# Spin wave vortex from the scattering on Bloch point solitons

V. L. Carvalho-Santos<EMAIL_ADDRESS>R. G. Elías
<EMAIL_ADDRESS>A. S. Núñez<EMAIL_ADDRESS>Instituto Federal de
Educação, Ciência e Tecnologia Baiano - Campus Senhor do Bonfim,
Km 04 Estrada da Igara, 48970-000 Senhor do Bonfim, Bahia, Brazil.
Departamento de Física, Universidad de Santiago de Chile and CEDENNA, Avda.
Ecuador 3493, Santiago, Chile.  Departamento de Física, Facultad de Ciencias
Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile

###### Abstract

The interaction of a spin wave with a stationary Bloch point is studied. The
topological non-trivial structure of the Bloch point manifests in the
propagation of spin waves endowing them with a gauge potential that resembles
the one associated with the interaction of a magnetic monopole and an
electron. By pursuing this analogy, we are led to the conclusion that the
scattering of spin waves and Bloch points is accompanied by the creation of a
magnon vortex. Interference between such a vortex and a plane wave leads to
dislocations in the interference pattern that can be measurable by means of
magnon holography.

###### keywords:

Spin waves, Magnonics, Bloch point, Magnetic solitons.

††journal: Annals of physics

##  1 Introduction

In a magnetic material textures in the local magnetic ordering spread across
in the form of spin waves (SWs). Such disturbances correspond to elementary
excitations with zero associated charge and net spin equals to unity. The
field of solid state physics concerning the manipulation, detection and
dynamics of the SWs in a magnetic system [26] has been dubbed magnonics. The
field of magnonics has grown into a well established realm of magnetism and
opened new paths in the understanding of magnetization dynamics of complex
structures. The wide frequency range that magnonic excitations can display
(from GHz to THz [7]), the possibility of tailoring the SW spectrum in the so-
called magnonic crystals [25], the possibility of manipulating magnetic
textures by SWs, [20] the interference pattern of SWs in the presence of
geometrical constrictions or of magnetic solitons, [21] and the observation of
the particular geometry of the scattered SWs after passing through magnetic
solitons [14] are simple examples of the rich tapestry of phenomena in the
fields. Interestingly, quantum effects can play a crucial role in the physics
of macroscopic assemblies of magnons, leading to the collective behaviour in
the form of Bose-Einstein condensates even at room temperature. [6, 30, 33,
32] Such a variety of phenomena has attracted interest from a diversity of
sources regarding its potential applications in technological SW devices. [36,
24, 10, 8] The different contributions of the magnetic energy play a different
role depending on the scales associated with the SW phenomena. It is from this
perspective that the SW wavelength $\lambda$ can be used to separate the
behaviour into two regimes. [19] Short wavelengths ($\lambda<1\;\mu m$) are
dominated by the short range exchange interaction. This kind to SWs are named
exchange SWs. On the other hand, in the behaviour of SWs of longer wavelengths
the physics is dominated by the long range dipolar interaction, in this regime
we talk about dipolar or magnetostatic SWs. In this paper we will study SW
phenomena of short wavelength and the physical behaviour will be dominated by
the exchange interaction. Our basic model will, therefore, start with an
analysis of the SW phenomena in presence only of the exchange interaction,
leaving all the effects arising from other energy contributions in the form of
perturbations.

Our main focus will be in the interplay between SWs and a background magnetic
texture. This effect has been studied in several papers in the context of SW
propagation across domain walls, vortices and skyrmions. [18, 28, 13, 22, 23]
A smooth texture affects the dynamics of a SW by changing the basic wave
equation that rules the SW behaviour. A very interesting idea has been put
forward by Dugaev et al. [13] that showed that the magnetic texture induces
Berry phases [4] and Berry-curvatures and applied these ideas to study the
Aharanov-Bohm-like [1] effect of the magnetization in a magnetic ring with a
non-trivial magnetic configuration. Our analysis start from the exchange
Hamiltonian and derive an effective Hamiltonian for the SWs that is equivalent
to a $O(2)$ gauge theory. The role of the Berry curvature predicted by Dugaev
et al. enters in our theory in the form a $O(2)$ vector potential. In addition
to the Berry connection induced by the background spin texture we find an
additional term that resembles a scalar potential. From this perspective we
derive a Noether current that is conserved. We study under what circumstances
the behaviour of the effective theory can be cast in the form of a
Schrödinger-like equation for the magnon amplitude. In this way we can easily
derive the corresponding effective magnetic field felt by the SWs because of
the texture. After a revision of the insights gained putting SWs in a gauge-
invariant form, we focus on the particular texture given by a three
dimensional soliton, with non zero topological charge, known as Bloch point
(BP). [12, 15] After evaluating the different potentials, we conclude that the
basic physics of the SWs in presence of a BP corresponds to an effective
theory of an electron in presence of a magnetic monopole. Our main result is
that in the context of the interaction of SWs with a BP the outgoing pattern
correspond to a SW vortex. This interesting result can be studied from an
experimental point of view by using the technique of magnon-holography [24].

Figure 1: Different Bloch point singularities. (a) Bloch point with
$\gamma=0$, (b) Bloch point with $\gamma=\pi/2$ and (c) effective magnetic
field seen by the SWs. We have highlighted the singularities associated with
the Dirac strings along the $\pm\bm{z}$-axis.

##  2 Gauge theory from isotropic ferromagnet

We consider a system of classical spins parameterised by a dimensionless
vector field $\bm{S}(\bm{r})=S\bm{n}(\bm{r})$, where $\hbar S$ (with $\hbar$
the reduced Planck constant) is the molecular spin. We will restrict ourself
to the exchange energy describing an isotropic ferromagnet and neglecting the
dipolar and boundary terms. The exchange energy is explicitly ${\cal
E}=\int({\rm d}V/2a)J(\nabla\bm{S})^{2}$, with $J$ being the exchange energy
constant, $a$ the lattice parameter and the integral extends over the whole
volume. Assuming a given magnetic texture, $\bm{n}_{0}(\bm{r})$, to be a
solution that minimizes the energy functional, we are interested in the
fluctuations (SWs) around it. We consider the field
$\bm{n}(\bm{r},t)=\bm{n}_{0}(\bm{r})+\bm{\eta}(\bm{r},t)$, where
$\bm{\eta}(\bm{r},t)$ can be considered small and will encode the SWs
dynamics. As fluctuations of a normalised vector field SWs are always in the
plane normal to the magnetization. To take advantage of this constraint we use
the local transformation $\pazocal{R}^{-1}$ that rotates the magnetic texture
and put it in the $z$-axis as $\pazocal{R}^{-1}\bm{n}_{0}=\hat{\bm{z}}$. After
the transformation, the fluctuations are in the normal plane to $z$ and can be
written using a three-dimensional vector
$\bm{\omega}=(\omega_{1},\omega_{2},\omega_{3})$ defined by the relation
$\pazocal{R}^{-1}\bm{\eta}=\hat{\bm{z}}\times\bm{\omega}=(-\omega_{2},\omega_{1},0)$.
This relation shows us that, as a consequence of the normalization, there are
only two-components of $\bm{\omega}$ that counts for the SW dynamics and
therefore it is enough to considering a two component field
$\bm{\omega}\equiv(\omega_{1},\omega_{2})$. In this way we can express the
normalised magnetization vector as
$\bm{n}=\pazocal{R}\hat{\bm{z}}+\pazocal{R}(\hat{\bm{z}}\times\bm{\omega})$
and expand the exchange density $\epsilon=(JS^{2}/2a)(\nabla\bm{n})^{2}$ until
the second order in the fluctuations. Using the fact that $\bm{n}_{0}$ is a
local minimum of the functional and the unitarity of the rotation matrix, that
is to say $\pazocal{R}^{T}\pazocal{R}=I$ we find the following SW Hamiltonian:

$\pazocal{E}=\frac{\hbar^{2}}{2m^{*}}((-i\vec{\nabla}+\vec{\pazocal{A}})\bm{\omega})^{2}-\bm{\omega}^{T}\pazocal{V}\bm{\omega},$
(1)

where $m^{*}=\hbar^{2}/(4JSa^{2})$ is the effective mass and the gauge
(tensor) potential $\vec{\pazocal{A}}=-i\pazocal{R}_{\alpha
2}(\vec{\nabla}\pazocal{R}_{\alpha 1})\tau$ (summation over repeated indices
is assumed), with $\tau=-i\sigma_{y}$ where $\sigma_{y}$ is the corresponding
Pauli matrix. In this way $\tau$ the generator of the group O(2). It is worth
to note that in Eq. (1) the square means the vector product and not the square
of the complex norm. Explicitly the $\pazocal{V}=(\hbar^{2}/4m^{*})\nu$
potential is

$\nu=\left(\begin{array}[]{ c c }(\pazocal{R}_{\alpha
2}\vec{\nabla}\pazocal{R}_{\alpha 1})^{2}-(\vec{\nabla}\pazocal{R}_{\alpha
2})^{2}&-\vec{\nabla}\pazocal{R}_{\alpha
1}\cdot\vec{\nabla}\pazocal{R}_{\alpha 2}\\\ -\vec{\nabla}\pazocal{R}_{\alpha
1}\cdot\vec{\nabla}\pazocal{R}_{\alpha 2}&(\pazocal{R}_{\alpha
2}\vec{\nabla}\pazocal{R}_{\alpha 1})^{2}-(\vec{\nabla}\pazocal{R}_{\alpha
1})^{2}\end{array}\right)$

We can exploit the symmetry of the problem writing the rotation matrix
$\pazocal{R}$ in function of the Euler rotation angles $\Theta$, $\Phi$ and
$\Psi$ of the original magnetic texture
$\bm{n}_{0}=(\sin\Theta\cos\Phi,\sin\Theta\sin\Phi,\cos\Theta)$. In this
representation both potentials take a simple form:

$\vec{\pazocal}{A}=i\bm{A}\tau,$ (2)

where explicitly, the gauge vector potential field $\bm{A}$ is:

$\bm{A}=\cos\Theta\nabla\Phi+\nabla\Psi,$ (3)

and the $\pazocal{V}$ potential can be written as
$\pazocal{V}=S^{T}V(\Theta,\Phi)S$, where $S=e^{\Psi\tau}$ is a rotation in
the gauge coordinate $\Psi$ and the ¡¡scalar¿¿ potential is

$V=\frac{\hbar^{2}}{4m^{*}}\left(\begin{array}[]{ c c
}(\nabla\Theta)^{2}&-\sin\Theta\>\nabla\Theta\cdot\nabla\Phi\\\
-\sin\Theta\>\nabla\Theta\cdot\nabla\Phi&\sin^{2}\Theta(\nabla\Phi)^{2}\end{array}\right).$
(4)

In these equations the gauge symmetry is explicit, $\Psi$ is the gauge freedom
coming from the fact that we can locally rotate $\bm{n}_{0}$ around itself
without changing the physics. This symmetry endows a conservation law given by
the current:
$\vec{j}=(\hbar/m^{*})(i\vec{\nabla}-\vec{\pazocal{A}})\bm{\omega}^{T}\tau\bm{\omega}$.
This current is the best definition of the spin wave current.

