy functional, which will play the role of a Lyapunov function associated to the dynamical system (6). We show convergence of the generated trajectories and also a rate of order $\mathcal{O}(1/t^{2})$ for the decay $g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)-\min g.$ Further, we show that also the error $g(x(t))-\min g$ has a rate of order $\mathcal{O}(1/t^{2})$ . Finally, we show the convergence of the generated trajectories to a minimum of the objective function $g.$ In section 4 we conclude our paper and we present some possible related future researches.

2 Existence and uniqueness

2.1 On strong global solutions

The first step toward our existence and uniqueness result obtained in the present section concerns the definition of a strong global solution of the dynamical system (6).

Definition 1

We call the function $x:[t_{0},+\infty)\to\mathbb{R}^{m}$ a strong global solution of the dynamical system (6) if satisfies the following properties:

	$\displaystyle(i)\quad x,\dot{x}:[t_{0},+\infty)\to\mathbb{R}^{m}\text{ are locally absolutely continuous};$
	$\displaystyle(ii)\quad\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+\nabla g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)=0\text{ for almost every }t\geq t_{0};$
	$\displaystyle(iii)\quad x(t_{0})=u_{0}\text{ and }\dot{x}(t_{0})=v_{0}.$

For brevity reasons, we recall that a mapping $x:[t_{0},+\infty)\to\mathbb{R}^{m}$ is called locally absolutely continuous if it is absolutely continuous on every compact interval $[t_{0},T]$ , where $T>t_{0}$ . Further, we have the following equivalent characterizations for an absolutely continuous function $x:[t_{0},T]\longrightarrow\mathbb{R}^{m}$ , (see, for instance, [13, 5]):

(a)

there exists $y:[t_{0},T]\to\mathbb{R}^{m}$ an integrable function, such that

$x(t)=x(t_{0})+\int\limits_{t_{0}}^{t}y(t)ds,\forall t\in[t_{0},T];$
(b)

$x$ is a continuous function and its distributional derivative is Lebesque integrable on the interval $[t_{0},T];$

(c)

for every $\varepsilon>0$ , there exists $\eta>0$ , such that for every finite family $I_{k}=(a_{k},b_{k})$ from $[t_{0},T]$ , the following implication is valid :

\displaystyle\left[I_{k}\cap I_{j}=\emptyset\text{ and }\sum\limits_{k}|b_{k}-a_{k}|<\eta\right]\Rightarrow\left[\sum\limits_{k}\|x(b_{k})-x(a_{k})\|<\varepsilon\right].

Remark 6

Let $x:[t_{0},+\infty)\to\mathbb{R}^{m}$ be a locally absolutely continuous function. Then $x$ is differentiable almost everywhere and its derivative coincides with its distributional derivative almost everywhere. On the other hand, we have the equality $\dot{x}(t)=y(t)$ for almost every $t\in[t_{0},+\infty)$ , where $y=y(t)$ is defined at the integration formula (a).

The first result of the present section concerns the existence and uniqueness of the trajectories generated by the dynamical system (6). We prove existence and uniqueness of a strong global solution of (6) by making use of the Cauchy-Lipschitz-Picard Theorem for absolutely continues trajectories (see for example [25, Proposition 6.2.1], [32, Theorem 54]). The key argument is that one can rewrite (6) as a particular first order dynamical system in a suitably chosen product space (see also [6, 9, 20, 21]).

Theorem 7

Let $(u_{0},v_{0})\in\mathbb{R}^{m}\times\mathbb{R}^{m}$ . Then, the dynamical system (6) admits a unique strong global solution.

Proof.

By making use of the notation $X(t)=(x(t),\dot{x}(t))$ the system (6) can be rewritten as a first order dynamical system:

\left\{\begin{array}[]{ll}\dot{X}(t)=F(t,X(t))\\ X(t_{0})=(u_{0},v_{0}),\end{array}\right.

(11)

where $F:[t_{0},+\infty)\times\mathbb{R}^{m}\times\mathbb{R}^{m}\longrightarrow\mathbb{R}^{m}\times\mathbb{R}^{m},\,F(t,u,v)=\left(v,-\frac{\alpha}{t}v-\nabla g\left(u+\left(\gamma+\frac{\beta}{t}\right)v\right)\right).$

The existence and uniqueness of a strong global solution of (6) follows according to the Cauchy-Lipschitz-Picard Theorem applied to the first order dynamical system (11). In order to prove the existence and uniqueness of the trajectories generated by (11) we show the following:

(I) For every $t\in[t_{0},+\infty)$ the mapping $F(t,\cdot,\cdot)$ is $L(t)$ -Lipschitz continuous and $L(\cdot)\in L^{1}_{loc}([t_{0},+\infty))$ .

(II) For all $u,v\in{\mathbb{R}^{m}}$ one has $F(\cdot,u,v)\in L^{1}_{loc}([t_{0},+\infty),{\mathbb{R}^{m}}\times{\mathbb{R}^{m}})$ .

Let us prove (I). Let $t\in[t_{0},+\infty)$ be fixed and consider the pairs $(u,v)$ and $(\bar{u},\bar{v})$ from $\mathbb{R}^{m}\times\mathbb{R}^{m}$ . Using the Lipschitz continuity of ${\nabla}g$ and the obvious inequality $\|A+B\|^{2}\leq 2\|A\|^{2}+2\|B\|^{2}$ for all $A,B\in\mathbb{R}^{m}$ , we make the following estimations :

	$\displaystyle\\|F(t,u,v)-F(t,\overline{u},\overline{v})\\|$	$\displaystyle=\sqrt{\\|v-\overline{v}\\|^{2}+\left\\|\frac{\alpha}{t}(\overline{v}-v)+\nabla g\left(\overline{u}+\left(\gamma+\frac{\beta}{t}\right)\overline{v}\right)-\nabla g\left(u+\left(\gamma+\frac{\beta}{t}\right)v\right)\right\\|^{2}}$
		$\displaystyle\leq\sqrt{\left(1+2\left(\frac{\alpha}{t}\right)^{2}\right)\\|v-\overline{v}\\|^{2}+2L_{g}^{2}\left\\|(u-\overline{u})+\left(\gamma+\frac{\beta}{t}\right)(v-\overline{v})\right\\|^{2}}$
		$\displaystyle\leq\sqrt{\left(1+2\left(\frac{\alpha}{t}\right)^{2}+4L_{g}^{2}\left(\gamma+\frac{\beta}{t}\right)^{2}\right)\\|v-\overline{v}\\|^{2}+4L_{g}^{2}\left\\|u-\overline{u}\right\\|^{2}}$
		$\displaystyle\leq\sqrt{1+4L_{g}^{2}+2\left(\frac{\alpha}{t}\right)^{2}+4L_{g}^{2}\left(\gamma+\frac{\beta}{t}\right)^{2}}\sqrt{\\|v-\overline{v}\\|^{2}+\\|u-\overline{u}\\|^{2}}$
		$\displaystyle=\sqrt{1+4L_{g}^{2}+2\left(\frac{\alpha}{t}\right)^{2}+4L_{g}^{2}\left(\gamma+\frac{\beta}{t}\right)^{2}}\,\\|(u,v)-(\overline{u},\overline{v})\\|.$

By employing the notation $L(t)=\sqrt{1+4L_{g}^{2}+2\left(\frac{\alpha}{t}\right)^{2}+4L_{g}^{2}\left(\gamma+\frac{\beta}{t}\right)^{2}}$ , we have that

\|F(t,u,v)-F(t,\bar{u},\bar{v})\|\leq L(t)\cdot\|(u,v)-(\bar{u},\bar{v})\|.

Obviously the function $t\longrightarrow L(t)$ is continuous on $[t_{0},+\infty)$ , hence $L(\cdot)$ is integrable on $[t_{0},T]$ for all $t_{0}<T<+\infty$ .

For proving (II) consider $(u,v)\in\mathbb{R}^{m}\times\mathbb{R}^{m}$ a fixed pair of elements and let $T>t_{0}$ . We consider the following estimations:

	$\displaystyle\int_{t_{0}}^{T}\\|F(t,u,v)\\|dt$	$\displaystyle=\int_{t_{0}}^{T}\sqrt{\\|v\\|^{2}+\left\\|\frac{\alpha}{t}v+\nabla g\left(u+\left(\gamma+\frac{\beta}{t}\right)v\right)\right\\|^{2}}dt$
		$\displaystyle\leq\int_{t_{0}}^{T}\sqrt{\left(1+2\left(\frac{\alpha}{t}\right)^{2}\right)\\|v\\|^{2}+4\left\\|\nabla g\left(u+\left(\gamma+\frac{\beta}{t}\right)v\right)-{\nabla}g(u)\right\\|^{2}+4\\|{\nabla}g(u)\\|^{2}}dt$
		$\displaystyle\leq\int_{t_{0}}^{T}\sqrt{\left(1+2\left(\frac{\alpha}{t}\right)^{2}+4L_{g}^{2}\left(\gamma+\frac{\beta}{t}\right)^{2}\right)\\|v\\|^{2}+4\\|{\nabla}g(u)\\|^{2}}dt$
		$\displaystyle\leq\sqrt{\\|v\\|^{2}+\\|\nabla g(u)\\|^{2}}\int_{t_{0}}^{T}\sqrt{5+2\left(\frac{\alpha}{t}\right)^{2}+4L_{g}^{2}\left(\gamma+\frac{\beta}{t}\right)^{2}}dt$

and the conclusion follows by the continuity of the function $t\longrightarrow\sqrt{5+2\left(\frac{\alpha}{t}\right)^{2}+4L_{g}^{2}\left(\gamma+\frac{\beta}{t}\right)^{2}}$ on $[t_{0},T]$ .

