# Extragradient method with feasible inexact projection to variational
inequality problem

R. Díaz Millán<EMAIL_ADDRESS>School of Information Technology,
Deakin University, Geelong, Australia O.P. Ferreira<EMAIL_ADDRESS>IME,
Universidade Federal de Goiás, Goiânia, Brazil J. Ugon<EMAIL_ADDRESS>School of Information Technology, Deakin University, Geelong, Australia

###### Abstract

The variational inequality problem in finite-dimensional Euclidean space is
addressed in this paper, and two inexact variants of the extragradient method
are proposed to solve it. Instead of computing exact projections on the
constraint set, as in previous versions extragradient method, the proposed
methods compute feasible inexact projections on the constraint set using a
relative error criterion. The first version of the proposed method provided is
a counterpart to the classic form of the extragradient method with constant
steps. In order to establish its convergence we need to assume that the
operator is pseudo-monotone and Lipschitz continuous, as in the standard
approach. For the second version, instead of a fixed step size, the method
presented finds a suitable step size in each iteration by performing a line
search. Like the classical extragradient method, the proposed method does just
two projections into the feasible set in each iteration. A full convergence
analysis is provided, with no Lipschitz continuity assumption of the operator
defining the variational inequality problem.

keywords: Variational inequality problem, Extragradient method, Frank-Wolfe
algorithm, conditional gradient method, feasible inexact projection. MSC 2020:
65K05, 90C30, 90C25

##  1 Introduction

This paper addresses the variational inequality problem in finite-dimensional
Euclidean space. This problem is formally stated as follows: Let
$F:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ be an operator and ${\cal
C}\subset{\mathbb{R}}^{n}$ be a nonempty and closed convex set. The
variational inequality problem (VIP($F,{\cal C}$)) associated with $F$ and
${\cal C}$ consists in finding a $x^{*}\in{\cal C}$ such that

$\left\langle F({x^{*}}),x-{x^{*}}\right\rangle\geq
0,\qquad\forall~{}x\in{\cal C}.$ (1)

We denote by ${\cal C}^{*}$ the solution set of problem 1, which we will
assume to be nonempty. The variational inequality problem is an issue that has
attracted the interest of the mathematics programming community not only by
their own interest but because it is an abstract model for several families of
problems in nonlinear analysis and its applications. For instance, if
$F=\nabla f$, where $f:{\mathbb{R}}^{n}\to{\mathbb{R}}$ is a differentiable
function, then VIP($F,{\cal C}$) correspond to the problem of minimizing the
function $f$ constrained the set ${\cal C}$. When ${\cal C}$ is a cone ${\cal
K}$, the VIP($F,{\cal C})$ is a complementary problem, which is stated in the
following form: Compute $x^{*}\in{\mathbb{R}}^{n}$ such that $x^{*}\in{\cal
K}$, $F(x^{*})\in{\cal K}^{*}$ and $\left\langle
F({x^{*}}),{x^{*}}\right\rangle=0$, where ${\cal K}^{*}$ denotes the dual of
${\cal K}$. For a comprehensive study of theory and applications of
variational inequality, see [17, 16].

The extragradient method was proposed in [24] in the 1970s and continues to
attract the interest of variational inequality experts, see [10, 25, 29, 3],
and the references therein. This method is attractive due to the fact that it
requires only two operator evaluations for each iteration, making it
numerically stable and hence potentially suited for addressing large-scale
problems. Apart from the projection needed in its definition, which is
responsible for approximately all of the computational demands if the
constraint set projection is difficult to compute, it is a relatively simple
method. In addition, the method converges with mild assumptions. All these
features motivate the study of it, resulting in many different versions of the
method throughout the years culminating in a large body of literature on the
subject, including [16, 17, 12, 31, 7, 9], and the references therein.

Another subject that has motivated the development of numerical methods for
addressing constrained problems is dealing with the computation of the
projection, which is the step that accounts for practically all of the
computing demands of methods that utilize projections, like the Extragradient
method. In general, computing the projection requires solving a quadratic
problem constrained to the feasible set at each iteration, which can
significantly raise the cost per iteration if the number of unknowns is large.
In light of this, it may not be reasonable to do exact projections when the
iterates of the method are distant from the solution of the problem under
consideration. Through the years, various inexact procedures that become
increasingly accurate as the solution of the problem under consideration is
approached have been proposed in an effort to reduce the computational cost
required for projections, leading to more effective projection-based methods;
see for example [5, 6, 19, 20, 27, 30, 26, 15, 14].