In a family of textures one can go farther along this analogy. If the
equations $(\nabla\Theta)^{2}=\sin^{2}\Theta(\nabla\Phi)^{2}$ and
$\nabla\Theta\cdot\nabla\Phi=0$ are satisfied, the potential (4) becomes a
multiple of the identity. These identities hold for the case of a Bloch point
that we are investigating. Interestingly, if that is the case, if we define
$\Psi=\omega_{1}+i\omega_{2}$ we can write an equation for a unique complex
field without anomalous terms as a Schrödinger-like functional:

$\hat{H}_{\text{eff}}=\frac{\hbar^{2}}{2m^{*}}|(-i\nabla+\bm{A})\Psi|^{2}-V|\Psi|^{2}$
(5)

where the gauge field $\bm{A}$ can be considered a vector potential giving an
effective magnetic field felt by a quantum particle.

The conditions over the magnetization field that lead to $V\propto I$ are
stringent. Nevertheless there are many particular cases where this condition
holds and it has been exploited in the literature before. [18, 22] It is
interesting to point out that this kind of analogy has been exploited even in
fluid dynamics. [9] Put it in the present form it is easy to recognize in
which cases we can write a Schrödinger equations for SWs. For the effective
Hamiltonian Eq. (5) there is a Noether conserved current done by
$\bm{j}=(\hbar/m^{*})[2\operatorname{Im}(\Psi^{\dagger}\nabla\Psi)-\bm{A}|\Psi|^{2}]$.

##  3 Bloch point SWs and the Aharonov-Bohm effect

We can address now the problem of the scattering and SWs interference around a
Bloch point (BP). The BP is characterised by a magnetization field that covers
the whole sphere around the center of the soliton, which is, therefore,
singular at that point for a normalised magnetization field. Its form, see
Fig.(2), is given by the field $\Theta(\bm{r})=p\theta+\gamma$,
$\Phi(\bm{r})=q\phi$, where $(\theta,\phi)$ are the spherical coordinates of
the space and $p$ and $q$ are integers that give the topological charge $Q=pq$
of the BP. [12]

Here we will choose for simplicity $p=1$, so $Q=q$. The conditions given a
Schrödinger equations are satisfied for BPs and the effective Hamiltonian for
the linear oscillations of SWs around the BP can be written as in the form of
Eq. (5), with $\mathbf{A}=\frac{q\cot\theta}{r}\hat{\phi}$ and
$V(r)=-\frac{\hbar^{2}}{4m}\frac{q^{2}}{r^{2}}$. [14] In this way, around the
BP, we have that the effective magnetic field is given by a monopole-like
magnetic field (known as Dirac monopole [11]) generated by a ¡¡magnetic
charge¿¿ $q$. The vector potential is singular on a line whose location
depends on the chosen gauge. In the present gauge we have
$\nabla\cdot\bm{A}=0$, the so-called Coulomb gauge, the singularity line is
along the whole $z$ axis and the potential is symmetric, see Fig. (2), this is
known as the Schwinger’s potential. [29, 31]

We can use the fact that when SW propagates the wave function acquires a
magnetic phase. [1] The phase difference between two interfering arbitrary
paths can be calculated considering the integral of the potential vector:

$\Delta\phi_{AB}=\oint\bm{A}\cdot\bm{ds},$ (6)

where $\bm{ds}$ is an infinitesimal part of the trajectory. From Stoke’s
theorem the expression in Eq. (6) is the magnetic flux through the surface
defined by the closed loop. The two paths can be continuously deformed in
order to enclose a solid angle $\Omega$ of a spherical surface. Thus, the
magnetic flux enclosed by the two paths can be given by $\Delta\Phi=\int
qd\Omega=q\phi\int\sin\theta d\theta$, where $\theta$ is the polar angle. It
has been shown [35, 27, 34, 17, 3] that this quantity is directly related with
the magnetic charge, giving

$\Delta\phi_{AB}=2q\phi,$ (7)

which is equivalent in our case to the statement that there is an Aharonov-
Bohm (AB) phase directly proportional to the topological charge of the BP.

Figure 2: Scattering of a spin wave by a magnetic Bloch point. The problem is
equivalent to the scattering of a charged object by a magnetic monopole. In
both cases the outgoing wave has the form of a vortex.

The BP field leads to an integer $m$ ($q=m/2$), resulting in a perfect phase
vortex of topological charge $m$. Thus, the interaction of a plane SW with a
BP enables the typical azimuthal AB phase shift to occur and vortex SW states
to be created. The Noether current can be readily calculated
$\bm{j}_{\phi}=-(\hbar/m^{*})|\Psi|^{2}(q(1-\cos\theta)/(r\sin\theta))$.

The analytical solution to this problem is known from the theory of magnetic
monopoles [29, 5] and can be written as
$\Psi(r,\theta)=\sum_{l=q}^{\infty}\sum_{m=-l}^{l}F_{l}(r,\theta)e^{im\phi}$,
where $F_{l}(r,\theta)$ is a complicated function including the generalised
spherical harmonics and the spherical Bessel functions (see [14] and the
citations there in) and $m$ an integer in the shown interval. It is worth to
note that $m$ is directly related with the topological charge of the BP as the
sum begins with $l=q$, meaning that in a non-trivial topology ($q\neq 0$)
there are always modes with $m\neq 0$. Just as in the monopole case, [3] the
SW wave function can be written as: [29]

$\displaystyle\Psi_{\text{out}}(\bm{r})\sim\frac{1}{r}e^{ikr}f(\theta)e^{iq\phi}=\Psi(r,\theta)e^{im\phi},$
(8)

known as a vortex state.

##  4 Magnon holography

In order to test these results, an experiment using holography can be used.
Holography is the technique in which we produce the interference between an
arbitrary beam with a coherent one, which serves as reference. [16] The
interference pattern allows us to obtain the whole information about the
original beam. Magnon holography is based upon the same principle by using SWs
and it has recently appeared as a convenient technique to study SWs and to the
rise of new technologies for data processing. [24] In the present case we take
advantage of the known properties of the interference of SWs to propose a
mechanism to detect BPs. To do so, let us suppose and incident plane wave (IW)
of SWs has been prepared to propagate in the $z$ direction and to pass across
the BP. The phase distribution on this plane changes according to Eq. (8) when
a BP is set in the middle of the path. This change could be recorded by using
the hologram principle by making the incident wave (IW) interfere with a
reference plane wave (RW) $\psi_{R}=Ae^{i\bm{k}\cdot\bm{r}}$ that does not
pass in the BP region and having the same momentum and SW amplitude. The RW
propagates in a different direction, with an angle $\varphi$ with respect to
the SW passing through the BP. Reconstruction of the IW in an optical-like
form could be performed such as a hologram by using Brillouin light
scattering. [2] The phase of the IW is also reproduced and could be visualised
by means of the interference between this wave and another optical wave.
Visualisation is in the form of a set of equal-phase lines, i.e., a contour
map of the spatial distribution of the magnetostatic volume wave modes that
is, the ¡¡phase changes¿¿ when the RW is adjusted to be the same as the object
wave for an empty specimen (in the present case, a ferromagnetic state). Then,
the comparison wave is tilted and as well as an electron interference pattern,
[3] we have an interferogram consisting of fringes. Thus, the magnon vortices
could be seen by using an apparatus similar to that one proposed by Fukuhara
et al.[17] in the context of electron scattering, associated to the
experimental arrangement given in Ref. [2], which uses the Brillouin
scattering setup for the optical detection of the component of the
magnetization.

Considering that the IW and the RW have the same amplitude and initial phases,
and assuming that IW is perpendicularly incident to the recording plane,
parameterised by the coordinates $(x,y)$, the intensity of interference
pattern is given, in the proximity of the z-axis is:

$\frac{I}{2\mathscr{A}}=1+\cos(kx\sin\varphi-q\phi),$ (9)

where $\mathscr{A}$ is twice the square of the amplitude of the waves (that we
considerer the same for both) and the direction of propagation of the IW is
the axis $z$ and the k-vector of the RW lies, without lose of generality, in
the $(x,z)$ plane (see Fig. 2). The interference patterns for different angles
between the beams are shown in Fig. 3. It can be noted that the interference
pattern looks like a dislocation defect in a solid, showing the presence of a
singularity in the system. Such singularity is associated with the string
appearing in the potential vector. If the RW is going in the opposite
direction of the IW ($\varphi=\pi$), the interference pattern is radial,
showing the singularity in the center of the scattered SW vortex. The
$\varphi$ angle between the two waves changes the scale of the pattern, but
the dislocation remains, which makes the dislocation pattern robust.

Figure 3: (a) Magnon current associated with a magnon vortex. The plane of the
vortex is perpendicular to the incoming wave. (b) Interference pattern between
the magnon scattered by a BP for $q=1$ and a free magnon forming with the
former an angle $\varphi=\pi/2$. $\lambda$ is ten times the wavelength of the
waves.

##  5 Conclusions

In this paper we predict the appearance of a SW vortex for SWs propagating in
the presence of a BP and we propose a way to detect using magnon holography.
To study the SWs in the presence of a magnetic texture in a isotropic magnet
we work in the local rotated frame where magnetization is in the ferromagnetic
state. In this frame we were allowed to show that the effective theory for the
SWs is a gauge theory with O(2) symmetry. We showed in which cases the texture
support SWs that have a dynamics equivalent to that of a quantum particle one
where the information about the texture is in the scalar and vector potential
of the Schrödinger hamiltonian. For BPs the non-trivial topology of the
texture gives a singular potential of a magnetic monopole and the azimuthal
part of the scattered SW reflects the singularity by developing a vortex
state. This can be indirectly detected by the interference of a free SW with
the same wave vector, and the interference pattern has a robust and
recognisable structure. This measuring process can be applied to other SWs in
order to detect and control magnetic solitons.

##  6 Acknowledgments

R.G.E thanks Conicyt Pai/Concurso Nacional de Apoyo al Retorno de
Investigadores/as desde el Extranjero Folio 821320024. V.L.C.S. thanks the
Brazilian agency CNPq (Grant No. 229053/2013-0), for financial support. The
authors acknowledge funding from Proyecto Fondecyt No. 1150072, Proyecto Basal
FB0807-CEDENNA, and by Anillo de Ciencia y Tecnonología ACT 1117.