The Cauchy-Lipschitz-Picard theorem guarantees existence and uniqueness of the trajectory of the first order dynamical system (11) and thus of the second order dynamical system (6). $\blacksquare$

Remark 8

Note that we did not use the convexity assumption imposed on $g$ in the proof of Theorem 7. However, we emphasize that according to the proof of Theorem 7, the assumption that $g$ has a Lipschitz continuous gradient is essential in order to obtain existence and uniqueness of the trajectories generated by the dynamical system (6).

2.2 On the third order derivatives

In this section we show that the third order derivative of a strong global solution of the dynamical system (6) exists almost everywhere on $[t_{0},+\infty)$ . Further we give an upper bound estimate for the third order derivative in terms of velocity and acceleration of a strong global solution of (6). For simplicity, in the proof of the following sequel we employ the $1$ -norm on $\mathbb{R}^{m}\times\mathbb{R}^{m}$ , defined as $\|(x_{1},x_{2})\|_{1}=\|x_{1}\|+\|x_{2}\|$ , for all $x_{1},x_{2}\in\mathbb{R}^{m}$ . Obviously one has

\frac{1}{\sqrt{2}}\|(x_{1},x_{2})\|_{1}\leq\|(x_{1},x_{2})\|=\sqrt{\|x_{1}\|^{2}+\|x_{2}\|^{2}}\leq\|(x_{1},x_{2})\|_{1},\mbox{ for all }x_{1},x_{2}\in\mathbb{R}^{m}.

Proposition 9

For the starting points $(u_{0},v_{0})\in\mathbb{R}^{m}\times\mathbb{R}^{m}$ let $x$ be the unique strong global solution of the dynamical system (6). Then, $\ddot{x}$ is locally absolutely continuous on $[t_{0},+\infty)$ , consequently the third order derivative $x^{(3)}$ exists almost everywhere on $[t_{0},+\infty)$ .

Proof.

We show that $\dot{X}(t)=(\dot{x}(t),\ddot{x}(t))$ is locally absolutely continuous, hence $\ddot{x}$ is also locally absolutely continuous. This implies by Remark 6 that $x^{(3)}$ exists almost everywhere on $[t_{0},+\infty)$ .

Let $T>t_{0}$ and $s,t\in[t_{0},T]$ . We consider the following chain of inequalities :

	$\displaystyle\\|\dot{X}(s)-\dot{X}(t)\\|_{1}$	$\displaystyle=\\|F(s,X(s))-F(t,X(t))\\|_{1}$
		$\displaystyle=\left\\|\left(\dot{x}(s)-\dot{x}(t),\frac{\alpha}{t}\dot{x}(t)-\frac{\alpha}{s}\dot{x}(t)+{\nabla}g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)-{\nabla}g\left(x(s)+\left(\gamma+\frac{\beta}{s}\right)\dot{x}(s)\right)\right)\right\\|_{1}$
		$\displaystyle\leq\\|\dot{x}(s)-\dot{x}(t)\\|+\left\\|\frac{\alpha}{s}\dot{x}(s)-\frac{\alpha}{t}\dot{x}(t)\right\\|$
		$\displaystyle+\left\\|{\nabla}g\left(x(s)+\left(\gamma+\frac{\beta}{s}\right)\dot{x}(s)\right)-{\nabla}g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)\right\\|$

and by using the $L_{g}$ -Lipschitz continuity of ${\nabla}g$ we obtain

	$\displaystyle\\|\dot{X}(s)-\dot{X}(t)\\|_{1}$	$\displaystyle\leq\\|\dot{x}(s)-\dot{x}(t)\\|+\left\\|\frac{\alpha}{s}\dot{x}(s)-\frac{\alpha}{t}\dot{x}(t)\right\\|+L_{g}\\|x(s)-x(t)\\|+L_{g}\left\\|\left(\gamma+\frac{\beta}{s}\right)\dot{x}(s)-\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right\\|$
		$\displaystyle\leq\\|\dot{x}(s)-\dot{x}(t)\\|+\dfrac{\alpha}{s}\\|\dot{x}(s)-\dot{x}(t)\\|+\left\|\dfrac{\alpha}{s}-\frac{\alpha}{t}\right\|\\|\dot{x}(t)\\|+L_{g}\\|x(s)-x(t)\\|$
		$\displaystyle+L_{g}\left(\gamma+\frac{\beta}{s}\right)\\|\dot{x}(s)-\dot{x}(t)\\|+L_{g}\left\|\frac{\beta}{s}-\frac{\beta}{t}\right\|\cdot\\|\dot{x}(t)\\|$
		$\displaystyle=\left(1+\dfrac{\alpha}{s}+L_{g}\left(\gamma+\frac{\beta}{s}\right)\right)\\|\dot{x}(s)-\dot{x}(t)\\|+L_{g}\\|x(s)-x(t)\\|+\left(\alpha+L_{g}\|\beta\|\right)\cdot\left\|\dfrac{1}{s}-\dfrac{1}{t}\right\|\\|\dot{x}(t)\\|.$

Further, let us introduce the following additional notations:

L_{1}:=\max\limits_{s\in[t_{0},T]}\left(1+\dfrac{\alpha}{s}+L_{g}\left(\gamma+\frac{\beta}{s}\right)\right)=1+\dfrac{\alpha}{t_{0}}+L_{g}\left(\gamma+\frac{\beta}{t_{0}}\right)\mbox{ and }L_{2}:=\left(\alpha+L_{g}|\beta|\right)\max\limits_{t\in[t_{0},T]}\|\dot{x}(t)\|.

Then, one has

\displaystyle\|\dot{X}(s)-\dot{X}(t)\|\leq\|\dot{X}(s)-\dot{X}(t)\|_{1}\leq L_{1}\|\dot{x}(s)-\dot{x}(t)\|+L_{g}\|x(s)-x(t)\|+L_{2}\left|\dfrac{1}{s}-\dfrac{1}{t}\right|.

By the fact that $x$ is the strong global solution for the dynamical system (6), it follows that $x$ and $\dot{x}$ are absolutely continuous on the interval $[t_{0},T]$ . Moreover, the function $t\longrightarrow\dfrac{1}{t}$ belongs to $C^{1}([t_{0},T],\mathbb{R})$ , hence it is also absolutely continuous on the interval $[t_{0},T]$ . Let $\varepsilon>0$ . Then, there exists $\eta>0$ , such that for $I_{k}=(a_{k},b_{k})\subseteq[t_{0},T]$ satisfying $I_{k}\cap I_{j}=\emptyset$ and $\sum\limits_{k}|b_{k}-a_{k}|<\eta$ , we have that

\sum\limits_{k}\|\dot{x}(b_{k})-\dot{x}(a_{k})\|<\dfrac{\varepsilon}{3L_{1}},\,\sum\limits_{k}\|x(b_{k})-x(a_{k})\|<\dfrac{\varepsilon}{3L_{g}}\mbox{ and }\sum\limits_{k}\Big{|}\dfrac{1}{b_{k}}-\dfrac{1}{a_{k}}\Big{|}<\dfrac{\varepsilon}{3L_{2}}.

Summing all up, we obtain

\displaystyle\sum\limits_{k}\|\dot{X}(b_{k})-\dot{X}(a_{k})\|\leq L_{1}\sum\limits_{k}\|\dot{x}(b_{k})-\dot{x}(a_{k})\|+L_{g}\sum\limits_{k}\|x(b_{k})-x(a_{k})\|+L_{2}\sum\limits_{k}\left|\dfrac{1}{b_{k}}-\dfrac{1}{a_{k}}\right|<\varepsilon,

consequently $\dot{X}$ is absolutely continuous on $[t_{0},T]$ . and the conclusion follows. $\blacksquare$

Concerning an upper bound estimate of the third order derivative $x^{(3)}$ the following result holds.

Lemma 10

For the initial values $(u_{0},v_{0})\in\mathbb{R}^{m}\times\mathbb{R}^{m}$ consider $x$ the unique strong global solution of the second-order dynamical system (6). Then, there exists $K>0$ such that for almost every $t\in[t_{0},+\infty)$ , we have that :

\|x^{(3)}(t)\|\leq K(\|\dot{x}(t)\|+\|\ddot{x}(t)\|).

(12)

Proof.

For $h>0$ we consider the following inequalities :

	$\displaystyle\\|\dot{X}(t+h)-X(t)\\|_{1}$	$\displaystyle=\\|F(t+h,X(t+h))-F(t,X(t))\\|_{1}$
		$\displaystyle\leq\left\\|\left(\dot{x}(t+h)-\dot{x}(t)\right)\right\\|+\alpha\left\\|\dfrac{1}{t+h}\dot{x}(t+h)-\dfrac{1}{t}\dot{x}(t)\right\\|$
		$\displaystyle+\left\\|\nabla g\left(x(t+h)+\left(\gamma+\dfrac{\beta}{t+h}\right)\dot{x}(t+h)\right)-\nabla g\left(x(t)+\left(\gamma+\dfrac{\beta}{t}\right)\dot{x}(t)\right)\right\\|$
		$\displaystyle\leq\left\\|\left(\dot{x}(t+h)-\dot{x}(t)\right)\right\\|+\alpha\left\\|\dfrac{1}{t+h}\dot{x}(t+h)-\dfrac{1}{t}\dot{x}(t)\right\\|$
		$\displaystyle+L_{g}\\|x(t+h)-x(t)\\|+L_{g}\left\\|\left(\gamma+\dfrac{\beta}{t+h}\right)\dot{x}(t+h)-\left(\gamma+\dfrac{\beta}{t}\right)\dot{x}(t)\right\\|.$

Now, dividing by $h>0$ and taking the limit ${h\longrightarrow 0}$ , it follows that

\displaystyle\|\ddot{X}(t)\|_{1}\leq\|\ddot{x}(t)\|+\alpha\left\|\left(\dfrac{1}{t}\dot{x}(t)\right)^{\prime}\right\|+L_{g}\|\dot{x}(t)\|+L_{g}\left\|\left(\left(\gamma+\dfrac{\beta}{t}\right)\dot{x}(t)\right)^{\prime}\right\|.