The purpose of this paper is to present two variants of the extragradient
method which employ the use of feasible inexact projections. In the variants
of the extragradient method that we propose, we will employ a version of the
scheme proposed in [30, Example 1] in which the inexact projection over the
feasible set is calculated allowing an appropriate relative error tolerance.
Firstly, we present a variation of the extragradient method with constant
stepsize and show that it preserves the same convergence result as the classic
method, see [17, 24]. We show that if $F$ is a pseudo monotone operator on
${\cal C}$ with respect to ${\cal C}^{*}$ and Lipschitz continuous, the
sequence generated converges to a solution of VIP($F,{\cal C}$). It is
important to note in this version that the Lipschitz constant is required to
compute the stepsize. Considering that the Lipschitz constant is not
accessible or difficult to compute in almost every application, we propose and
analyse a feasible inexact projection version of the extragradient method
using an Armiljo-type line search. It is worth noting that, like the classical
extragradient method, the method does just two projections into the feasible
set in each iteration. The full convergence of the sequence to a solution is
shown, with $F$ being a pseudo monotone operator on ${\cal C}$ with respect to
${\cal C}^{*}$ and no Lipschitz continuity assumption, which is the same
results as the version with exact projection, see [3, 8, 32, 23, 28].

The organization of the paper is as follows. In section 2, we present some
notation and basic results used throughout the paper. In Section 3 we will
revisit the concept of feasible inexact projection onto a closed and convex
set and describe some new properties of the feasible inexact projection.
Section 4 describe and analyze the extragradient method with a feasible
inexact projection for solving problem (1). In Section 5 is introduced and
analyzed an inexact variant of the extragradient method with line search for
solve VIP($F,{\cal C}$). Finally, some concluding remarks are made in Section
7.

##  2 Preliminaries

In this section, we present some preliminary results used throughout the
paper. We denote: ${\mathbb{N}}:=\\{1,2,3,\ldots\\}$,
$\langle\cdot,\cdot\rangle$ is the usual inner product and $\|\cdot\|$ is the
Euclidean norm. Let ${\cal C}\subset\mathbb{R}^{n}$ be closed, convex and
nonempty set, the projection is the map ${\cal{P}}_{\cal
C}:\mathbb{R}^{n}\to{\cal C}$ defined by

${\cal{P}}_{\cal C}(v):=\arg\min_{z\in{\cal C}}\|v-z\|.$

In the next lemma, we present some important properties of the projection
mapping.

######  Lemma 1.

Given a convex and closed set ${\cal C}\subset\mathbb{R}^{n}$ and
$v\in\mathbb{R}^{n}$, the following properties hold:

  1. (i)

$\langle v-{\cal{P}}_{\cal C}(v),z-{\cal{P}}_{\cal C}(v)\rangle\leq 0$, for
all $z\in{\cal C}$;

  2. (ii)

$\|{\cal{P}}_{\cal C}(v)-z\|^{2}\leq\|v-z\|^{2}-\|{\cal{P}}_{\cal
C}(v)-v\|^{2}$, for all $z\in{\cal C}$.

###### Proof.

The item (i) is proved in [2, Theorem 3.14]. For item (ii), combine
$\|v-z\|^{2}=\|{\cal{P}}_{\cal C}(v)-v\|^{2}+\|{\cal{P}}_{\cal
C}(v)-z\|^{2}-2\langle{\cal{P}}_{\cal C}(v)-v,{\cal{P}}_{\cal C}(v)-z\rangle$
with item (i). ∎

For the formula in the next proposition see, for example, [2, Example 3.21].