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# Convergence Analysis of Projection Method for Variational Inequalities

Yekini Shehu111Department of Mathematics, Zhejiang Normal University, Jinhua,
321004, People’s Republic of China; Institute of Science and Technology (IST),
Am Campus 1, 3400, Klosterneuburg, Vienna, Austria; e-mail:
<EMAIL_ADDRESS>Olaniyi. S. Iyiola222Department of Mathematics,
Minnesota State University-Moorhead, Minnesota, USA; e-mail:
<EMAIL_ADDRESS>Xiao-Huan Li333College of Science, Civil Aviation
University of China, Tianjin 300300, China.; e-mail<EMAIL_ADDRESS>and Qiao-Li Dong444College of Science, Civil Aviation University of China,
Tianjin 300300, China.; e-mail<EMAIL_ADDRESS>
(January 22, 2021)

###### Abstract

The main contributions of this paper are the proposition and the convergence
analysis of a class of inertial projection-type algorithm for solving
variational inequality problems in real Hilbert spaces where the underline
operator is monotone and uniformly continuous. We carry out a unified analysis
of the proposed method under very mild assumptions. In particular, weak
convergence of the generated sequence is established and nonasymptotic
$O(1/n)$ rate of convergence is established, where $n$ denotes the iteration
counter. We also present some experimental results to illustrate the profits
gained by introducing the inertial extrapolation steps.

##  1 Introduction

We first state the formal definition of some classes of functions that play an
essential role in this paper.

Let $H$ be a real Hilbert space and $X\subseteq H$ be a nonempty subset.

######  Definition 1.1.

A mapping $F:X\to H$ is called

  * (a)

monotone on $X$ if $\langle F(x)-F(y),x-y\rangle\geq 0$ for all $x,y\in X$;

  * (b)

Lipschitz continuous on $X$ if there exists a constant $L>0$ such that

$\|F(x)-F(y)\|\leq L\|x-y\|,\ \forall x,y\in X.$

  * (c)

sequentially weakly continuous if for each sequence $\\{x_{n}\\}$ we have:
$\\{x_{n}\\}$ converges weakly to $x$ implies $\\{F(x_{n})\\}$ converges
weakly to $F(x)$.

Let $C$ be a nonempty, closed and convex subset of $H$ and $F:C\rightarrow H$
be a continuous mapping. The variational inequality problem (for short,
VI($F,C$)) is defined as: find $x\in C$ such that

$\displaystyle\langle F(x),y-x\rangle\geq 0,\quad\forall y\in C.$ (1)

Let SOL denote the solution set of VI($F,C$) (1). Variational inequality
theory is an important tool in economics, engineering mechanics, mathematical
programming, transportation, and so on (see, for example, [7, 8, 22, 29, 30,
31, 38]).

A well-known projection-type method for solving VI($F,C$) (1) is the
extragradient method introduced by Korpelevich in [32]. It is well known that
the extragradient method requires two projections onto the set $C$ and two
evaluations of $F$ per iteration.

One important hallmark in the design of numerical methods related to the
extragradient method is to minimize the number of evaluations of $P_{C}$ per
iteration because if $C$ is a general closed and convex set, then a minimal
distance problem has to be solved (twice) in order to obtain the next iterate.
This has the capacity to seriously affect the efficiency of the extragradient
method in a situation, where a projection onto $C$ is hard to evaluate and
therefore computationally costly.

An attempt in this direction was initiated by Censor et al.[18], who modified
extragradient method by replacing the second projection onto the closed and
convex subset $C$ with the one onto a subgradient half-space. Their method,
which therefore uses only one projection onto $C$, is called the subgradient
extragradient method: $x_{1}\in H$,

$\displaystyle\left\\{\begin{array}[]{llll}&y_{n}=P_{C}(x_{n}-\lambda
F(x_{n})),\\\ &T_{n}:=\\{w\in H:\langle x_{n}-\lambda
F(x_{n})-y_{n},w-y_{n}\rangle\leq 0\\},\\\ &x_{n+1}=P_{T_{n}}(x_{n}-\lambda
F(y_{n})).\end{array}\right.$ (5)

Using (5), Censor et al. [18] proved weak convergence result for VI($F,C$) (1)
with a monotone and $L$-Lipschitz-continuous mapping $F$ where
$\lambda\in(0,\frac{1}{L})$. Several other related methods to extragradient
method and (5) for solving VI($F,C$) (1) in real Hilbert spaces when $F$ is
monotone and $L$-Lipschitz-continuous mapping have been studied in the
literature (see, for example, [15, 16, 17, 21, 26, 35, 36, 37, 40, 46]).

Motivated the result of Alvarez and Attouch in [2] and Censor et al. in [18],
Thong and Hieu [45] introduced an algorithm which is a combination of (5) and
inertial method for solving VI($F,C$) (1) in real Hilbert space:
$x_{0},x_{1}\in H$,

$\displaystyle\left\\{\begin{array}[]{llll}&w_{n}=x_{n}+\alpha_{n}(x_{n}-x_{n-1}),\\\
&y_{n}=P_{C}(w_{n}-\lambda F(w_{n})),\\\ &T_{n}:=\\{w\in H:\langle
w_{n}-\lambda F(w_{n})-y_{n},w-y_{n}\rangle\leq 0\\},\\\
&x_{n+1}=P_{T_{n}}(w_{n}-\lambda F(y_{n})).\end{array}\right.$ (10)

Thong and Hieu [45] proved that the sequence $\\{x_{n}\\}$ generated by (10)
converges weakly to a solution of VI($F,C$) (1) with a monotone and
$L$-Lipschitz-continuous mapping $F$ where $0<\lambda
L\leq\frac{\frac{1}{2}-2\alpha-\frac{1}{2}\alpha^{2}-\delta}{\frac{1}{2}-\alpha+\frac{1}{2}\alpha^{2}}$
for some $0<\delta<\frac{1}{2}-2\alpha-\frac{1}{2}\alpha^{2}$ and
$\\{\alpha_{n}\\}$ is a non-decreasing sequence with
$0\leq\alpha_{n}\leq\alpha<\sqrt{5}-2$.

One the main features of the above mentioned methods (5), (10) and other
related methods is the computational issue, for example, step-sizes. The step-
sizes in these above methods are bounded by the inverse of the Lipschitz
constant which is quite inefficient, because in most cases a global Lipschitz
constant (if it indeed exists) of $F$ cannot be accurately estimated, and is
usually overestimated, thereby resulting in too small step-sizes. This, of
course, is not practical. Therefore, algorithms (5) and (10) are not
applicable in most cases of interest. The usual approach to overcome this
difficulty consists in some prediction of a step-size with its further
correction (see [29, 38]) or in a usage of an Armijo type line search
procedure along a feasible direction (see [43]). In terms of computations, the
latter approach is more effective, since very often the former approach
requires too many projections onto the feasible set per iteration.

This paper focuses on the analysis and development of computational
projection-type algorithm with inertial extrapolation step for solving
VI($F,C$) (1) when the underline operator $F$ is monotone and uniformly
continuous when the feasible set $C$ is a nonempty closed affine subset. We
obtain weak convergence of the sequence generated by our method. We provide
theoretical analysis of our result with weaker assumption on the underline
operator $F$ unlike [17, 18, 35, 36] and many other related results on
monotone variational inequalities. We also establish the nonasymptotic
$O(1/n)$ rate of convergence, which is not given before in other previous
inertial type projection methods for VI($F,C$) (1) (see, e.g.,[21, 45]) and
give carefully designed computational experiments to illustrate our results.
Our computational results show that our proposed methods outperform the
iterative methods (5) and (10). Furthermore, our result complements some
recent results on inertial type algorithms (see, e.g., [2, 3, 4, 5, 6, 10, 12,
13, 19, 33, 34, 41, 42]).

The paper is organized as follows: We first recall some basic definitions and
results in Section 2. Some discussions about the proposed inertial projection-
type method are given in Section 3. The weak convergence analysis of our
algorithm is then investigated in Section 4. We give the rate of convergence
of our proposed method in Section 5 and some numerical experiments can be
found in Section 6. We conclude with some final remarks in Section 7.

##  2 Preliminaries

First, we recall some properties of the projection, cf. [9] for more details.
For any point $u\in H$, there exists a unique point $P_{C}u\in C$ such that

$\|u-P_{C}u\|\leq\|u-y\|,\leavevmode\nobreak\ \forall y\in C.$

$P_{C}$ is called the metric projection  of $H$ onto $C$. We know that $P_{C}$
is a nonexpansive mapping of $H$ onto $C$. It is also known that $P_{C}$
satisfies

$\langle
x-y,P_{C}x-P_{C}y\rangle\geq\|P_{C}x-P_{C}y\|^{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \forall x,y\in H.$ (11)

In particular, we get from (11) that

$\langle x-y,x-P_{C}y\rangle\geq\|x-P_{C}y\|^{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \forall x\in C,y\in H.$ (12)

Furthermore, $P_{C}x$ is characterized by the properties

$P_{C}x\in C\quad\text{and}\quad\langle x-P_{C}x,P_{C}x-y\rangle\geq
0,\leavevmode\nobreak\ \forall y\in C.$ (13)

Further properties of the metric projection can be found, for example, in
Section 3 of [23].

The following lemmas will be used in our convergence analysis.

######  Lemma 2.1.

The following statements hold in $H$:

  * (a)

$\|x+y\|^{2}=\|x\|^{2}+2\langle x,y\rangle+\|y\|^{2}$ for all $x,y\in H$;

  * (b)

$2\langle x-y,x-z\rangle=\|x-y\|^{2}+\|x-z\|^{2}-\|y-z\|^{2}$ for all
$x,y,z\in H$;

  * (c)

$\|\alpha
x+(1-\alpha)y\|^{2}=\alpha\|x\|^{2}+(1-\alpha)\|y\|^{2}-\alpha(1-\alpha)\|x-y\|^{2}$
for all $x,y\in H$ and $\alpha\in\mathbb{R}$.

######  Lemma 2.2.

(see [1, Lem. 3]) Let $\\{\psi_{n}\\}$, $\\{\delta_{n}\\}$ and
$\\{\alpha_{n}\\}$ be the sequences in $[0,+\infty)$ such that
$\psi_{n+1}\leq\psi_{n}+\alpha_{n}(\psi_{n}-\psi_{n-1})+\delta_{n}$ for all
$n\geq 1$, $\sum_{n=1}^{\infty}\delta_{n}<+\infty$ and there exists a real
number $\alpha$ with $0\leq\alpha_{n}\leq\alpha<1$ for all $n\geq 1$. Then the
following hold:
$(i)\leavevmode\nobreak\ \leavevmode\nobreak\ \sum_{n\geq
1}[\psi_{n}-\psi_{n-1}]_{+}<+\infty$, where $[t]_{+}=\max\\{t,0\\}$;
(ii) there exists $\psi^{*}\in[0,+\infty)$ such that
$\lim_{n\rightarrow+\infty}\psi_{n}=\psi^{*}$.

######  Lemma 2.3.

(see [9, Lem. 2.39]) Let $C$ be a nonempty set of $H$ and $\\{x_{n}\\}$ be a
sequence in $H$ such that the following two conditions hold:
(i) for any $x\in C$, $\lim_{n\rightarrow\infty}\|x_{n}-x\|$ exists;
(ii) every sequential weak cluster point of $\\{x_{n}\\}$ is in $C$.
Then $\\{x_{n}\\}$ converges weakly to a point in $C$.

The following lemmas were given in $\mathbb{R}^{n}$ in [25]. The proof of the
lemmas are the same if given in infinite dimensional real Hilbert spaces.
Hence, we state the lemmas and omit the proof in real Hilbert spaces.