Consequently,

\displaystyle\|x^{(3)}(t)\|\leq\alpha\left\|-\dfrac{1}{t^{2}}\dot{x}(t)+\dfrac{1}{t}\ddot{x}(t)\right\|+L_{g}\|\dot{x}(t)\|+L_{g}\left\|-\dfrac{\beta}{t^{2}}\dot{x}(t)+\left(\gamma+\dfrac{\beta}{t}\right)\ddot{x}(t)\right\|.

Finally,

\|x^{(3)}(t)\|\leq\left(L_{g}+\dfrac{\alpha+L_{g}|\beta|}{t^{2}}\right)\|\dot{x}(t)\|+\left(\frac{\alpha}{t}+L_{g}\left|\gamma+\dfrac{\beta}{t}\right|\right)\|\ddot{x}(t)\|\leq K(\|\dot{x}(t)\|+\|\ddot{x}(t)\|),

where $K=\max\left\{\max_{t\geq t_{0}}\left(L_{g}+\dfrac{\alpha+L_{g}|\beta|}{t^{2}}\right),\max_{t\geq t_{0}}\left(\frac{\alpha}{t}+L_{g}\left|\gamma+\dfrac{\beta}{t}\right|\right)\right\}$ . $\blacksquare$

3 Convergence analysis

3.1 On a general energy functional associated to the dynamical system (6)

In order to obtain convergence rates for the function values in the trajectories generated by the dynamical system (6), we need to introduce an appropriate energy functional which will play the role of a Lyapunov function. The form of such an energy functional associated to heavy ball system with vanishing damping and its extensions is well known in the literature, see for instance [10], [12], [1] and [2]. However, the novelty of the dynamical system (6), compared with the extended/perturbed variants of the heavy ball system studied in the above mentioned papers, consists in the fact that in system (6) the perturbation is carried out in the argument of the gradient of the objective function. This seems to be a new approach in the literature, therefore the previously mentioned energy functionals are not suitable for a valuable convergence analysis of the dynamical system (6). Hence, let us denote $\alpha(t)=\frac{\alpha}{t}$ and $\beta(t)=\gamma+\frac{\beta}{t}$ , and assume that $\operatorname*{argmin}g\neq\emptyset.$ Further, let $g^{*}=\min g=g(x^{*}),\,x^{*}\in\operatorname*{argmin}g.$ In connection to the dynamical system (6), we introduce the general energy functional

\mathcal{E}(t)=a(t)(g(x(t)+\beta(t)\dot{x}(t))-g^{*})+\frac{1}{2}\|b(t)(x(t)-x^{*})+c(t)\dot{x}(t)\|^{2}+\frac{d(t)}{2}\|x(t)-x^{*}\|^{2},

(13)

which can be seen as an extension of the energy function studied in [10] in connection to the heavy ball system with vanishing damping.

Our purpose is to define the non-negative functions $a(t),b(t),c(t),d(t)$ such that $\dot{\mathcal{E}}(t)\leq 0$ , that is, the function $\mathcal{E}(t)$ is non-increasing after a $t_{1}\geq t_{0}.$ Indeed, if $\mathcal{E}(t)$ is non-increasing for $t\geq t_{1},$ then

a(t)(g(x(t)+\beta(t)\dot{x}(t))-g^{*})\leq\mathcal{E}(t)\leq\mathcal{E}(t_{1}),

in other words

g(x(t)+\beta(t)\dot{x}(t))-g^{*}\leq\frac{\mathcal{E}(t_{1})}{a(t)},\mbox{ for all }t\geq t_{1}.

In what follows we derive the conditions which must be imposed on the positive functions $a(t),b(t),c(t),d(t)$ in order to obtain $\dot{\mathcal{E}}(t)\leq 0$ for every $t\geq t_{1}.$ We have,

\dot{\mathcal{E}}(t)=a^{\prime}(t)(g(x(t)+\beta(t)\dot{x}(t))-g^{*})+a(t)\langle{\nabla}g(x(t)+\beta(t)\dot{x}(t)),\beta(t)\ddot{x}(t)+(\beta^{\prime}(t)+1)\dot{x}(t)\rangle+

\langle b^{\prime}(t)(x(t)-x^{*})+(b(t)+c^{\prime}(t))\dot{x}(t)+c(t)\ddot{x}(t),b(t)(x(t)-x^{*})+c(t)\dot{x}(t)\rangle+

\frac{d^{\prime}(t)}{2}\|x(t)-x^{*}\|^{2}+d(t)\langle\dot{x}(t),x(t)-x^{*}\rangle.

Now, from (6) we get

\ddot{x}(t)=-\alpha(t)\dot{x}(t)-{\nabla}g(x(t)+\beta(t)\dot{x}(t)),

hence

\dot{\mathcal{E}}(t)=a^{\prime}(t)(g(x(t)+\beta(t)\dot{x}(t))-g^{*})+

a(t)\left\langle{\nabla}g(x(t)+\beta(t)\dot{x}(t)),-\beta(t){\nabla}g(x(t)+\beta(t)\dot{x}(t))+\left(-\beta(t)\alpha(t)+\beta^{\prime}(t)+1\right)\dot{x}(t)\right\rangle+

\left\langle b^{\prime}(t)(x(t)-x^{*})+\left(b(t)+c^{\prime}(t)-c(t)\alpha(t)\right)\dot{x}(t)-c(t){\nabla}g(x(t)+\beta(t)\dot{x}(t)),b(t)(x(t)-x^{*})+c(t)\dot{x}(t)\right\rangle+

\frac{d^{\prime}(t)}{2}\|x(t)-x^{*}\|^{2}+d(t)\langle\dot{x}(t),x(t)-x^{*}\rangle.

Consequently,

\dot{\mathcal{E}}(t)=a^{\prime}(t)(g(x(t)+\beta(t)\dot{x}(t))-g^{*})+

(14)

-a(t)\beta(t)\|{\nabla}g(x(t)+\beta(t)\dot{x}(t))\|^{2}+\left(-a(t)\alpha(t)\beta(t)+a(t)\beta^{\prime}(t)+a(t)-c^{2}(t)\right)\langle{\nabla}g(x(t)+\beta(t)\dot{x}(t)),\dot{x}(t)\rangle+

\left(b^{\prime}(t)b(t)+\frac{d^{\prime}(t)}{2}\right)\|x(t)-x^{*}\|^{2}+\left(b^{2}(t)+b(t)c^{\prime}(t)+b^{\prime}(t)c(t)-b(t)c(t)\alpha(t)+d(t)\right)\langle\dot{x}(t),x(t)-x^{*}\rangle+

c(t)\left(b(t)+c^{\prime}(t)-c(t)\alpha(t)\right)\|\dot{x}(t)\|^{2}-b(t)c(t)\langle{\nabla}g(x(t)+\beta(t)\dot{x}(t)),x(t)-x^{*}\rangle.

But

\langle{\nabla}g(x(t)+\beta(t)\dot{x}(t)),x(t)-x^{*}\rangle=\langle{\nabla}g(x(t)+\beta(t)\dot{x}(t)),x(t)+\beta(t)\dot{x}(t)-x^{*}\rangle-\langle{\nabla}g(x(t)+\beta(t)\dot{x}(t)),\beta(t)\dot{x}(t)\rangle

and by the convexity of $g$ we have

\langle{\nabla}g(x(t)+\beta(t)\dot{x}(t)),x(t)+\beta(t)\dot{x}(t)-x^{*}\rangle\geq g(x(t)+\beta(t)\dot{x}(t))-g^{*},

hence

-b(t)c(t)\langle{\nabla}g(x(t)+\beta(t)\dot{x}(t)),x(t)-x^{*}\rangle\leq

-b(t)c(t)(g(x(t)+\beta(t)\dot{x}(t))-g^{*})+b(t)c(t)\beta(t)\langle{\nabla}g(x(t)+\beta(t)\dot{x}(t)),\dot{x}(t)\rangle.