######  Proposition 2.

Let $a,v\in\mathbb{R}^{n}$ and $H=\\{x\in\mathbb{R}^{n}:~{}\langle
v,x-a\rangle\leq 0\\}$. If ${\bar{x}}\notin H$, then

${\cal{P}}_{H}({\bar{x}})={\bar{x}}-\frac{1}{\|v\|^{2}}\langle
v,{\bar{x}}-a\rangle v.$

Let $F:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ be an operator, ${\cal
C}\subset{\mathbb{R}}^{n}$ be a nonempty and closed convex set. The operator
$F$ is said to be pseudo monotone on ${\cal C}$ with respect to the solution
set ${\cal C}^{*}$ of problem 1 if the set ${\cal C}^{*}$ is nonempty and, for
every $x^{*}\in{\cal C}^{*}$, there holds:

$\left\langle F(x),x-{x^{*}}\right\rangle\geq 0,\qquad\forall x\in{\cal C}.$

######  Definition 1.

Let $S$ be a nonempty subset of $\mathbb{R}^{n}$. A sequence
$(v_{k})_{k\in\mathbb{N}}\subset\mathbb{R}^{n}$ is said to be quasi-Fejér
convergent to $S$, if and only if, for all $v\in S$ there exists
${\bar{k}}\geq 0$ and a summable sequence $(\epsilon_{k})_{k\in\mathbb{N}}$,
such that $\|v_{k+1}-v\|^{2}\leq\|v_{k}-v\|^{2}+\epsilon_{k}$ for all
$k\geq{\bar{k}}$.

In the following lemma, we state the main properties of quasi-Fejér sequences
that we will need; a comprehensive study on this topic can be found in [11].

######  Lemma 3.

Let $S$ be a nonempty subset of $\mathbb{R}^{n}$ and
$(v_{k})_{k\in\mathbb{N}}$ be a quasi-Fejér sequence convergent to $S$. Then,
the following conditions hold:

  1. (i)

the sequence $(v_{k})_{k\in\mathbb{N}}$ is bounded;

  2. (ii)

if a cluster point ${\bar{v}}$ of $(v_{k})_{k\in\mathbb{N}}$ belongs to $S$,
then $(v_{k})_{k\in\mathbb{N}}$ converges to ${\bar{v}}$.

##  3 Feasible inexact projection

In this section, we will revisit the concept of feasible inexact projection
onto a closed and convex set. This concept has already been utilized in [1,
13, 14, 15]. We also describe some new properties of the feasible inexact
projection, which is employed throughout the work. The definition of feasible
inexact projection is as follows.

######  Definition 2.

Let ${\cal C}\subset{\mathbb{R}}^{n}$ be a closed convex set and
$\gamma\in{\mathbb{R}}_{+}$ is a given error tolerance vector. The feasible
inexact projection mapping relative to $u\in{\cal C}$ with error tolerance
vector ${\gamma}$, denoted by ${\cal P}^{\gamma}_{\cal
C}(u,\cdot):{\mathbb{R}}^{n}\rightrightarrows{\cal C}$, is the set-valued
mapping defined as follows

${\cal P}^{\gamma}_{\cal C}(u,v):=\left\\{w\in{\cal
C}:~{}\big{\langle}v-w,y-w\big{\rangle}\leq\gamma\|w-u\|^{2},~{}\forall
y\in{\cal C}\right\\}.$ (2)

Each point $w\in{\cal P}^{\gamma}_{\cal C}(u,v)$ is called a feasible inexact
projection of $v$ onto ${\cal C}$ relative to $u$ with error tolerance
$\varphi_{\gamma}$.

The feasible inexact projection generalizes the concept of usual projection.
In the following, we present some remarks about this concept.

######  Remark 1.

Let $\gamma\in{\mathbb{R}}_{+}$ be error tolerance vector, ${\cal
C}\subset{\mathbb{R}}^{n}$, $u\in{\cal C}$ and ${\gamma}$ be as in Definition
2. For all $v\in{\mathbb{R}}^{n}$, it follows from (2) that ${\cal
P}^{0}_{\cal C}(u,v)$ is the exact projection of $v$ onto ${\cal C}$; see [4,
Proposition 2.1.3, p. 201]. Moreover, ${\cal P}^{0}_{\cal C}(u,v)\subset{\cal
P}^{\gamma}_{\cal C}(u,v)$ which implies that ${\cal P}^{\gamma}_{\cal
C}(u,v)\neq\varnothing$, for all $u\in{\cal C}$ and $v\in{\mathbb{R}}^{n}$. In
general, if $\gamma\leq{\bar{\gamma}}$ then ${\cal P}^{\gamma}_{\cal
C}(u,v)\subset{\cal P}^{\bar{\gamma}}_{\cal C}(u,v)$.

Below we present a particular counterpart of the firm non-expansiveness of the
projection operator to a feasible inexact projection operator, its proof
follows the same idea of [14].