######  Lemma 2.4.

Let $C$ be a nonempty closed and convex subset of $H$. Let $h$ be a real-
valued function on $H$ and define $K:=\\{x:h(x)\leq 0\\}$. If $K$ is nonempty
and $h$ is Lipschitz continuous on $C$ with modulus $\theta>0$, then

${\rm dist}(x,K)\geq\theta^{-1}\max\\{h(x),0\\},\leavevmode\nobreak\ \forall
x\in C,$

where ${\rm dist}(x,K)$ denotes the distance function from $x$ to $K$.

######  Lemma 2.5.

Let $C$ be a nonempty closed and convex subset of $H$, $y:=P_{C}(x)$ and
$x^{*}\in C$. Then

$\|y-x^{*}\|^{2}\leq\|x-x^{*}\|^{2}-\|x-y\|^{2}.$ (14)

The following lemma was stated in [28, Prop. 2.11], see also [27, Prop. 4].

######  Lemma 2.6.

Let $H_{1}$ and $H_{2}$ be two real Hilbert spaces. Suppose
$F:H_{1}\rightarrow H_{2}$ is uniformly continuous on bounded subsets of
$H_{1}$ and $M$ is a bounded subset of $H_{1}$. Then $F(M)$ is bounded.

Finally, the following result states the equivalence between a primal and a
weak form of variational inequality for continuous, monotone operators.

######  Lemma 2.7.

([44, Lem. 7.1.7]) Let $C$ be a nonempty, closed, and convex subset of $H$.
Let $F:C\rightarrow H$ be a continuous, monotone mapping and $z\in C$. Then

$z\in{\rm SOL}\Longleftrightarrow\langle F(x),x-z\rangle\geq 0\quad\text{for
all }x\in C.$

##  3 Inertial Projection-type Method

Let us first state the assumptions that we will assume to hold for the rest of
this paper.

######  Assumption 3.1.

Suppose that the following hold:

  * (a)

The feasible set $C$ is a nonempty closed affine subset of the real Hilbert
space $H$.

  * (b)

$F:C\to H$ is monotone and uniformly continuous on bounded subsets of $H$.

  * (c)

The solution set SOL of VI$(F,C)$ is nonempty.

######  Assumption 3.2.

Suppose the real sequence $\\{\alpha_{n}\\}$ and constants
$\beta,\delta,\sigma>0$ satisfy the following conditions:

  * (a)

$\\{\alpha_{n}\\}\subset(0,1)$ with
$0\leq\alpha_{n}\leq\alpha_{n+1}\leq\alpha<1$ for all $n$.

  * (b)

$\delta>\frac{\alpha(1+\alpha)(\alpha+\delta\sigma)+\alpha\sigma\delta(\alpha+\delta\sigma)}{\sigma}$
and
$\beta<\frac{\delta\sigma}{\alpha+\delta\sigma}-\alpha(1+\alpha)-\alpha\sigma\delta$.

Let

$r(x):=x-P_{C}(x-F(x))$

stand for the residual equation.

Observe that if we take $y=x-F(x)$ in (12), then we have

$\langle F(x),r(x)\rangle\geq\|r(x)\|^{2},\leavevmode\nobreak\ \forall x\in
C.$ (15)

We now introduce our proposed method below.

Algorithm 1 Inertial Projection-type Method

1:Choose sequence $\\{\alpha_{n}\\}$ and $\sigma\in(0,1)$ such that the
conditions from Assumption 3.2 hold, and take $\gamma\in(0,1)$. Let
$x_{0}=x_{1}\in H$ be a given starting point. Set $n:=1$.

2:Set  $w_{n}:=x_{n}+\alpha_{n}(x_{n}-x_{n-1}).$ Compute
$z_{n}:=P_{C}(w_{n}-F(w_{n}))$. If $r(w_{n})=w_{n}-z_{n}=0$: STOP.

3:Compute $y_{n}=w_{n}-\gamma^{m_{n}}r(w_{n})$, where $m_{n}$ is the smallest
nonnegative integer satisfying  $\langle
F(y_{n}),r(w_{n})\rangle\geq\frac{\sigma}{2}\|r(w_{n})\|^{2}.$ (16) Set
$\eta_{n}:=\gamma^{m_{n}}$.

4:Compute  $x_{n+1}=P_{C_{n}}(w_{n}),$ (17) where $C_{n}=\\{x:h_{n}(x)\leq
0\\}$ and  $h_{n}(x):=\langle F(y_{n}),x-y_{n}\rangle.$ (18)

5:Set $n\leftarrow n+1$ and goto 2.

It is clear that $r(w_{n})=0$ implies that we are at a solution of the
variational inequality. In our convergence theory, we will implicitly assume
that this does not occur after finitely many iterations, so that Algorithm 1
generates an infinite sequence satisfying, in particular, $r(w_{n})\neq 0$ for
all $n\in\mathbb{N}$. We will see that this property implies that Algorithm 1
is well defined.

######  Remark 3.3.

(a) Algorithm 1 requires, at each iteration, only one projection onto the
feasible set $C$ and another projection onto the half-space $C_{n}$ (see [14]
for formula for computing projection onto half-space), which is less expensive
than the extragradient method especially for the case when computing the
projection onto the feasible set $C$ is a dominating task during iteration.

(b) Our Algorithm 1 is much more applicable than (5) and (10) in the sense
that algorithm (5) and (10) are applicable only for monotone and
$L$-Lipschitz-continuous mapping $F$. Thus, the $L$-Lipschitz constant of $F$
or an estimate of it is needed in order to implement the iterative method (5)
but our Algorithm 1 is applicable for a much more general class of monotone
and uniformly continuous mapping $F$.

(c) We observe that the step-size rule in Step 3 involves a couple of
evaluations of $F$, but these are often much less expensive than projections
onto $C$ which was considered in [29, 38]. Furthermore, using the fact that
$F$ is continuous and (15), we can see that Step 3 in Algorithm 1 is well-
defined. $\Diamond$

######  Lemma 3.4.

Let the function $h_{n}$ be defined by (18). Then

$h_{n}(w_{n})\geq\frac{\sigma\eta_{n}}{2}\|w_{n}-z_{n}\|^{2}.$

In particular, if $w_{n}\neq z_{n}$, then $h_{n}(w_{n})>0$. If
$x^{*}\in\text{SOL}$, then $h_{n}(x^{*})\leq 0$.

###### Proof.

Since $y_{n}=w_{n}-\eta_{n}(w_{n}-z_{n})$, using (16) we have

$\displaystyle h_{n}(w_{n})$ $\displaystyle=\langle
F(y_{n}),w_{n}-y_{n}\rangle$ $\displaystyle=\eta_{n}\langle
F(y_{n}),w_{n}-z_{n}\rangle\geq\eta_{n}\frac{\sigma}{2}\|w_{n}-z_{n}\|^{2}\geq
0.$

If $w_{n}\neq z_{n}$, then $h_{n}(w_{n})>0$. Furthermore, suppose
$x^{*}\in\text{SOL}$. Then by Lemma 2.7 we have $\langle
F(x),x-x^{*}\rangle\geq 0\quad\text{for all }x\in C.$ In particular, $\langle
F(y_{n}),y_{n}-x^{*}\rangle\geq 0$ and hence $h_{n}(x^{*})\leq 0.$ ∎

Observe that, in finding $\eta_{n}$, the operator $F$ is evaluated (possibly)
many times, but no extra projections onto the set $C$ are needed. This is in
contrast to a couple of related algorithms for the solution of monotone
variational inequalities where the calculation of a suitable step-size
requires (possibly) many projections onto $C$, see, e.g., [20, 29, 46].

##  4 Convergence Analysis

We present our main result in this section. To this end, we begin with a
result that shows that the sequence $\\{x_{n}\\}$ generated by Algorithm 1 is
bounded under the given assumptions.

######  Lemma 4.1.

Let $\\{x_{n}\\}$ be generated by Algorithm 1. Then under Assumptions 3.1 and
3.2, we have that $\\{x_{n}\\}$ is bounded.

###### Proof.

Let $x^{*}\in\text{SOL}$. By Lemma 2.5 we get (since $x^{*}\in C_{n}$) that

$\displaystyle\|x_{n+1}-x^{*}\|^{2}$ $\displaystyle=$
$\displaystyle\|P_{C_{n}}(w_{n})-x^{*}\|^{2}\leq\|w_{n}-x^{*}\|^{2}-\|x_{n+1}-w_{n}\|^{2}$
$\displaystyle=$ $\displaystyle\|w_{n}-x^{*}\|^{2}-{\rm
dist}^{2}(w_{n},C_{n}).$

Now, using Lemma 2.1 (c), we have

$\displaystyle\|w_{n}-x^{*}\|^{2}$ $\displaystyle=$
$\displaystyle\|(1+\alpha_{n})(x_{n}-x^{*})-\alpha_{n}(x_{n-1}-x^{*})\|^{2}$
(20) $\displaystyle=$
$\displaystyle(1+\alpha_{n})\|x_{n}-x^{*}\|^{2}-\alpha_{n}\|x_{n-1}-x^{*}\|$
$\displaystyle+\alpha_{n}(1+\alpha_{n})\|x_{n}-x_{n-1}\|^{2}.$

Substituting (20) into (4), we have

$\displaystyle\|x_{n+1}-x^{*}\|^{2}$ $\displaystyle\leq$
$\displaystyle(1+\alpha_{n})\|x_{n}-x^{*}\|^{2}-\alpha_{n}\|x_{n-1}-x^{*}\|^{2}+\alpha_{n}(1+\alpha_{n})\|x_{n}-x_{n-1}\|^{2}$
(21) $\displaystyle-\|x_{n+1}-w_{n}\|^{2}.$

We also have (using Lemma 2.1 (a))

$\displaystyle\|x_{n+1}-w_{n}\|^{2}$ $\displaystyle=$
$\displaystyle\|(x_{x_{n}+1}-x_{n})-\alpha_{n}(x_{n}-x_{n-1})\|^{2}$ (22)
$\displaystyle=$
$\displaystyle\|x_{x_{n}+1}-x_{n}\|^{2}+\alpha^{2}\|x_{n}-x_{n-1}\|^{2}$
$\displaystyle-2\alpha_{n}\langle x_{x_{n}+1}-x_{n},x_{n}-x_{n-1}\rangle$
$\displaystyle\geq$
$\displaystyle\|x_{x_{n}+1}-x_{n}\|^{2}+\alpha^{2}\|x_{n}-x_{n-1}\|^{2}$
$\displaystyle+\alpha_{n}\Big{(}-\rho_{n}\|x_{x_{n}+1}-x_{n}\|^{2}-\frac{1}{\rho_{n}}\|x_{n}-x_{n-1}\|^{2}\Big{)},$

where $\rho_{n}:=\frac{1}{\alpha_{n}+\delta\sigma}$. Combining (21) and (22),
we get