Therefore, (14) becomes

\dot{\mathcal{E}}(t)\leq(a^{\prime}(t)-b(t)c(t))(g(x(t)+\beta(t)\dot{x}(t))-g^{*})-a(t)\beta(t)\|{\nabla}g(x(t)+\beta(t)\dot{x}(t))\|^{2}+

(15)

\left(-a(t)\alpha(t)\beta(t)+a(t)\beta^{\prime}(t)+a(t)-c^{2}(t)+b(t)c(t)\beta(t)\right)\langle{\nabla}g(x(t)+\beta(t)\dot{x}(t)),\dot{x}(t)\rangle+

\left(b^{\prime}(t)b(t)+\frac{d^{\prime}(t)}{2}\right)\|x(t)-x^{*}\|^{2}+\left(b^{2}(t)+b(t)c^{\prime}(t)+b^{\prime}(t)c(t)-b(t)c(t)\alpha(t)+d(t)\right)\langle\dot{x}(t),x(t)-x^{*}\rangle+

c(t)\left(b(t)+c^{\prime}(t)-c(t)\alpha(t)\right)\|\dot{x}(t)\|^{2}

In order to have $\dot{\mathcal{E}}(t)\leq 0$ for all $t\geq t_{1},\,t_{1}\geq t_{0},$ one must assume that for all $t\geq t_{1}$ the following inequalities hold:

	$\displaystyle a^{\prime}(t)-b(t)c(t)\leq 0,$		(16)
	$\displaystyle-a(t)\beta(t)\leq 0,$		(17)
	$\displaystyle-a(t)\alpha(t)\beta(t)+a(t)\beta^{\prime}(t)+a(t)-c^{2}(t)+b(t)c(t)\beta(t)=0,$		(18)
	$\displaystyle b^{\prime}(t)b(t)+\frac{d^{\prime}(t)}{2}\leq 0,$		(19)
	$\displaystyle b^{\prime}(t)c(t)+b(t)\left(b(t)+c^{\prime}(t)-c(t)\alpha(t)\right)+d(t)=0,$		(20)
	$\displaystyle c(t)\left(b(t)+c^{\prime}(t)-c(t)\alpha(t)\right)\leq 0.$		(21)

Remark 11

Observe that (17) implies that $\beta(t)\geq 0$ for all $t\geq t_{1},\,t_{1}\geq t_{0}$ and this shows that in dynamical system (6), one must have $\beta\geq 0$ whenever $\gamma=0.$ Further, (19) is satisfied whenever $b(t)$ and $d(t)$ are constant functions. It is obvious that there exists $t_{1}$ such that for all $t\geq t_{1}$ we have $-\alpha(t)\beta(t)+\beta^{\prime}(t)+1=1-\frac{\alpha\gamma}{t}-\frac{\alpha\beta+\beta}{t^{2}}>0,$ hence from (18) we get

a(t)=\frac{c^{2}(t)-b(t)c(t)\beta(t)}{-\alpha(t)\beta(t)+\beta^{\prime}(t)+1}.

Since (20) and (21) do not depend by $\beta(t),$ it seem natural to choose $c(t)$ , $b(t)$ and $d(t)$ the same as in case of heavy ball system with vanishing damping (see [10]), that is, $c(t)=t,\,b(t)=b\in(0,\alpha-1]$ and $d(t)=b(\alpha-1-b),$ for all $t\geq t_{0},$ provided $\alpha>1.$ Now, an easy computation shows that in this case

a(t)=t^{2}+\gamma(\alpha-b)t+\beta+(\beta+\alpha\gamma^{2})(\alpha-b)+

\frac{(\alpha\beta\gamma+(\gamma\beta+2\alpha\beta\gamma+\alpha^{2}\gamma^{3})(\alpha-b))t+(\alpha\beta+\beta)(\beta+(\beta+\alpha\gamma^{2})(\alpha-b))}{t^{2}-\alpha\gamma t-(\alpha\beta+\beta)},

hence (16) is satisfied whenever $b>2,$ which implies that $\alpha>3.$

However, if $\gamma=0$ then

a(t)=t^{2}+\beta+\beta(\alpha-b)+\frac{((\alpha\beta+\beta)(\beta+\beta(\alpha-b))}{t^{2}-(\alpha\beta+\beta)},

hence (16) holds also for $b=2$ and $\alpha=3.$

3.2 Error estimates for the values

In this section we obtain convergence rate of order $\mathcal{O}(1/t^{2}),\,t\longrightarrow+\infty$ for the difference $g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)-g^{*}$ where $g^{*}=g(x^{*})=\min g,\,x^{*}\in\operatorname*{argmin}g\neq\emptyset.$ From here we are able to show that $g(x(t))-g^{*}$ also has a convergence rate of order $\mathcal{O}(1/t^{2}),\,t\longrightarrow+\infty.$ However, just as in the case of heavy ball system with vanishing damping, in order to obtain these rates, it is necessary to assume $\alpha\geq 3$ in our system (6). We have the following result.

Theorem 12

Let $x$ be the unique strong global solution of (6) and assume that $\operatorname*{argmin}g\neq\emptyset.$ Assume further that $\alpha>3$ , $\gamma>0$ and $\beta\in\mathbb{R}$ .

Then, there exists $K\geq 0$ and $t_{1}\geq t_{0}$ such that

g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)-\min g\leq\frac{K}{t^{2}},\,\mbox{for all }t\geq t_{1}.

(22)

Further,

\int_{t_{0}}^{+\infty}t\left(g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)-\min g\right)dt<+\infty,

(23)

and

\int_{t_{0}}^{+\infty}t^{2}\left\|{\nabla}g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)\right\|^{2}dt<+\infty.

(24)

Proof.

Let $\min g=g^{*}=g(x^{*}),\,x^{*}\in\operatorname*{argmin}g.$ Consider the energy functional (13) with

b(t)=\alpha-1>2,\,c(t)=t,\,d(t)=0.

According to Remark 11 we have

a(t)=t^{2}+\gamma t+2\beta+\alpha\gamma^{2}+\frac{(3\alpha\beta\gamma+\alpha^{2}\gamma^{3}+\beta\gamma)t+\beta(\alpha+1)(2\beta+\alpha\gamma^{2})}{t^{2}-\alpha\gamma t-\beta(\alpha+1)}

and the conditions (18)-(21) are satisfied with equality for every $t\geq t_{0}$ .

Obviously, if $t$ is big enough on has that $a(t)>0$ and since $\gamma>0$ it holds that $\gamma+\frac{\beta}{t}>0$ for $t$ big enough, even if $\beta<0$ . Hence, there exists $t^{\prime}\geq t_{0}$ such (17) is satisfied for every $t\geq t^{\prime}.$

Now, $a^{\prime}(t)-b(t)c(t)=(3-\alpha)t+\gamma+\mathcal{O}\left(\frac{1}{t^{2}}\right)$ and by taking into account that $\alpha>3$ we obtain that there exists $t^{\prime\prime}\geq t_{0}$ such that (16) holds for every $t\geq t^{\prime\prime}.$

Let $t^{\prime\prime\prime}=\max(t^{\prime},t^{\prime\prime}).$ We conclude that, $\dot{\mathcal{E}}(t)\leq 0$ for all $t\geq t^{\prime\prime\prime},$ hence $\mathcal{E}(t)$ is nonincreasing, i.e.

a(t)\left(g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)-\min g\right)\leq\mathcal{E}(t)\leq\mathcal{E}(t^{\prime\prime\prime}),\mbox{ for all }t\geq t^{\prime\prime\prime}.

But, obviously there exists $t_{1}\geq t^{\prime\prime\prime}$ such that $a(t)\geq t^{2}$ for all $t\geq t_{1}$ and by denoting $K=\mathcal{E}(t^{\prime\prime\prime})$ we obtain

g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)-\min g\leq\frac{K}{t^{2}},\mbox{ for all }t\geq t_{1}.

Next we prove (23). Since $a^{\prime}(t)-b(t)c(t)=(3-\alpha)t+\gamma+\mathcal{O}\left(\frac{1}{t^{2}}\right)$ there exist $t_{2}\geq t_{1}$ such that

a^{\prime}(t)-b(t)c(t)\leq\frac{(3-\alpha)t}{2},\mbox{ for all }t\geq t_{2}.

Hence,

\dot{\mathcal{E}}(t)\leq\frac{(3-\alpha)t}{2}\left(g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)-\min g\right),\mbox{ for all }t\geq t_{2}

and by integrating from $t_{2}$ to $T>t_{2}$ we get

\int_{t_{2}}^{T}t\left(g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)-\min g\right)dt\leq\frac{2}{3-\alpha}(\mathcal{E}(T)-\mathcal{E}(t_{2})).

Now, by letting $T\longrightarrow+\infty$ and taking into account that $\mathcal{E}$ is nonincreasing, the conclusion follows.

For proving (24) observe that there exists $t_{3}\geq t_{1}$ such that $-a(t)\left(\gamma+\frac{\beta}{t}\right)\leq-\frac{\gamma}{2}t^{2}$ for all $t\geq t_{3}.$ Consequently

\dot{\mathcal{E}}(t)\leq-\frac{\gamma}{2}t^{2}\left\|{\nabla}g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)\right\|^{2},\mbox{ for all }t\geq t_{3}.

Integrating from $t_{3}$ to $T>t_{3}$ and letting $T\longrightarrow+\infty$ we obtain the desired conclusion. $\blacksquare$

Remark 13

Note that according to Remark 11 the condition (17) holds even if $\gamma=0$ and $\beta\geq 0.$ Consequently, even in the case $\alpha>3,\,\gamma=0,\,\beta\geq 0$ the conclusions (22) and (23) in Theorem 12 hold. However, if $\beta=0$ then we cannot obtain (24). Nevertheless, if $\beta>0$ then (24) becomes:

\int_{t_{0}}^{+\infty}t\left\|{\nabla}g\left(x(t)+\frac{\beta}{t}\dot{x}(t)\right)\right\|^{2}dt<+\infty.