######  Proposition 4.

Let $v\in{\mathbb{R}}^{n}$ and $\gamma\geq 0$. If
${w}\in{\cal{P}}^{\gamma}_{C}({u},v)$ and
${\bar{w}}={\cal{P}}_{C}({\bar{v}})$, then

$\|{w}-{\bar{w}}\|^{2}\leq\|{v}-{\bar{v}}\|^{2}-\|({v}-{\bar{v}})-({w}-{\bar{w}})\|^{2}+2\gamma\|{w}-{u}\|^{2}.$

###### Proof.

Since ${w}\in{\cal{P}}^{\gamma}_{C}({u},v)$ and
${\bar{w}}={\cal{P}}_{C}({\bar{v}})$, it follows from (2) and Lemma 1 that

$\big{\langle}v-w,{\bar{w}}-w\big{\rangle}\leq\gamma\|w-u\|^{2},\qquad\big{\langle}{\bar{v}}-{\bar{w}},w-{\bar{w}}\big{\rangle}\leq
0$

By adding the last two inequalities, some algebraic manipulations yield

$-\big{\langle}{\bar{v}}-{v},{\bar{w}}-w\big{\rangle}+\|{w}-{\bar{w}}\|^{2}\leq\gamma\|w-u\|^{2}.$

Since
$\|({\bar{v}}-{v})-({\bar{w}}-w)\|^{2}=\|{\bar{v}}-{v}\|^{2}-2\big{\langle}{\bar{v}}-{v},{\bar{w}}-w\big{\rangle}+\|{\bar{w}}-{w}\|^{2}$,
the desired inequality follows by combination with the last inequality. ∎

######  Lemma 5.

Let $F:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ be a operator, ${\cal
C}\subset{\mathbb{R}}^{n}$ be a nonempty, closed, and convex set, $x\in{\cal
C}$ and $0\leq\gamma<1$. Take $z\in{\mathbb{R}}^{n}$ and any inexact
projection

$w(\alpha)\in{\cal P}^{\gamma}_{\cal C}(x,x-\alpha
F(z)),\qquad\alpha\in(0,+\infty).$

Then, there hold:

  1. (i)

$\big{\langle}F(z),w(\alpha)-x\big{\rangle}\leq\dfrac{\gamma-1}{\alpha}\|w(\alpha)-x\|^{2}$;

  2. (ii)

$\|w(\alpha)-x\|\leq\dfrac{\alpha}{1-\gamma}\|F(z)\|$.

###### Proof.

Since $w(\alpha)\in{\cal P}^{\gamma}_{\cal C}(x,x-\alpha F(z))$ we obtain
$\big{\langle}x-\alpha
F(z)-w(\alpha),x-w(\alpha)\big{\rangle}\leq\gamma\|w(\alpha)-x\|^{2}$, which
after some algebraic manipulations yields

$\|w(\alpha)-x\|^{2}-\alpha\big{\langle}F(z),x-w(\alpha)\big{\rangle}\leq\gamma\|w(\alpha)-x\|^{2}.$

Thus, item $(i)$ follows from the last inequality.

We proceed to prove the tem $(ii)$. For that, first note that the item $(i)$
is equivalent to

$0\leq\dfrac{1-{\gamma}}{\alpha}\|w(\alpha)-x\|^{2}\leq\big{\langle}F(z),x-w(\alpha)\big{\rangle}.$
(3)

If $w(\alpha)=x$, then the inequality holds trivially. Assume that
$w(\alpha)\neq x$. Thus, the inequality in item (ii) follows by combining the
inequality (3) with $\langle
F(z),x-w(\alpha)\rangle\leq\|F(z)\|\|x-w(\alpha)\|$. ∎

######  Corollary 6.

The following statements are equivalent:

  1. (i)

$x$ is a solution of the VIP(F,${\cal C}$);

  2. (ii)

$x\in{\cal P}^{\gamma}_{\cal C}(x,x-\alpha F(x))$, for all
$\alpha\in(0,+\infty)$;

  3. (iii)

there exists ${\bar{\alpha}}>0$ such that $\langle
F(x),w({\bar{\alpha}})-x\rangle\geq 0$ for $w({\bar{\alpha}})\in{\cal
P}^{\gamma}_{\cal C}(x,x-{\bar{\alpha}}F(x))$.