$\displaystyle\|x_{n+1}-x^{*}\|^{2}-(1+\alpha_{n})\|x_{n}-x^{*}\|^{2}+\alpha_{n}\|x_{n-1}-x^{*}\|^{2}$
(23) $\displaystyle\leq$
$\displaystyle(\alpha_{n}\rho_{n}-1)\|x_{n+1}-x_{n}\|^{2}+\lambda_{n}\|x_{n}-x_{n-1}\|^{2},$

where

$\displaystyle\lambda_{n}:=\alpha_{n}(1+\alpha_{n})+\alpha_{n}\frac{1-\alpha_{n}\rho_{n}}{\rho_{n}}\geq
0$ (24)

since $\alpha_{n}\rho_{n}<1$. Taking into account the choice of $\rho_{n}$, we
have

$\delta=\frac{1-\alpha_{n}\rho_{n}}{\sigma\rho_{n}}$

and from (24), it follows that

$\displaystyle\lambda_{n}$ $\displaystyle=$
$\displaystyle\alpha_{n}(1+\alpha_{n})+\alpha_{n}\frac{1-\alpha_{n}\rho_{n}}{\rho_{n}}$
(25) $\displaystyle\leq$ $\displaystyle\alpha(1+\alpha)+\alpha\sigma\delta.$

Following the same arguments as in [1, 2, 11], we define
$\varphi_{n}:=\|x_{n}-x^{*}\|^{2},n\geq 1$ and
$\varepsilon_{n}:=\varphi_{n}-\alpha_{n}\varphi_{n-1}+\lambda_{n}\|x_{n}-x_{n-1}\|^{2},n\geq
1.$ By the monotonicity of $\\{\alpha_{n}\\}$ and the fact that
$\varphi_{n}\geq 0$, we have

$\varepsilon_{n+1}-\varepsilon_{n}\leq\varphi_{n+1}-(1+\alpha_{n})\varphi_{n}+\alpha_{n}\varphi_{n-1}+\lambda_{n+1}\|x_{n+1}-x_{n}\|^{2}-\lambda_{n}\|x_{n}-x_{n-1}\|^{2}.$

Using (23), we have

$\displaystyle\varepsilon_{n+1}-\varepsilon_{n}$ $\displaystyle\leq$
$\displaystyle(\alpha_{n}\rho_{n}-1)\|x_{n+1}-x_{n}\|^{2}+\lambda_{n}\|x_{n}-x_{n-1}\|^{2}$
(26)
$\displaystyle+\lambda_{n+1}\|x_{n+1}-x_{n}\|^{2}-\lambda_{n}\|x_{n}-x_{n-1}\|^{2}$
$\displaystyle=$
$\displaystyle(\alpha_{n}\rho_{n}-1+\lambda_{n+1})\|x_{n+1}-x_{n}\|^{2}.$

We now claim that

$\displaystyle\alpha_{n}\rho_{n}-1+\lambda_{n+1}\leq-\beta.$ (27)

Indeed by the choice of $\rho_{n}$, we have

$\displaystyle\alpha_{n}\rho_{n}-1+\lambda_{n+1}\leq-\beta$
$\displaystyle\Leftrightarrow$
$\displaystyle\alpha_{n}\rho_{n}-1+\lambda_{n+1}+\beta\leq 0$
$\displaystyle\Leftrightarrow$
$\displaystyle\lambda_{n+1}+\beta+\frac{\alpha_{n}}{\alpha_{n}+\delta\sigma}-1\leq
0$ $\displaystyle\Leftrightarrow$
$\displaystyle\lambda_{n+1}+\beta-\frac{\delta\sigma}{\alpha_{n}+\delta\sigma}\leq
0$ $\displaystyle\Leftrightarrow$
$\displaystyle(\alpha_{n}+\delta\sigma)(\lambda_{n+1}+\beta)\leq\delta\sigma$

Now, using (25), we have

$(\alpha_{n}+\delta\sigma)(\lambda_{n+1}+\beta)\leq((\alpha+\delta\sigma)(\alpha(1+\alpha)\alpha\delta\sigma+\beta)\leq\delta\sigma,$

where the last inequality follows from Assumption 3.2 (b). Hence, the claim in
(27) is true.

Thus, it follows from (26) and (27) that

$\displaystyle\varepsilon_{n+1}-\varepsilon_{n}\leq-\beta\|x_{n+1}-x_{n}\|^{2}.$
(28)

The sequence $\\{\varepsilon_{n}\\}$ is non-increasing and the bounds of
$\\{\alpha_{n}\\}$ delivers

$\displaystyle-\alpha\varphi_{n-1}\leq\varphi_{n}-\alpha\varphi_{n-1}\leq\varepsilon_{n}\leq\varepsilon_{1},n\geq
1.$ (29)

It then follows that

$\displaystyle\varphi_{n}\leq\alpha^{n}\varphi_{0}+\varepsilon_{1}\sum_{k=0}^{n-1}\alpha^{k}\leq\alpha^{n}\varphi_{0}+\frac{\varepsilon_{1}}{1-\alpha},n\geq
1.$ (30)

Combining (28) and (34), we get

$\displaystyle\beta\sum_{k=1}^{n}\|x_{k+1}-x_{k}\|^{2}$ $\displaystyle\leq$
$\displaystyle\varepsilon_{1}-\varepsilon_{n+1}$ (31) $\displaystyle\leq$
$\displaystyle\varepsilon_{1}+\alpha\varphi_{n}$ $\displaystyle\leq$
$\displaystyle\alpha^{n+1}\varphi_{0}+\frac{\varepsilon_{1}}{1-\alpha}$
$\displaystyle\leq$
$\displaystyle\varphi_{0}+\frac{\varepsilon_{1}}{1-\alpha},$

which shows that

$\displaystyle\sum_{k=1}^{\infty}\|x_{k+1}-x_{k}\|^{2}<\infty.$ (32)

Thus, $\underset{n\rightarrow\infty}{\lim}\|x_{n+1}-x_{n}\|=0$. From
$w_{n}=x_{n}+\alpha_{n}(x_{n}-x_{n-1})$, we have

$\displaystyle\|w_{n}-x_{n}\|$ $\displaystyle\leq$
$\displaystyle\alpha_{n}\|x_{n}-x_{n-1}\|$ $\displaystyle\leq$
$\displaystyle\alpha\|x_{n}-x_{n-1}\|\rightarrow 0,n\rightarrow\infty.$

Similarly,

$\|x_{n+1}-w_{n}\|\leq\|x_{n+1}-x_{n}\|+\|x_{n}-w_{n}\|\rightarrow
0,n\rightarrow\infty.$

Using Lemma 2.2, (23), (25) and (32), we have that
$\underset{n\rightarrow\infty}{\lim}\|x_{n}-x^{*}\|$ exists. Hence,
$\\{x_{n}\\}$ is bounded. ∎

In the next two lemmas, we show that certain subsequences obtained in
Algorithm 1 are null subsequences. These two lemmas are necessary in order to
show that the weak limit of $\\{x_{n}\\}$ is an element of $SOL$ and for our
weak convergence in Theorem 4.4 below.

######  Lemma 4.2.

Let $\\{x_{n}\\}$ generated by Algorithm 1 above and Assumptions 3.1 and 3.2
hold. Then

  * (a)

$\displaystyle\lim_{n\rightarrow\infty}\eta_{n}\|w_{n}-z_{n}\|^{2}=0$;

  * (b)

$\displaystyle\lim_{n\rightarrow\infty}\|w_{n}-z_{n}\|=0.$

###### Proof.

Let $x^{*}\in\text{SOL}$. Since $F$ is uniformly continuous on bounded subsets
of $X$, then $\\{F(x_{n})\\},\\{z_{n}\\},\\{w_{n}\\}$ and $\\{F(y_{n})\\}$ are
bounded. In particular, there exists $M>0$ such that $\|F(y_{n})\|\leq M$ for
all $n\in\mathbb{N}$. Combining Lemma 2.4 and Lemma 3.4, we get

$\displaystyle\|x_{n+1}-x^{*}\|^{2}$ $\displaystyle=$
$\displaystyle\|P_{C_{n}}(w_{n})-x^{*}\|^{2}\leq\|w_{n}-x^{*}\|^{2}-\|x_{n+1}-w_{n}\|^{2}$
(33) $\displaystyle=$ $\displaystyle\|w_{n}-x^{*}\|^{2}-{\rm
dist}^{2}(w_{n},C_{n})$ $\displaystyle\leq$
$\displaystyle\|w_{n}-x^{*}\|^{2}-\Big{(}\frac{1}{M}h_{n}(w_{n})\Big{)}^{2}$
$\displaystyle\leq$
$\displaystyle\|w_{n}-x^{*}\|^{2}-\Big{(}\frac{1}{2M}\sigma\eta_{n}\|r(w_{n})\|^{2}\Big{)}^{2}$
$\displaystyle=$
$\displaystyle\|w_{n}-x^{*}\|^{2}-\Big{(}\frac{1}{2M}\sigma\eta_{n}\|w_{n}-z_{n}\|^{2}\Big{)}^{2}.$

Since $\\{x_{n}\\}$ is bounded, we obtain from (33) that

$\displaystyle\Big{(}\frac{1}{2M}\sigma\eta_{n}\|w_{n}-z_{n}\|^{2}\Big{)}^{2}$
$\displaystyle\leq$ $\displaystyle\|w_{n}-x^{*}\|^{2}-\|x_{n+1}-x^{*}\|^{2}$
(34) $\displaystyle=$
$\displaystyle\Big{(}\|w_{n}-x^{*}\|-\|x_{n+1}-x^{*}\|\Big{)}\Big{(}\|w_{n}-x^{*}\|+\|x_{n+1}-x^{*}\|\Big{)}$
$\displaystyle\leq$ $\displaystyle\|w_{n}-x^{*}\|-\|x_{n+1}-x^{*}\|M_{1}$
$\displaystyle\leq$ $\displaystyle\|w_{n}-x_{n+1}\|M_{1},$

where $M_{1}:=\sup_{n\geq 1}\\{\|w_{n}-x^{*}\|+\|x_{n+1}-x^{*}\|\\}$. This
establishes (a).

To establish (b), We distinguish two cases depending on the behaviour of (the
bounded) sequence of step-sizes $\\{\eta_{n}\\}$.

Case 1: Suppose that $\liminf_{n\to\infty}\eta_{n}>0$. Then

$0\leq\|r(w_{n})\|^{2}=\frac{\eta_{n}\|r(w_{n})\|^{2}}{\eta_{n}}$

and this implies that

$\displaystyle\limsup_{n\to\infty}\|r(w_{n})\|^{2}$
$\displaystyle\leq\limsup_{n\to\infty}\bigg{(}\eta_{n}\|r(w_{n})\|^{2}\bigg{)}\bigg{(}\limsup_{n\to\infty}\frac{1}{\eta_{n}}\bigg{)}$
$\displaystyle=\bigg{(}\limsup_{n\to\infty}\eta_{n}\|r(w_{n})\|^{2}\bigg{)}\frac{1}{\liminf_{n\to\infty}\eta_{n}}$
$\displaystyle=0.$

Hence, $\limsup_{n\to\infty}\|r(w_{n})\|=0$. Therefore,

$\lim_{n\rightarrow\infty}\|w_{n}-z_{n}\|=\lim_{n\rightarrow\infty}\|r(w_{n})\|=0.$

Case 2: Suppose that $\liminf_{n\to\infty}\eta_{n}=0$. Subsequencing if
necessary, we may assume without loss of generality that
$\lim_{n\to\infty}\eta_{n}=0$ and
$\lim_{n\rightarrow\infty}\|w_{n}-z_{n}\|=a\geq 0$.