Remark 14

Concerning the case $\alpha=3$ , note that (17) is satisfied if one assumes that $\gamma=0$ but $\beta\geq 0.$ Moreover, in this case (16) is also satisfied since, one has $a^{\prime}(t)-b(t)c(t)=-\frac{4\beta^{2}(\alpha+1)}{(t^{2}-\beta(\alpha+1))^{2}}\leq 0.$ Hence, also in the case $\alpha=3,\,\gamma=0,\,\beta\geq 0$ (22) in the conclusion of Theorem 12 holds.

Moreover, if we assume $\beta>0$ then (24) becomes:

\int_{t_{0}}^{+\infty}t\left\|{\nabla}g\left(x(t)+\frac{\beta}{t}\dot{x}(t)\right)\right\|^{2}dt<+\infty.

Next we show that also the error $g\left(x(t)\right)-\min g$ is of order $\mathcal{O}(1/t^{2}).$ For obtaining this result we need the Descent Lemma, see [28].

Lemma 15

Let $g:\mathbb{R}^{m}\longrightarrow\mathbb{R}$ be a Frèchet differentiable function with $L_{g}$ Lipschitz continuous gradient. Then one has

g(y)\leq g(x)+\langle{\nabla}g(x),y-x\rangle+\frac{L_{g}}{2}\|y-x\|^{2},\,\forall x,y\in\mathbb{R}^{m}.

Theorem 16

Let $x$ be the unique strong global solution of (6) and assume that $\operatorname*{argmin}g\neq\emptyset.$ If $\alpha>3,\,\gamma>0$ and $\beta\in\mathbb{R}$ , then $x$ is bounded and there exists $K\geq 0$ and $t_{1}\geq t_{0}$ such that

\|\dot{x}(t)\|\leq\frac{K}{t},\mbox{ for all }t\geq t_{1}.

(25)

Further,

\int_{t_{0}}^{+\infty}t\|\dot{x}(t)\|^{2}dt\leq+\infty

(26)

and there exists $K_{1}\geq 0$ and $t_{2}\geq t$ such that

g\left(x(t)\right)-\min g\leq\frac{K_{1}}{t^{2}},\mbox{ for all }t\geq t_{2}.

(27)

Proof.

For $x^{*}\in\operatorname*{argmin}g$ let $g^{*}=g(x^{*})=\min g$ and consider the energy function (13) with $b(t)=b,$ where $2<b<\alpha-1$ , $c(t)=t$ , $d(t)=b(\alpha-1-b)>0$ and

a(t)=\frac{c^{2}(t)-b(t)c(t)\left(\gamma+\frac{\beta}{t}\right)}{1-\frac{\beta}{t^{2}}-\frac{\alpha}{t}\left(\gamma+\frac{\beta}{t}\right)}=\frac{(t^{2}-b\gamma t-b\beta)t^{2}}{t^{2}-\alpha\gamma t-\beta(\alpha+1)}=\left(1+\frac{(\alpha-b)\gamma t-\beta(\alpha+1-b)}{t^{2}-\alpha\gamma t-\beta(\alpha+1)}\right)t^{2}.

According to Remark 11 there exists $t_{1}\geq t_{0}$ such that the conditions (16)-(21) are satisfied.

From the definition of $\mathcal{E}$ one has

\|b(t)(x(t)-x^{*})+c(t)\dot{x}(t)\|\leq\sqrt{2\mathcal{E}(t)},

and

\|x(t)-x^{*}\|\leq\sqrt{\frac{2}{d(t)}\mathcal{E}(t)}

that is

\|b(x(t)-x^{*})+t\dot{x}(t)\|\leq\sqrt{2\mathcal{E}(t)},

and

\|x(t)-x^{*}\|\leq\sqrt{\frac{2}{b(\alpha-1-b)}\mathcal{E}(t)}.

By using the fact that $\mathcal{E}$ nonincreasing on an interval $[t_{1},+\infty)$ the latter inequality assures that $x$ is bounded.

Now, by using the inequality $\|X-Y\|\geq\|X\|-\|Y\|$ we get

\|b(x(t)-x^{*})+t\dot{x}(t)\|\geq t\|\dot{x}(t)\|-\|b(x(t)-x^{*})\|,

hence for all $t\geq t_{1}$ one has

t\|\dot{x}(t)\|\leq\sqrt{2\mathcal{E}(t)}+\|b(x(t)-x^{*})\|\leq\left(1+\sqrt{\frac{1}{\alpha-1-b}}\right)\sqrt{2\mathcal{E}(t)}\leq K,

where $K=\left(1+\sqrt{\frac{1}{\alpha-1-b}}\right)\sqrt{2\mathcal{E}(t_{1})}.$

Further, (21) becomes $(b+1-\alpha)t<0,$ hence for all $t\geq t_{1}$ one has

\dot{\mathcal{E}}(t)\leq(b+1-\alpha)t\|\dot{x}(t)\|^{2}.

By integrating from $t_{1}$ to $T>t_{1}$ one gets

\int_{t_{1}}^{T}t\|\dot{x}(t)\|^{2}dt\leq\frac{1}{\alpha-1-b}(\mathcal{E}(t_{1})-\mathcal{E}(T))\leq\frac{1}{\alpha-1-b}\mathcal{E}(t_{1}).

By letting $T\longrightarrow+\infty$ we obtain

\int_{t_{0}}^{+\infty}t\|\dot{x}(t)\|^{2}dt\leq+\infty.

Now, by using Lemma 15 with $y=x(t)$ and $x=x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)$ we obtain

g(x(t))-g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)\leq

(28)

\left\langle{\nabla}g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right),-\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right\rangle+\frac{L_{g}}{2}\left\|\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right\|^{2}\leq

\frac{|\gamma t+\beta|}{t}\left\|{\nabla}g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)\right\|\left\|\dot{x}(t)\right\|+\frac{L_{g}}{2}\left(\gamma+\frac{\beta}{t}\right)^{2}\|\dot{x}(t)\|^{2}.

Now according to (24) there exists $t^{\prime}\geq t_{0}$ and $K^{\prime}>0$ such that

\left\|{\nabla}g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)\right\|\leq\frac{K^{\prime}}{t},\mbox{ for all }t\geq t^{\prime}.

Further (25) assures that there exists $t_{1}\geq t_{0}$ and $K>0$ such that

\|\dot{x}(t)\|\leq\frac{K}{t},\mbox{ for all }t\geq t_{1}.

Obviously,

\left|\gamma+\frac{\beta}{t}\right|\leq\max\left(\gamma,\gamma+\frac{\beta}{t_{0}}\right),\mbox{ for all }t\geq t_{0},

hence (28) assures that there exists $K_{1}^{\prime}>0$ and $t_{2}^{\prime}\geq t_{0}$ such that

g(x(t))-g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)\leq\frac{K_{1}^{\prime}}{t^{2}},\mbox{ for all }t\geq t_{2}^{\prime}.

(29)

Now, by adding (22) and (29) we get that there exists $K_{1}>0$ and $t_{2}\geq t_{0}$ such that

\left(g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)-\min g\right)+\left(g(x(t))-g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)\right)\leq\frac{K_{1}}{t^{2}},\mbox{ for all }t\geq t_{2}

that is

g(x(t))-\min g\leq\frac{K_{1}}{t^{2}},\mbox{ for all }t\geq t_{2}.

$\blacksquare$

Remark 17

Note that if we assume that $\beta\geq 0$ then one can allow $\gamma=0$ in the hypotheses of Theorem 16, since in this case condition (17) is satisfied and the conclusions of Theorem 16 hold.

Remark 18

Note that (24), which holds whenever $\alpha>3$ and $\gamma>0$ , assures that

t\left\|{\nabla}g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)\right\|\in L^{2}([t_{0},+\infty),\mathbb{R}).

Further, (26) assures that

\sqrt{t}\|\dot{x}(t)\|\in L^{2}([t_{0},+\infty),\mathbb{R}).

(30)

Consequently, the system (6) leads to

\|t\ddot{x}(t)\|=\left\|\alpha\dot{x}(t)+t{\nabla}g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)\right\|\leq\alpha\|\dot{x}(t)\|+t\left\|{\nabla}g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)\right\|

hence,

t\|\ddot{x}(t)\|\in L^{2}([t_{0},+\infty),\mathbb{R}).

(31)

Now, from (12) we have

\sqrt{t}\|x^{(3)}(t)\|\leq K(\sqrt{t}\|\dot{x}(t)\|+\sqrt{t}\|\ddot{x}(t)\|)

for some $K>0$ and for almost every $t\geq t_{0}$ , which combined with (30) and (31) gives

\sqrt{t}\|x^{(3)}(t)\|\in L^{2}([t_{0},+\infty),\mathbb{R}).

(32)

Remark 19

Notice that (30), (31) and (32) assure in particular that

\lim_{t\longrightarrow+\infty}\|\dot{x}(t)\|=\lim_{t\longrightarrow+\infty}\|\ddot{x}(t)\|=\lim_{t\longrightarrow+\infty}\|{x}^{(3)}(t)\|=0.

(33)

Remark 20

In the case $\gamma=0$ and $\beta\geq 0$ according to Remark 13 one has

\sqrt{t}\left\|{\nabla}g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)\right\|\in L^{2}([t_{0},+\infty),\mathbb{R}).

Hence, as in Remark 18 we derive that

\sqrt{t}\|\ddot{x}(t)\|,\,\sqrt{t}\|x^{(3)}(t)\|\in L^{2}([t_{0},+\infty),\mathbb{R})

and consequently (33) holds.

3.3 The convergence of the generated trajectories

In this section we show convergence of the generated trajectories to a minimum point of the objective $g.$ The main tool that we use in order to attain this goal will be the following continuous version of Opial Lemma, (see [34, 30, 10].

Lemma 21

Let $H$ to be a separable Hilbert space and $S\subseteq H$ , with $S\neq\emptyset$ . Further, consider $x:[t_{0},+\infty)\to H$ a given map. Suppose that the following conditions are satisfied :

	$\displaystyle(i)\quad\forall z\in S,\,\exists\lim\limits_{t\longrightarrow+\infty}\\|x(t)-z\\|$
	$\displaystyle(ii)\quad\text{every weak sequentially limit point of }x(t)\text{ belongs to the set }S.$

Then, $x(t)$ converges weakly to a point of $S$ as $t\to+\infty$ .

Remark 22

In the setting of the proof concerning the convergence of the generated trajectories, we consider the set $S$ to be $argmin(g)$ . Moreover, the Hilbert space $H$ is $\mathbb{R}^{m}$ , and this implies that we actually deduce strong convergence of a strong global solution of the dynamical system (6) to a minimum of the objective function $g.$

Consider now the set of limit points of the trajectory $x$ , that is

\omega(x)=\{\overline{x}\in\mathbb{R}^{m}:\exists(t_{n})_{n\in\mathbb{N}}\subseteq\mathbb{R},\,t_{n}\longrightarrow+\infty,\,n\longrightarrow+\infty\text{ such that }x(t_{n})\longrightarrow\overline{x},\,n\longrightarrow+\infty\}.

We show that $\omega(x)\subseteq\operatorname*{argmin}g.$ We emphasize that since $g$ is convex one has

\operatorname*{argmin}g=\operatorname*{crit}g:=\{\overline{x}\in\mathbb{R}^{m}:{\nabla}g(\overline{x})=0.\}.

We have the following result.

Lemma 23

Let $x$ be the unique strong global solution of (6) and assume that $\operatorname*{argmin}g\neq\emptyset.$ If $\alpha>3,\,\gamma>0$ and $\beta\in\mathbb{R}$ , then the following assumptions hold.

(i)

$\omega(x)=\omega\left(x(\cdot)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(\cdot)\right)$ ;
(ii)

$\omega(x)\subseteq\operatorname*{argmin}g.$

Proof.

Indeed (33) assures that $\lim_{t\longrightarrow+\infty}\left(\gamma+\dfrac{\beta}{t}\right)\dot{x}(t)=0$ , which immediately proves (i).

For proving (ii) consider $\overline{x}\in\omega(x).$ Then, there exists $(t_{n})_{n\in\mathbb{N}}\subseteq\mathbb{R},\,t_{n}\longrightarrow+\infty,\,n\longrightarrow+\infty$ such that $\lim_{n\longrightarrow+\infty}x(t_{n})=\overline{x}.$ Now, since ${\nabla}g$ is continuous and $\lim_{n\longrightarrow+\infty}\left(x(t_{n})+\left(\gamma+\frac{\beta}{t_{n}}\right)\dot{x}(t_{n})\right)=\overline{x}$ one has

\lim_{n\longrightarrow+\infty}{\nabla}g\left(x(t_{n})+\left(\gamma+\frac{\beta}{t_{n}}\right)\dot{x}(t_{n})\right)={\nabla}g(\overline{x}).

Further, according to (33)

\lim_{n\longrightarrow+\infty}\left(\ddot{x}(t_{n})+\frac{\alpha}{t_{n}}\dot{x}(t_{n})\right)=0.

Now, the system (6) gives

0=\lim_{n\longrightarrow+\infty}\left(\ddot{x}(t_{n})+\frac{\alpha}{t_{n}}\dot{x}(t_{n})+{\nabla}g\left(x(t_{n})+\left(\gamma+\frac{\beta}{t_{n}}\right)\dot{x}(t_{n})\right)\right)={\nabla}g(\overline{x})

that is $\overline{x}\in\operatorname*{argmin}g$ and this proves (ii). $\blacksquare$

Remark 24

Obviously, according to Remark 20 the conclusion of Lemma 23 remains valid also in the case $\alpha>3,\,\gamma=0$ and $\beta\geq 0.$ Note that (ii) from the conclusion of Lemma 23 is actually the condition (ii) from Lemma 21. For proving (i) from Lemma 21, that is, the limit $\lim_{t\longrightarrow+\infty}\|x(t)-x^{*}\|$ exists for every $x^{*}\in\operatorname*{argmin}g,$ we need the following result from [10].

Lemma 25

(Lemma A.4. [10]) Let $t_{0}>0$ , and let $w:[t_{0},+\infty)\longrightarrow\mathbb{R}$ be a continuously differentiable function which is bounded from below. Assume that

t\ddot{w}(t)+\alpha\dot{w}(t)\leq G(t)

for some $\alpha>1,$ almost every $t>t_{0}$ , and some nonnegative function $G\in L^{1}(t_{0},+\infty).$ Then, the positive part $[\dot{w}]_{+}$ of $\dot{w}$ belongs to $L^{1}(t_{0},+\infty)$ and limit $\lim_{t\longrightarrow+\infty}w(t)$ exists.

Now we can prove the following.

Lemma 26

Let $x$ be the unique strong global solution of (6) and assume that $\operatorname*{argmin}g\neq\emptyset.$ If $\alpha>3,\,\gamma>0$ and $\beta\in\mathbb{R}$ , then for every $x^{*}\in\operatorname*{argmin}g$ there exists the limit

\lim_{t\longrightarrow+\infty}\|x(t)-x^{*}\|.

Proof.

Let $x^{*}\in\operatorname*{argmin}g$ and define the function $h_{x^{*}}:[t_{0},+\infty)\longrightarrow\mathbb{R},\,h_{x^{*}}(t)=\dfrac{1}{2}\|x(t)-x^{*}\|^{2}$ . Using the chain rule with respect to the differentiation of $h_{x^{*}}$ , we obtain that

\displaystyle\dot{h}_{x^{*}}(t)=\langle\dot{x}(t),x(t)-x^{*}\rangle

and that

\displaystyle\ddot{h}_{x^{*}}(t)=\langle\ddot{x}(t),x(t)-x^{*}\rangle+\|\dot{x}(t)\|^{2}.

Let us denote $g^{\ast}=g(x^{\ast})=min(g)$ . Using the dynamical system (6), one has that

\displaystyle\ddot{h}_{x^{*}}(t)+\dfrac{\alpha}{t}\dot{h}_{x^{*}}(t)=-\left\langle\nabla g\left(x(t)+\left(\gamma+\dfrac{\beta}{t}\right)\dot{x}(t)\right),x(t)-x^{\ast}\right\rangle+\|\dot{x}(t)\|^{2}.

This means that

\ddot{h}_{x^{*}}(t)+\dfrac{\alpha}{t}\dot{h}_{x^{*}}(t)=-\left\langle\nabla g\left(x(t)+\left(\gamma+\dfrac{\beta}{t}\right)\dot{x}(t)\right),x(t)+\left(\gamma+\dfrac{\beta}{t}\right)\dot{x}(t)-x^{*}\right\rangle+\|\dot{x}(t)\|^{2}

(34)

+\left\langle\nabla g\left(x(t)+\left(\gamma+\dfrac{\beta}{t}\right)\dot{x}(t)\right),\left(\gamma+\dfrac{\beta}{t}\right)\dot{x}(t)\right\rangle.

By the convexity of the mapping $g$ and taking into account that $g^{*}=g(x^{*})=\min g$ , we obtain

-\left\langle\nabla g\left(x(t)+\left(\gamma+\dfrac{\beta}{t}\right)\dot{x}(t)\right),x(t)+\left(\gamma+\dfrac{\beta}{t}\right)\dot{x}(t)-x^{*}\right\rangle\leq g^{*}-g\left(x(t)+\left(\gamma+\dfrac{\beta}{t}\right)\dot{x}(t)\right)\leq 0,

hence by using (6) the inequality (34) becomes

\displaystyle\ddot{h}_{x^{*}}(t)+\dfrac{\alpha}{t}\dot{h}_{x^{*}}(t)\leq g^{*}-g\left(x(t)+\left(\gamma+\dfrac{\beta}{t}\right)\dot{x}(t)\right)+\left(1-\dfrac{\alpha}{t}\left(\gamma+\dfrac{\beta}{t}\right)\right)\|\dot{x}(t)\|^{2}-\left(\gamma+\dfrac{\beta}{t}\right)\langle\dot{x}(t),\ddot{x}(t)\rangle.

Consequently, one has

t\ddot{h}_{x^{*}}(t)+\alpha\dot{h}_{x^{*}}(t)\leq\left(t-\alpha\left(\gamma+\dfrac{\beta}{t}\right)\right)\|\dot{x}(t)\|^{2}-\left(\gamma t+\beta\right)\langle\dot{x}(t),\ddot{x}(t)\rangle.

(35)

Now, according to (30) one has

t\|\dot{x}(t)\|^{2}\in L^{1}([t_{0},+\infty),\mathbb{R}).

Consequently,

\left(t-\alpha\left(\gamma+\dfrac{\beta}{t}\right)\right)\|\dot{x}(t)\|^{2}\in L^{1}([t_{0},+\infty),\mathbb{R}).