Define
$\bar{y}_{n}:=\frac{1}{\gamma}\eta_{n}z_{n}+\Big{(}1-\frac{1}{\gamma}\eta_{n}\Big{)}w_{n}$
or, equivalently, $\bar{y}_{n}-w_{n}=\frac{1}{\gamma}\eta_{n}(z_{n}-w_{n})$.
Since $\\{z_{n}-w_{n}\\}$ is bounded and since $\lim_{n\to\infty}\eta_{n}=0$
holds, it follows that

$\lim_{n\to\infty}\|\bar{y}_{n}-w_{n}\|=0.$ (35)

From the step-size rule and the definition of $\bar{y}_{k}$, we have

$\langle
F(\bar{y}_{n}),w_{n}-z_{n}\rangle<\frac{\sigma}{2}\|w_{n}-z_{n}\|^{2},\
\forall n\in\mathbb{N},$

or equivalently

$2\langle F(w_{n}),w_{n}-z_{n}\rangle+2\langle
F(\bar{y}_{n})-F(w_{n}),w_{n}-z_{n}\rangle<\sigma\|w_{n}-z_{n}\|^{2},\ \forall
n\in\mathbb{N}.$

Setting $t_{n}:=w_{n}-F(w_{n})$, we obtain form the last inequality that

$2\langle w_{n}-t_{n},w_{n}-z_{n}\rangle+2\langle
F(\bar{y}_{n})-F(w_{n}),w_{n}-z_{n}\rangle<\sigma\|w_{n}-z_{n}\|^{2},\ \forall
n\in\mathbb{N}.$

Using Lemma 2.1 (b) we get

$2\langle
w_{n}-t_{n},w_{n}-z_{n}\rangle=\|w_{n}-z_{n}\|^{2}+\|w_{n}-t_{n}\|^{2}-\|z_{n}-t_{n}\|^{2}.$

Therefore,

$\|w_{n}-t_{n}\|^{2}-\|z_{n}-t_{n}\|^{2}<(\sigma-1)\|w_{n}-z_{n}\|^{2}-2\langle
F(\bar{y}_{n})-F(w_{n}),w_{n}-z_{n}\rangle\ \forall n\in\mathbb{N}.$

Since $F$ is uniformly continuous on bounded subsets of $H$ and (35), if $a>0$
then the right hand side of the last inequality converges to $(\sigma-1)a<0$
as $n\to\infty$. From the last inequality we have

$\limsup_{n\to\infty}\left(\|w_{n}-t_{n}\|^{2}-\|z_{n}-t_{n}\|^{2}\right)\leq(\sigma-1)a<0.$

For $\epsilon=-(\sigma-1)a/2>0$, there exists $N\in\mathbb{N}$ such that

$\|w_{n}-t_{n}\|^{2}-\|z_{n}-t_{n}\|^{2}\leq(\sigma-1)a+\epsilon=(\sigma-1)a/2<0\quad\forall
n\in\mathbb{N},n\geq N,$

leading to

$\|w_{n}-t_{n}\|<\|z_{n}-t_{n}\|\quad\forall n\in\mathbb{N},n\geq N,$

which is a contradiction to the definition of $z_{n}=P_{C}(w_{n}-F(w_{n}))$.
Hence $a=0$, which completes the proof. ∎

The boundedness of the sequence $\\{x_{n}\\}$ implies that there is at least
one weak limit point. We show that such weak limit point belongs to $SOL$ in
the next result.

######  Lemma 4.3.

Let Assumptions 3.1 and 3.2 hold. Furthermore let $\\{x_{n_{k}}\\}$ be a
subsequence of $\\{x_{n}\\}$ converging weakly to a limit point $p$. Then
$p\in\text{SOL}$.

###### Proof.

By the definition of $z_{n_{k}}$ together with (13), we have

$\langle w_{n_{k}}-F(w_{n_{k}})-z_{n_{k}},x-z_{n_{k}}\rangle\leq 0,\ \forall
x\in C,$

which implies that

$\langle w_{n_{k}}-z_{n_{k}},x-z_{n_{k}}\rangle\leq\langle
F(w_{n_{k}}),x-z_{n_{k}}\rangle,\ \forall x\in C.$

Hence,

$\langle w_{n_{k}}-z_{n_{k}},x-z_{n_{k}}\rangle+\langle
F(w_{n_{k}}),z_{n_{k}}-w_{n_{k}}\rangle\leq\langle
F(w_{n_{k}}),x-w_{n_{k}}\rangle,\ \forall x\in C.$ (36)

Fix $x\in C$ and let $k\rightarrow\infty$ in (47). Since
$\lim_{k\to\infty}\|w_{n_{k}}-z_{n_{k}}\|=0$, we have

$0\leq\liminf_{k\to\infty}\langle F(w_{n_{k}}),x-w_{n_{k}}\rangle$ (37)

for all $x\in C$. It follows from (47) and the monotonicity of $F$ that

$\displaystyle\langle w_{n_{k}}-z_{n_{k}},x-z_{n_{k}}\rangle+\langle
F(w_{n_{k}}),z_{n_{k}}-w_{n_{k}}\rangle$ $\displaystyle\leq$
$\displaystyle\langle F(w_{n_{k}}),x-w_{n_{k}}\rangle$ $\displaystyle\leq$
$\displaystyle\langle F(x),x-w_{n_{k}}\rangle\quad\forall x\in C.$

Letting $k\to+\infty$ in the last inequality, remembering that
$\lim_{k\to\infty}\|w_{n_{k}}-z_{n_{k}}\|=0$ for all $k$, we have

$\langle F(x),x-p\rangle\geq 0\quad\forall x\in C.$

In view of Lemma 2.7, this implies $p\in\text{SOL}$. ∎

All is now set to give the weak convergence result in the theorem below.

######  Theorem 4.4.

Let Assumptions 3.1 and 3.2 hold. Then the sequence $\\{x_{n}\\}$ generated by
Algorithm 1 weakly converges to a point in SOL.

###### Proof.

We have shown that
(i) $\lim_{n\to\infty}\|x_{n}-x^{*}\|$ exists;
(ii) $\omega_{w}(x_{n})\subset\text{SOL}$, where
$\omega_{w}(x_{n}):=\\{x:\exists x_{n_{j}}\rightharpoonup x\\}$ denotes the
weak $\omega$-limit set of $\\{x_{n}\\}$.
Then, by Lemma 2.3, we have that $\\{x_{n}\\}$ converges weakly to a point in
SOL. ∎

We give some discussions on further contributions of this paper in the remark
below.

######  Remark 4.5.

(a) Our iterative Algorithm 1 is more applicable than some recent results on
projection type methods with inertial extrapolation step for solving VI($F,C$)
(1) in real Hilbert spaces. For instance, the proposed method in [21] can only
be applied for a case when $F$ is monotone and $L$-Lipschitz continuous.
Moreover, the Lipschitz constant or an estimate of it has to be known when
implementing the Algorithm 3.1 of [21]. In this result, Algorithm 1 is
applicable when $F$ is uniformly continuous and monotone operator.

(b) In finite-dimensional spaces, the assumption that $F$ is uniformly
continuous on bounded subsets of $C$ automatically holds when $F$ is
continuous. Moreover, in this case, only continuity of $F$ is required and our
weak convergence in Theorem 4.4 coincides with global convergence of sequence
of iterates $\\{x_{n}\\}$ in $\mathbb{R}^{n}$.

(c) Lemmas 3.5, 4.1, 4.2 and Theorem 4.4 still hold for a more general case of
$F$ pseudo-monotone (i.e., for all $x,y\in H$, $\langle F(x),y-x\rangle\geq
0\Longrightarrow\langle F(y),y-x\rangle\geq 0;$). We give a version of Lemma
4.3 for the case of $F$ pseudo-monotone in the Appendix. $\Diamond$

##  5 Rate of Convergence

In this section we give the rate of convergence of the iterative method 1
proposed in Section 3. We show that the proposed method has sublinear rate of
convergence and establish the nonasymptotic $O(1/n)$ convergence rate of the
proposed method. To the best of our knowledge, there is no convergence rate
result known in the literature without stronger assumptions for inertial
projection-type Algorithm 1 for VI$(F,C)$ (1) in infinite dimensional Hilbert
spaces.

######  Theorem 5.1.

Let Assumptions 3.1 and 3.2 hold. Let the sequence $\\{x_{n}\\}$ be generated
by Algorithm 1 and $x_{0}=x_{1}$. Then for any $x^{*}\in\text{SOL}$ and for
any positive integer $n$, it holds that

$\underset{1\leq i\leq
n}{\min}\|x_{i+1}-w_{i}\|^{2}\leq\frac{\Big{[}1+\Big{(}\frac{\alpha}{1-\alpha}+\frac{\alpha(1-\alpha)}{\beta}\Big{)}\Big{(}1+\frac{1-\alpha_{0}}{1-\alpha}\Big{)}\Big{]}\|x_{0}-x^{*}\|^{2}}{n}.$

###### Proof.

From (21), we have

$\displaystyle\|x_{n+1}-x^{*}\|^{2}-\|x_{n}-x^{*}\|^{2}-\alpha_{n}(\|x_{n}-x^{*}\|^{2}-\|x_{n-1}-x^{*}\|^{2})$
(38) $\displaystyle\leq$
$\displaystyle\alpha_{n}(1+\alpha_{n})\|x_{n}-x_{n-1}\|^{2}-\|x_{n+1}-w_{n}\|^{2}.$

This implies that

$\displaystyle\|x_{n+1}-w_{n}\|^{2}$ $\displaystyle\leq$
$\displaystyle\varphi_{n}-\varphi_{n+1}+\alpha_{n}(\varphi_{n}-\varphi_{n-1})+\delta_{n}$
(39) $\displaystyle\leq$
$\displaystyle\varphi_{n}-\varphi_{n+1}+\alpha[V_{n}]_{+}+\delta_{n},$

where $\delta_{n}:=\alpha_{n}(1+\alpha_{n})\|x_{n}-x_{n-1}\|^{2}$,
$V_{n}:=\varphi_{n}-\varphi_{n-1}$, $[V_{n}]_{+}:=\max\\{V_{n},0\\}$ and
$\varphi_{n}:=\|x_{n}-x^{*}\|^{2}$.