(36)

Further,

-\left(\gamma t+\beta\right)\langle\dot{x}(t),\ddot{x}(t)\rangle\leq\frac{1}{2}\|\dot{x}(t)\|^{2}+\frac{1}{2}\left(\gamma t+\beta\right)^{2}\|\ddot{x}(t)\|^{2}

and (31) assures that

t^{2}\|\ddot{x}(t)\|^{2}\in L^{1}([t_{0},+\infty),\mathbb{R}),

hence

-\left(\gamma t+\beta\right)\langle\dot{x}(t),\ddot{x}(t)\rangle\in L^{1}([t_{0},+\infty),\mathbb{R}).

(37)

According to (36) and (37) we get that the function

G(t)=\left(t-\alpha\left(\gamma+\dfrac{\beta}{t}\right)\right)\|\dot{x}(t)\|^{2}+\frac{1}{2}\|\dot{x}(t)\|^{2}+\frac{1}{2}\left(\gamma t+\beta\right)^{2}\|\ddot{x}(t)\|^{2}\in L^{1}([t_{0},+\infty),\mathbb{R}).

Moreover, it is obvious that there exists $t_{1}\geq t_{0}$ such that $G$ is non negative for every $t\geq t_{1}.$ From (35) one has

t\ddot{h}_{x^{*}}(t)+\alpha\dot{h}_{x^{*}}(t)\leq G(t),\mbox{ for every }t\geq t_{0},

(38)

hence Lemma 25 leads to the existence of the limit

\lim_{t\longrightarrow+\infty}h_{x^{*}}(t)

and consequently the limit

\lim_{t\longrightarrow+\infty}\|x(t)-x^{*}\|

also exists. $\blacksquare$

Now, we present the main result of this subsection regarding the convergence of the solution of the dynamical system (6) as $t\to+\infty$ .

Theorem 27

Let $x$ be the unique strong global solution of the dynamical system (6) and assume that $\operatorname*{argmin}g\neq\emptyset.$ If $\alpha>3,\,\gamma>0$ and $\beta\in\mathbb{R}$ , then $x(t)$ converges to a point in $\operatorname*{argmin}g$ as $t\longrightarrow+\infty$ .

Proof.

Taking into account that the conclusion (ii) in Lemma 23 and the conclusion of Lemma 26 are exactly the conditions in the hypothesis of Lemma 21 with $S=\operatorname*{argmin}g,$ the conclusion of the theorem is straightforward. $\blacksquare$

Remark 28

As it was expected, Theorem 27 remains true if in its hypothesis we assume only that $\alpha>3,\,\gamma=0$ and $\beta\geq 0.$ Indeed, note that under these assumptions the conclusion of Lemma 26 holds, since $G$ from its proof becomes

G(t)=\left(t-\dfrac{\alpha\beta}{t}\right)\|\dot{x}(t)\|^{2}+\frac{1}{2}\|\dot{x}(t)\|^{2}+\frac{1}{2}\beta^{2}\|\ddot{x}(t)\|^{2}

and according to Remark 20 $G(t)\in L^{1}([t_{0},+\infty),\mathbb{R}).$ Moreover, it is obvious that there exists $t_{1}\geq t_{0}$ such that $G$ is non negative for every $t\geq t_{1}.$

This fact combined with Remark 24 lead to the desired conclusion.

4 Conclusions

In this paper we study a second order dynamical system which can be viewed as an extension of the heavy ball system with vanishing damping. This dynamical system is actually a perturbed version of the heavy ball system with vanishing damping, but the perturbation is made in the argument of the gradient of the objective function. Numerical experiments show that this perturbation brings a smoothing effect in the behaviour of the energy error $g(x(t))-\min g$ and also in the behaviour of the absolute error $\|x(t)-x^{*}\|$ , where $x(t)$ is a generated trajectory by our dynamical system and $x^{*}$ is a minimum of the objective $g.$ Another novelty of our system that via explicit discretization leads to inertial algorithms. A related future research is the convergence analysis of algorithm (10), since this algorithm contains as particular case the famous Nesterov algorithm. However, since Algorithm (10) may allow different inertial steps, its area of applicability can be considerable.

We have shown existence and uniqueness of the trajectories generated by our dynamical system even for the case the objective function $g$ is non-convex. Further, we treated the cases when the energy error $g(x(t))-\min g$ is of order $\mathcal{O}\left(\frac{1}{t^{2}}\right)$ and we obtained the convergence of a generated trajectory to a minimum of the objective function $g.$ Another related research is the convergence analysis of the generated trajectories in the case when the objective function $g$ is possible non-convex. This would be a novelty in the literature even for the case $\alpha>3,\,\gamma=\beta=0$ , that is, for the case of the heavy ball system with vanishing damping.

We underline that the dynamical system (6) can easily be extended to proximal-gradient dynamical systems (see [18, 21] and the references therein). Indeed, let $f:\mathbb{R}^{m}\longrightarrow\overline{\mathbb{R}}$ be a proper convex and lower semicontinuous function and let $g:\mathbb{R}^{m}\longrightarrow\mathbb{R}$ be a possible non-convex smooth function with $L_{g}$ Lipschitz continuous gradient.

We recall that the proximal point operator of the convex function $\lambda f$ is defined as

\operatorname*{prox}\nolimits_{\lambda f}:{\mathbb{R}^{m}}\rightarrow{\mathbb{R}^{m}},\quad\operatorname*{prox}\nolimits_{\lambda f}(x)=\operatorname*{argmin}_{y\in{\mathbb{R}^{m}}}\left\{f(y)+\frac{1}{2\lambda}\|y-x\|^{2}\right\}.

Consider the optimization problem

\inf_{x\in\mathbb{R}^{m}}f(x)+g(x).

One can associate to this optimization problem the following second order proximal-gradient dynamical system.

\left\{\begin{array}[]{lll}\displaystyle\ddot{x}(t)+\left(\gamma+\frac{\alpha+\beta}{t}\right)\dot{x}(t)+x(t)=\operatorname*{prox}\nolimits_{\lambda f}\left(\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)-\lambda{\nabla}g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)\right),\\ \\ \displaystyle x(t_{0})=u_{0},\,\dot{x}(t_{0})=v_{0},\end{array}\right.

(39)

where $\alpha>0,\,\gamma>0,\,\lambda>0,\,\beta\in\mathbb{R}$ and $t_{0}>0.$

Obviously, when $f\equiv 0$ and $\lambda=1$ then (39) becomes the dynamical system (6) studied in the present paper.

The discretization of the the dynamical system (39) leads to proximal-gradient inertial algorithms, similar to the modified FISTA algorithm studied by Chambolle and Dossal in [24], (see also [15]).

Indeed, the explicit discretization of (6) with respect to the time variable $t$ , with step size $h_{n}$ and initial points $x_{0}:=u_{0},\,x_{1}:=v_{0}$ yields the iterative scheme

\frac{x_{n+1}-2x_{n}+x_{n-1}}{h_{n}^{2}}+\left(\gamma+\frac{\alpha+\beta}{nh_{n}}\right)\frac{x_{n}-x_{n-1}}{h_{n}}+x_{n+1}=

\operatorname*{prox}\nolimits_{\lambda f}\left(x_{n}+\left(\gamma+\frac{\beta}{nh_{n}}\right)\frac{x_{n}-x_{n-1}}{h_{n}}-\lambda\nabla g\left(x_{n}+\left(\gamma+\frac{\beta}{nh_{n}}\right)\frac{x_{n}-x_{n-1}}{h_{n}}\right)\right)\ \forall n\geq 1.

The latter can be expressed as

(1+h_{n}^{2})x_{n+1}=x_{n}+\left(1-\gamma h_{n}-\frac{\alpha+\beta}{n}\right)(x_{n}-x_{n-1})+

h_{n}^{2}\operatorname*{prox}\nolimits_{\lambda f}\left(x_{n}+\left(\gamma+\frac{\beta}{nh_{n}}\right)\frac{x_{n}-x_{n-1}}{h_{n}}-\lambda\nabla g\left(x_{n}+\left(\gamma+\frac{\beta}{nh_{n}}\right)\frac{x_{n}-x_{n-1}}{h_{n}}\right)\right).

Consequently, the dynamical system (39) leads to the algorithm: For $x_{0},\,x_{1}\in\mathbb{R}^{m}$ consider

\left\{\begin{array}[]{llll}\displaystyle y_{n}=x_{n}+\left(1-\gamma h_{n}-\frac{\alpha+\beta}{n}\right)(x_{n}-x_{n-1}),\\ \\ \displaystyle z_{n}=x_{n}+\left(\frac{\gamma}{h_{n}}+\frac{\beta}{nh_{n}^{2}}\right)(x_{n}-x_{n-1}),\\ \\ \displaystyle x_{n+1}=\frac{1}{1+h_{n}^{2}}y_{n}+\frac{h_{n}^{2}}{1+h_{n}^{2}}\operatorname*{prox}\nolimits_{\lambda f}(z_{n}-\lambda{\nabla}g(z_{n})),\end{array}\right.

(40)

where $\alpha\geq 0,\,\gamma\geq 0$ and $\beta\in\mathbb{R}.$

Now, the simplest case is obtained by taking in (40) constant step size $h_{n}\equiv 1$ . By denoting $\alpha+\beta$ still with $\alpha\in\mathbb{R}$ , the Algorithm (40) becomes: For $x_{0},x_{1}\in\mathbb{R}^{m}$ and for every $n\geq 1$ consider

\left\{\begin{array}[]{llll}\displaystyle y_{n}=x_{n}+\frac{(1-\gamma)n-\alpha}{n}(x_{n}-x_{n-1}),\\ \\ \displaystyle z_{n}=x_{n}+\frac{\gamma n+\beta}{n}(x_{n}-x_{n-1}),\\ \\ \displaystyle x_{n+1}=\frac{1}{2}y_{n}+\frac{1}{2}\operatorname*{prox}\nolimits_{\lambda f}(z_{n}-\lambda{\nabla}g(z_{n})),\end{array}\right.