Observe from (31) that

$\sum_{n=1}^{\infty}\|x_{n+1}-x_{n}\|^{2}\leq\frac{1}{\beta}\Big{[}\varphi_{0}+\frac{\varepsilon_{1}}{1-\alpha}\Big{]}.$

So,

$\displaystyle\sum_{n=1}^{\infty}\delta_{n}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\alpha_{n}(1+\alpha_{n})\|x_{n}-x_{n-1}\|^{2}$
(40) $\displaystyle\leq$
$\displaystyle\sum_{n=1}^{\infty}\alpha(1+\alpha)\|x_{n}-x_{n-1}\|^{2}$
$\displaystyle=$
$\displaystyle\alpha(1+\alpha)\sum_{n=1}^{\infty}\|x_{n}-x_{n-1}\|^{2}$
$\displaystyle\leq$
$\displaystyle\frac{\alpha(1+\alpha)}{\beta}\Big{[}\varphi_{0}+\frac{\varepsilon_{1}}{1-\alpha}\Big{]}:=C_{1}.$

The inequality (38) implies that

$\displaystyle V_{n+1}$ $\displaystyle\leq$
$\displaystyle\alpha_{n}V_{n}+\delta_{n}$ $\displaystyle\leq$
$\displaystyle\alpha[V_{n}]_{+}+\delta_{n}.$

Therefore,

$\displaystyle[V_{n+1}]_{+}$ $\displaystyle\leq$
$\displaystyle\alpha[V_{n}]_{+}+\delta_{n}$ (41) $\displaystyle\leq$
$\displaystyle\alpha^{n}[V_{1}]_{+}+\sum_{j=1}^{n}\alpha^{j-1}\delta_{n+1-j}.$

Note that by our assumption $x_{0}=x_{1}$. This implies that
$V_{1}=[V_{1}]_{+}=0$ and $\delta_{1}=0$. From (41), we get

$\displaystyle\sum_{n=2}^{\infty}[V_{n}]_{+}$ $\displaystyle\leq$
$\displaystyle\frac{1}{1-\alpha}\sum_{n=1}^{\infty}\delta_{n}$ (42)
$\displaystyle=$
$\displaystyle\frac{1}{1-\alpha}\sum_{n=2}^{\infty}\delta_{n}.$

From (39), we get

$\displaystyle\sum_{i=1}^{n}\|x_{i+1}-w_{i}\|^{2}$ $\displaystyle\leq$
$\displaystyle\varphi_{1}-\varphi_{n}+\alpha\sum_{i=1}^{n}[V_{i}]_{+}+\sum_{i=2}^{n}\delta_{i}$
(43) $\displaystyle\leq$ $\displaystyle\varphi_{1}+\alpha C_{2}+C_{1},$

where
$C_{2}=\frac{C_{1}}{1-\alpha}\geq\frac{1}{1-\alpha}\sum_{i=2}^{\infty}\delta_{i}\geq\sum_{i=2}^{\infty}[V_{i}]_{+}$
by (42). Now, since
$\varepsilon_{1}=\varphi_{1}-\alpha_{1}\varphi_{0}=(1-\alpha_{1})\varphi_{1}$,
we have

$\displaystyle\varphi_{1}+\alpha C_{2}+C_{1}$ $\displaystyle=$
$\displaystyle\varphi_{0}+\frac{\alpha C_{1}}{1-\alpha}$ (44)
$\displaystyle+\frac{\alpha(1+\alpha)}{\beta}\Big{[}\varphi_{0}+\frac{\varepsilon_{1}}{1-\alpha}\Big{]}$
$\displaystyle=$ $\displaystyle\varphi_{0}+\frac{\alpha C_{1}}{1-\alpha}$
$\displaystyle+\frac{\alpha(1+\alpha)}{\beta}\Big{[}1+\frac{1-\alpha_{0}}{1-\alpha}\Big{]}\varphi_{0}$
$\displaystyle=$
$\displaystyle\varphi_{0}+\frac{\alpha}{1-\alpha}\Big{[}1+\frac{1-\alpha_{0}}{1-\alpha}\Big{]}$
$\displaystyle+\frac{\alpha(1+\alpha)}{\beta}\Big{[}1+\frac{1-\alpha_{0}}{1-\alpha}\Big{]}\varphi_{0}$
$\displaystyle=$
$\displaystyle\Big{[}1+\Big{(}\frac{\alpha}{1-\alpha}+\frac{\alpha(1-\alpha)}{\beta}\Big{)}\Big{(}1+\frac{1-\alpha_{0}}{1-\alpha}\Big{)}\Big{]}\varphi_{0}.$

From (43) and (44), we obtain

$\displaystyle\underset{1\leq i\leq
n}{\min}\|x_{i+1}-w_{i}\|^{2}\leq\frac{\Big{[}1+\Big{(}\frac{\alpha}{1-\alpha}+\frac{\alpha(1-\alpha)}{\beta}\Big{)}\Big{(}1+\frac{1-\alpha_{0}}{1-\alpha}\Big{)}\Big{]}\|x_{0}-x^{*}\|^{2}}{n}.$
(45)

∎

######  Remark 5.2.

(a) Note that $x_{n+1}=w_{n}$ implies that $w_{n}\in C_{n}$, where $C_{n}$ is
as defined in Algorithm 1 and hence $h_{n}(w_{n})\leq 0$. By Lemma 3.4, we get
$\frac{\sigma\eta_{n}}{2}\|w_{n}-z_{n}\|^{2}\leq h_{n}(w_{n})$. Therefore,

$0\leq\frac{\sigma\eta_{n}}{2}\|w_{n}-z_{n}\|^{2}\leq h_{n}(w_{n})\leq 0,$

which implies that $w_{n}=z_{n}$. Thus, the equality $x_{n+1}=w_{n}$ implies
that $x_{n+1}$ is already a solution of VI$(F,C)$ (1). In this sense, the
error estimate given in Theorem 5.1 can be viewed as a convergence rate result
of the inertial projection-type method 1. In particular, (45) implies that, to
obtain an $\epsilon$-optimal solution in the sense that
$\|x_{n+1}-w_{n}\|^{2}<\epsilon$, the upper bound of iterations required by
inertial projection-type method 1 is
$\frac{\Big{[}1+\Big{(}\frac{\alpha}{1-\alpha}+\frac{\alpha(1-\alpha)}{\beta}\Big{)}\Big{(}1+\frac{1-\alpha_{0}}{1-\alpha}\Big{)}\Big{]}\|x_{0}-x^{*}\|^{2}}{\epsilon}$.
We note that with the ” $\min_{1\leq i\leq n}$”, a nonasymptotic $O(1/n)$
convergence rate implies that an $\epsilon$-accuracy solution, in the sense
that $\|x_{n+1}-w_{n}\|^{2}<\epsilon$, is obtainable within no more than
$O(1/\epsilon)$ iterations. Furthermore, if $\alpha_{n}=0$ for all $n$, then
the ”$\min_{1\leq i\leq n}$” can be removed by setting $i=n$ in Theorem 5.1.
$\Diamond$

##  6 Numerical Experiments

In this section, we discuss the numerical behaviour of Algorithm 1 using
different test examples taken from the literature which are describe below and
compare our method with (5), (10) and the original Algorithm (when
$\alpha_{n}=0$) of Algorithm 1.

######  Example 6.1.

This first example (also considered in [35, 36]) is a classical example for
which the usual gradient method does not converge. It is related to the
unconstrained case of VI($F,C$) (1) where the feasible set is
$C:=\mathbb{R}^{m}$ (for some positive even integer $m$) and
$F:=(a_{ij})_{1\leq i,j\leq m}$ is the square matrix $m\times m$ whose terms
are given by

$\displaystyle a_{ij}=\left\\{\begin{array}[]{llll}&-1,\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm if}\leavevmode\nobreak\ \leavevmode\nobreak\
j=m+1-i\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\
\leavevmode\nobreak\ j>i\\\ &1,\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
if}\leavevmode\nobreak\ \leavevmode\nobreak\ j=m+1-i\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\ \leavevmode\nobreak\ j<i\\\
&0\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm otherwise}\end{array}\right.$

The zero vector $z=(0,\ldots,0)$ is the solution of this test example.

The initial point $x_{0}$ is the unit vector. We choose $\gamma=0.1$,
$\sigma=0.8$ and $\alpha_{n}=0.6$, $m=500$.

Figure 1: Comparison of Algorithm 1 with the original Algorithm.

The numerical result is listed in Figure 1, which illustrates that Algorithm 1
highly improves the original Algorithm.

######  Example 6.2.

This example is taken from [24] and has been considered by many authors for
numerical experiments (see, for example, [26, 37, 43]). The operator $A$ is
defined by $A(x):=Mx+q$, where $M=BB^{T}+S+D$, where
$B,S,D\in\mathbb{R}^{m\times m}$ are randomly generated matrices such that $S$
is skew-symmetric (hence the operator does not arise from an optimization
problem), $D$ is a positive definite diagonal matrix (hence the variational
inequality has a unique solution) and $q=0$. The feasible set $C$ is described
by linear inequality constraints $Bx\leq b$ for some random matrix
$B\in\mathbb{R}^{k\times m}$ and a random vector $b\in\mathbb{R}^{k}$ with
nonnegative entries. Hence the zero vector is feasible and therefore the
unique solution of the corresponding variational inequality. These projections
are computed by solving a quadratic optimization problem using the MATLAB
solver quadprog. Hence, for this class of problems, the evaluation of $A$ is
relatively inexpensive, whereas projections are costly. We present the
corresponding numerical results (number of iterations and CPU times in
seconds) using four different dimensions $m$ and two different numbers of
inequality constraints $k$.

We compare our proposed Algorithm 1, original Algorithm, subgradient
extragradient method (5) and the inertial subgradient extragradient method
(10) using Example 6.2 and the numerical results are listed in Tables 1 -4 and
shown in Figures 2-5 below. We take the initial point $x_{0}$ to be the unit
vector in these algorithms. We use “OPM” to denote the original Algorithm,
“SPM” to denote the subgradient extragradient method (5) and “iSPM” to denote
inertial subgradient extragradient method (10).

We choose the stopping criterion as $\|x^{k}\|\leq\epsilon=0.001.$ The size
$k=30,50,80$ and $m=20,50,80,100$. The matrices $B,S,D$ and the vector $b$ are
generated randomly. We choose $\gamma=0.1$, $\sigma=0.8$ and $\alpha_{n}=0.1$
in Algorithm (1). In (5), we choose $\sigma=0.8$, $\rho=0.1$, $\mu=0.2$. In
iSPM (10),
$\alpha=0.2,L=\|M\|,\tau=0.5\frac{\frac{1}{2}-2\alpha-\frac{1}{2}\alpha^{2}}{\frac{1}{2}-\alpha+\frac{1}{2}\alpha^{2}}$.

We denote by “Iter.” the number of iterations and “InIt.” the number of total
iterations of finding suitable step size in Tables 1 -4 below.

Table 1: Comparison of Algorithm 1, original Algorithm and methods (5) and (10) for $k=50,\alpha_{n}=0.1.$ |  | Iter. |  |  |  |  | InIt. |  |  |  | CPU in second |
---|---|---|---|---|---|---|---|---|---|---|---|---
$m$ | Alg.1 | OPM | SPM | iSPM |  | Alg.1 | OPM | SPM |  | Alg.1 | OPM | SPM | iSPM |
20 | 1044 | 1162 | 2891 | 4783 |  | 2376 | 5022 | 4283 |  | 0.0781 | 0.2188 | 0.6563 | 0.5781 |
50 | 4829 | 5912 | 25544 | 29639 |  | 13809 | 30885 | 123982 |  | 0.5625 | 0.6094 | 7.5938 | 2.1875 |
80 | 19803 | 22129 | 19736 | 61322 |  | 74066 | 157061 | 93047 |  | 6.6094 | 7.0313 | 15.9063 | 13.4844 |
100 | 26821 | 31149 | 35520 | 92579 |  | 101925 | 220297 | 173670 |  | 12.1406 | 15.8438 | 67.4219 | 34.6250 |
Table 2: Comparison of Algorithm 1, the original Algorithm and methods (5) and (10) for $k=80,\alpha_{n}=0.1.$ |  | Iter. |  |  |  | InIt. |  |  |  | CPU in second |
---|---|---|---|---|---|---|---|---|---|---|---
$m$ | Alg.1 | OPM | SPM | iSPM |  | Alg.1 | OPM | SPM |  | Alg.1 | OPM | SPM | iSPM
20 | 1479 | 1651 | 5096 | 5306 |  | 3676 | 7783 | 19071 |  | 0.1875 | 0.2188 | 6.7869 | 0.7500
50 | 4152 | 5088 | 8144 | 28033 |  | 11955 | 26594 | 36342 |  | 1.0625 | 1.2344 | 8.2344 | 6.2813
80 | 22711 | 25864 | 22281 | 64588 |  | 85176 | 182177 | 105237 |  | 8.2188 | 9.3438 | 20.6719 | 16.1563
100 | 26314 | 30568 | 37588 | 88138 |  | 99998 | 216162 | 185430 |  | 13.0625 | 16.1250 | 118.4375 | 36.0625
Table 3: Comparison of Algorithm 1, the original Algorithm and methods (5) and (10) for $k=30,\alpha_{n}=0.6.$ |  | Iter. |  |  |  | InIt. |  |  |  | CPU in second |
---|---|---|---|---|---|---|---|---|---|---|---
$m$ | Alg.1 | OPM | SPM | iSPM |  | Alg.1 | OPM | SPM |  | Alg.1 | OPM | SPM | iSPM
10 | 197 | 469 | 509 | 1327 |  | 341 | 1370 | 1460 |  | 1.21884 | 0.0313 | 0.2969 | 0.2500
30 | 1548 | 4745 | 4347 | 13697 |  | 4243 | 17534 | 16880 |  | 10.8750 | 0.4063 | 1.0938 | 0.8750
50 | 1581 | 4899 | 8269 | 26393 |  | 4762 | 18859 | 37237 |  | 12.9688 | 0.5000 | 2.4688 | 1.6094
70 | 6192 | 6256 | 7826 | 56491 |  | 22381 | 81445 | 31406 |  | 62.4844 | 5.5781 | 9.3281 | 9.1813
Table 4: Comparison of Algorithm 1, the original Algorithm and methods (5) and (10) for $k=50,\alpha_{n}=0.6.$ |  | Iter. |  |  |  | InIt. |  |  |  | CPU in second |
---|---|---|---|---|---|---|---|---|---|---|---
$m$ | Alg.1 | OPM | SPM | iSPM |  | Alg.1 | OPM | SPM |  | Alg.1 | OPM | SPM | iSPM
10 | 147 | 419 | 579 | 1117 |  | 253 | 1024 | 1119 |  | 3 | 0.0313 | 0.5781 | 0.3594
30 | 1715 | 5110 | 6373 | 13934 |  | 4705 | 19018 | 25006 |  | 30.0469 | 0.4844 | 1.5469 | 1.1563
50 | 1308 | 4798 | 9227 | 31585 |  | 4062 | 17873 | 41084 |  | 41.4375 | 0.4844 | 3.5156 | 2.0469
70 | 5673 | 14944 | 8205 | 52124 |  | 20548 | 74974 | 33716 |  | 393 | 4.5469 | 7.6406 | 10.4375
Figure 2: Comparison of Algorithm 1, original Algorithm and methods (5) and
(10). $k=50,\alpha_{n}=0.1.$ Figure 3: Comparison of Algorithm 1, original
Algorithm and methods (5) and (10). $k=80,\alpha_{n}=0.1.$ Figure 4:
Comparison of Algorithm 1, original Algorithm and methods (5) and (10).
$k=30,\alpha_{n}=0.6.$ Figure 5: Comparison of Algorithm 1, original Algorithm
and methods (5) and (10). $k=50,\alpha_{n}=0.6.$

Tables 1 -4 and Figures 2-5 show that Algorithm 1 improves the original
Algorithm with respect to “Iter.”, “InIt.” and CPU time. It is also observed
from Tables 1 -4 and Figures 2-5 that our proposed Algorithm 1 outperform the
subgradient extragradient method (5) and the inertial subgradient
extragradient method (10) with respect to the CPU time and the number of
iterations when the feasible set $C$ is nonempty closed affine subset of $H$.

##  7 Final Remarks

This paper presents a weak convergence result with inertial projection-type
method for monotone variational inequality problems in real Hilbert spaces
under very mild assumptions. This class of method is of inertial nature
because at each iteration the projection-type is applied to a point
extrapolated at the current iterate in the direction of last movement. Our
proposed algorithm framework is not only more simple and intuitive, but also
more general than some already proposed inertial projection type methods for
solving variational inequality. Based on some pioneering analysis and
Algorithm 1, we established certain nonasymptotic $O(1/n)$ convergence rate
results. Our preliminary implementation of the algorithms and experimental
results have shown that inertial algorithms are generally faster than the
corresponding original un-accelerated ones. In our experiments, the
extrapolation step-length $\alpha_{n}$ was set to be constant. How to select
$\alpha_{n}$ adaptively such that the overall performance is stable and more
efficient deserves further investigation. Interesting topics for future
research may include relaxing the conditions on $\\{\alpha_{n}\\}$, improving
the convergence results, and proposing modified inertial-type algorithms so
that the extrapolation step-size can be significantly enlarged.

Acknowledgements The project of the first author has received funding from the
European Research Council (ERC) under the European Union’s Seventh Framework
Program (FP7 - 2007-2013) (Grant agreement No. 616160)

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##  8 Appendix

In this case, we present a version of Lemma 4.3 for the case when $F$ is
pseudo-monotone.

######  Lemma 8.1.

Let $F$ be pseudo-monotone, uniformly continuous and sequentially weakly
continuous on $H$. Assume that Assumption 3.2 holds. Furthermore let
$\\{x_{n_{k}}\\}$ be a subsequence of $\\{x_{n}\\}$ converging weakly to a
limit point $p$. Then $p\in\text{SOL}$.

###### Proof.

By the definition of $z_{n_{k}}$ together with (13), we have

$\langle w_{n_{k}}-F(w_{n_{k}})-z_{n_{k}},x-z_{n_{k}}\rangle\leq 0,\ \forall
x\in C,$

which implies that

$\langle w_{n_{k}}-z_{n_{k}},x-z_{n_{k}}\rangle\leq\langle
F(w_{n_{k}}),x-z_{n_{k}}\rangle,\ \forall x\in C.$

Hence,

$\langle w_{n_{k}}-z_{n_{k}},x-z_{n_{k}}\rangle+\langle
F(w_{n_{k}}),z_{n_{k}}-w_{n_{k}}\rangle\leq\langle
F(w_{n_{k}}),x-w_{n_{k}}\rangle,\ \forall x\in C.$ (47)

Fix $x\in C$ and let $k\rightarrow\infty$ in (47). Since
$\lim_{k\to\infty}\|w_{n_{k}}-z_{n_{k}}\|=0$, we have

$0\leq\liminf_{k\to\infty}\langle F(w_{n_{k}}),x-w_{n_{k}}\rangle$ (48)

for all $x\in C$. Now we choose a sequence $\\{\epsilon_{k}\\}_{k}$ of
positive numbers decreasing and tending to $0$. For each $\epsilon_{k}$, we
denote by $N_{k}$ the smallest positive integer such that

$\left\langle F(w_{n_{j}}),x-w_{n_{j}}\right\rangle+\epsilon_{k}\geq
0\quad\forall j\geq N_{k},$ (49)

where the existence of $N_{k}$ follows from (48). Since
$\left\\{\epsilon_{k}\right\\}$ is decreasing, it is easy to see that the
sequence $\left\\{N_{k}\right\\}$ is increasing. Furthermore, for each $k$,
$F(w_{N_{k}})\not=0$ and, setting

$v_{N_{k}}=\frac{F(w_{N_{k}})}{\|F(w_{N_{k}}\|^{2}},$

we have $\left\langle F(w_{N_{k}}),v_{N_{k}}\right\rangle=1$ for each $k$. Now
we can deduce from (49) that for each $k$

$\left\langle F(w_{N_{k}}),x+\epsilon_{k}v_{N_{k}}-w_{N_{k}}\right\rangle\geq
0,$

and, since $F$ is pseudo-monotone, that

$\left\langle
F(x+\epsilon_{k}v_{N_{k}}),x+\epsilon_{k}v_{N_{k}}-w_{N_{k}}\right\rangle\geq
0.$ (50)

On the other hand, we have that $\left\\{x_{n_{k}}\right\\}$ converges weakly
to $p$ when $k\to\infty$. Since $F$ is sequentially weakly continuous on $C$,
$\left\\{F(w_{n_{k}})\right\\}$ converges weakly to $F(p)$. We can suppose
that $F(p)\not=0$ (otherwise, $p$ is a solution). Since the norm mapping is
sequentially weakly lower semicontinuous, we have

$0<\|F(p)\|\leq\lim\inf_{k\to\infty}\|F(w_{n_{k}})\|.$

Since $\left\\{w_{N_{k}}\right\\}\subset\left\\{w_{n_{k}}\right\\}$ and
$\epsilon_{k}\to 0$ as $k\to\infty$, we obtain

$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\lim\sup_{k\to\infty}\|\epsilon_{k}v_{N_{k}}\|=\lim\sup_{k\to\infty}\Big{(}\frac{\epsilon_{k}}{\|F(w_{n_{k}})\|}\Big{)}$
$\displaystyle\leq$
$\displaystyle\frac{\lim\sup_{k\to\infty}\epsilon_{k}}{\lim\inf_{k\to\infty}\|F(w_{n_{k}})\|}\leq\frac{0}{\|F(p)\|}=0,$

which implies that $\lim_{k\to\infty}\|\epsilon_{k}v_{N_{k}}\|=0$. Hence,
taking the limit as $k\to\infty$ in (50), we obtain

$\left\langle F(x),x-p\right\rangle\geq 0.$

Now, using Lemma 2.2 of [39], we have that $p\in\text{SOL}$. ∎