(41)

where $\alpha\geq\beta,\,\gamma\geq 0$ and $\beta\in\mathbb{R}.$ The convergence of the sequences generated by Algorithm (41) to a critical point of the objective function $f+g$ would open the gate for the study of FISTA type algorithms with nonidentical inertial terms.

References

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	$\displaystyle\\|F(t,u,v)-F(t,\overline{u},\overline{v})\\|$	$\displaystyle=\sqrt{\\|v-\overline{v}\\|^{2}+\left\\|\frac{\alpha}{t}(\overline{v}-v)+\nabla g\left(\overline{u}+\left(\gamma+\frac{\beta}{t}\right)\overline{v}\right)-\nabla g\left(u+\left(\gamma+\frac{\beta}{t}\right)v\right)\right\\|^{2}}$
		$\displaystyle\leq\sqrt{\left(1+2\left(\frac{\alpha}{t}\right)^{2}\right)\\|v-\overline{v}\\|^{2}+2L_{g}^{2}\left\\|(u-\overline{u})+\left(\gamma+\frac{\beta}{t}\right)(v-\overline{v})\right\\|^{2}}$
		$\displaystyle\leq\sqrt{\left(1+2\left(\frac{\alpha}{t}\right)^{2}+4L_{g}^{2}\left(\gamma+\frac{\beta}{t}\right)^{2}\right)\\|v-\overline{v}\\|^{2}+4L_{g}^{2}\left\\|u-\overline{u}\right\\|^{2}}$
		$\displaystyle\leq\sqrt{1+4L_{g}^{2}+2\left(\frac{\alpha}{t}\right)^{2}+4L_{g}^{2}\left(\gamma+\frac{\beta}{t}\right)^{2}}\sqrt{\\|v-\overline{v}\\|^{2}+\\|u-\overline{u}\\|^{2}}$
		$\displaystyle=\sqrt{1+4L_{g}^{2}+2\left(\frac{\alpha}{t}\right)^{2}+4L_{g}^{2}\left(\gamma+\frac{\beta}{t}\right)^{2}}\,\\|(u,v)-(\overline{u},\overline{v})\\|.$

	$\displaystyle\int_{t_{0}}^{T}\\|F(t,u,v)\\|dt$	$\displaystyle=\int_{t_{0}}^{T}\sqrt{\\|v\\|^{2}+\left\\|\frac{\alpha}{t}v+\nabla g\left(u+\left(\gamma+\frac{\beta}{t}\right)v\right)\right\\|^{2}}dt$
		$\displaystyle\leq\int_{t_{0}}^{T}\sqrt{\left(1+2\left(\frac{\alpha}{t}\right)^{2}\right)\\|v\\|^{2}+4\left\\|\nabla g\left(u+\left(\gamma+\frac{\beta}{t}\right)v\right)-{\nabla}g(u)\right\\|^{2}+4\\|{\nabla}g(u)\\|^{2}}dt$
		$\displaystyle\leq\int_{t_{0}}^{T}\sqrt{\left(1+2\left(\frac{\alpha}{t}\right)^{2}+4L_{g}^{2}\left(\gamma+\frac{\beta}{t}\right)^{2}\right)\\|v\\|^{2}+4\\|{\nabla}g(u)\\|^{2}}dt$
		$\displaystyle\leq\sqrt{\\|v\\|^{2}+\\|\nabla g(u)\\|^{2}}\int_{t_{0}}^{T}\sqrt{5+2\left(\frac{\alpha}{t}\right)^{2}+4L_{g}^{2}\left(\gamma+\frac{\beta}{t}\right)^{2}}dt$

	$\displaystyle\\|\dot{X}(s)-\dot{X}(t)\\|_{1}$	$\displaystyle=\\|F(s,X(s))-F(t,X(t))\\|_{1}$
		$\displaystyle=\left\\|\left(\dot{x}(s)-\dot{x}(t),\frac{\alpha}{t}\dot{x}(t)-\frac{\alpha}{s}\dot{x}(t)+{\nabla}g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)-{\nabla}g\left(x(s)+\left(\gamma+\frac{\beta}{s}\right)\dot{x}(s)\right)\right)\right\\|_{1}$
		$\displaystyle\leq\\|\dot{x}(s)-\dot{x}(t)\\|+\left\\|\frac{\alpha}{s}\dot{x}(s)-\frac{\alpha}{t}\dot{x}(t)\right\\|$
		$\displaystyle+\left\\|{\nabla}g\left(x(s)+\left(\gamma+\frac{\beta}{s}\right)\dot{x}(s)\right)-{\nabla}g\left(x(t)+\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right)\right\\|$

	$\displaystyle\\|\dot{X}(s)-\dot{X}(t)\\|_{1}$	$\displaystyle\leq\\|\dot{x}(s)-\dot{x}(t)\\|+\left\\|\frac{\alpha}{s}\dot{x}(s)-\frac{\alpha}{t}\dot{x}(t)\right\\|+L_{g}\\|x(s)-x(t)\\|+L_{g}\left\\|\left(\gamma+\frac{\beta}{s}\right)\dot{x}(s)-\left(\gamma+\frac{\beta}{t}\right)\dot{x}(t)\right\\|$
		$\displaystyle\leq\\|\dot{x}(s)-\dot{x}(t)\\|+\dfrac{\alpha}{s}\\|\dot{x}(s)-\dot{x}(t)\\|+\left\|\dfrac{\alpha}{s}-\frac{\alpha}{t}\right\|\\|\dot{x}(t)\\|+L_{g}\\|x(s)-x(t)\\|$
		$\displaystyle+L_{g}\left(\gamma+\frac{\beta}{s}\right)\\|\dot{x}(s)-\dot{x}(t)\\|+L_{g}\left\|\frac{\beta}{s}-\frac{\beta}{t}\right\|\cdot\\|\dot{x}(t)\\|$
		$\displaystyle=\left(1+\dfrac{\alpha}{s}+L_{g}\left(\gamma+\frac{\beta}{s}\right)\right)\\|\dot{x}(s)-\dot{x}(t)\\|+L_{g}\\|x(s)-x(t)\\|+\left(\alpha+L_{g}\|\beta\|\right)\cdot\left\|\dfrac{1}{s}-\dfrac{1}{t}\right\|\\|\dot{x}(t)\\|.$

	$\displaystyle\\|\dot{X}(t+h)-X(t)\\|_{1}$	$\displaystyle=\\|F(t+h,X(t+h))-F(t,X(t))\\|_{1}$
		$\displaystyle\leq\left\\|\left(\dot{x}(t+h)-\dot{x}(t)\right)\right\\|+\alpha\left\\|\dfrac{1}{t+h}\dot{x}(t+h)-\dfrac{1}{t}\dot{x}(t)\right\\|$
		$\displaystyle+\left\\|\nabla g\left(x(t+h)+\left(\gamma+\dfrac{\beta}{t+h}\right)\dot{x}(t+h)\right)-\nabla g\left(x(t)+\left(\gamma+\dfrac{\beta}{t}\right)\dot{x}(t)\right)\right\\|$
		$\displaystyle\leq\left\\|\left(\dot{x}(t+h)-\dot{x}(t)\right)\right\\|+\alpha\left\\|\dfrac{1}{t+h}\dot{x}(t+h)-\dfrac{1}{t}\dot{x}(t)\right\\|$
		$\displaystyle+L_{g}\\|x(t+h)-x(t)\\|+L_{g}\left\\|\left(\gamma+\dfrac{\beta}{t+h}\right)\dot{x}(t+h)-\left(\gamma+\dfrac{\beta}{t}\right)\dot{x}(t)\right\\|.$

An extension of the second order dynamical system that models Nesterov’s convex gradient method

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An extension of the second order dynamical system that models Nesterov’s convex gradient method††thanks: This work was supported by a grant of Ministry of Research and Innovation, CNCS - UEFISCDI, project number PN-III-P1-1.1-TE-2016-0266.

1 Introduction

1.1 An extension of the heavy ball method and the Nesterov type algorithms obtained via explicit discretization

Remark 1

Remark 2

Remark 3

Remark 4

1.2 Some numerical experiments

Remark 5

1.3 The organization of the paper

2 Existence and uniqueness

2.1 On strong global solutions

Definition 1

Remark 6

Theorem 7

Proof.

Remark 8

2.2 On the third order derivatives

Proposition 9

Proof.

Lemma 10

Proof.

3 Convergence analysis

3.1 On a general energy functional associated to the dynamical system (6)

Remark 11

3.2 Error estimates for the values

Theorem 12

Proof.

Remark 13

Remark 14

Lemma 15

Theorem 16

Proof.

Remark 17

Remark 18

Remark 19

Remark 20

3.3 The convergence of the generated trajectories

Lemma 21

Remark 22

Lemma 23

Proof.

Remark 24

Lemma 25

Lemma 26

Proof.

Theorem 27

Proof.

Remark 28

4 Conclusions

References

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An extension of the second order dynamical system that models Nesterov’s convex gradient method^†^†thanks: This work was supported by a grant of Ministry of Research and Innovation, CNCS - UEFISCDI, project number PN-III-P1-1.1-TE-2016-0266.