###### Proof.

Proof of equivalence between item $(i)$ and item $(ii)$: We first assume that
item $(i)$ holds, i.e., $x$ is a solution for problem (1). In this case, by
taking $w(\alpha)\in{\cal P}^{\gamma}_{\cal C}(x,x-\alpha F(x))$, we find that
$w(\alpha)\in{\cal C}$. Consequently, we have
$\big{\langle}F(x),w(\alpha)-x\big{\rangle}\geq 0$. Considering that
$\alpha>0$ and $0\leq\gamma<1$, the last inequality, along with item $(i)$ of
Lemma 5 for $z=x$, implies that $w(\alpha)=x$. Hence, $x\in{\cal
P}^{\gamma}_{\cal C}(x,x-\alpha F(x))$, and item $(ii)$ also holds.
Reciprocally, assuming that item $(ii)$ holds, if $x\in{\cal P}^{\gamma}_{\cal
C}(x,x-\alpha F(x))$, then applying (2) with $w=x$, $v=x-\alpha F(x)$, and
$u=x$ yields $\big{\langle}x-\alpha F(x)-x,y-x\big{\rangle}\leq 0$, for all
$y\in{\cal C}$. Given that $\alpha>0$, the last inequality is equivalent to
$\big{\langle}F(x),y-x\big{\rangle}\geq 0$, for all $y\in{\cal C}$. Thus, $x$
is a solution for problem (1), and item $(i)$ holds as well.

Proof of equivalence between item $(ii)$ and item $(iii)$: Let us assume that
item $(ii)$ holds. Thus, item $(i)$ also holds, and $x$ is a solution for
problem (1), which implies that $\left\langle F(x),y-x\right\rangle\geq 0$ for
all $y\in{\cal C}$. Considering that for any $w({\bar{\alpha}})\in{\cal
P}^{\gamma}_{\cal C}(x,x-{\bar{\alpha}}F(x))$, we have
$w({\bar{\alpha}})\in{\cal C}$, it follows that $\langle
F(x),w({\bar{\alpha}})-x\rangle\geq 0$, and item $(iii)$ holds. Conversely, we
assume, for contradiction, that item $(ii)$ does not hold. Therefore,
$x\notin{\cal P}^{\gamma}_{\cal C}(x,x-\alpha F(x))$, and considering that
$w(\alpha)\in{\cal P}^{\gamma}_{\cal C}(x,x-\alpha F(x))$, we conclude that
$x\neq w(\alpha)$. As a result, because $\alpha>0$ and
$0<\gamma\leq\bar{\gamma}$, it follows from item $(i)$ of Lemma 5 that
$\big{\langle}F(x),w(\alpha)-x\big{\rangle}<0$, for all
$\alpha\in(0,+\infty)$. Thus, item $(iii)$ does not hold, which leads to a
contradiction. Therefore, $(iii)$ implies $(ii)$. ∎

######  Corollary 7.

Let $F:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ be a operator, ${\cal
C}\subset{\mathbb{R}}^{n}$ be a nonempty, closed, and convex set, $x\in{\cal
C}$, $0\leq\gamma<1$ and $\alpha>0$. If $y\in{\cal P}^{\gamma}_{\cal
C}(x,x-\alpha F(x))$ and $x^{+}\in{\cal P}^{\gamma}_{\cal C}(x,x-\alpha
F(y))$, then the following inequalities hold:

  1. (i)

$\|y-x\|\leq\dfrac{\alpha}{1-\gamma}\|F(x)\|$;

  2. (ii)

$\|x^{+}-x\|\leq\dfrac{\alpha}{1-\gamma}\|F(y)\|$.

As a consequence, if $F$ is Lipschitz continuous on ${\cal C}$ with constant
$L>0$, then it holds:

$\|x^{+}-x\|\leq\dfrac{\alpha(1-\gamma+\alpha L)}{(1-\gamma)^{2}}\|F(x)\|.$
(4)

###### Proof.

Applying item $(ii)$ of Lemma 5 with $w(\alpha)=y$ and $z=x$ we obtain item
$(i)$ and with $w(\alpha)=x^{+}$ and $z=y$ we obtain item (ii). We proceed to
prove (4). For that, we first note that due to $F$ be Lipschitz continuous on
${\cal C}$ with constant $L>0$, we obtain that

$\|F(y)\|\leq\|F(y)-F(x)\|+\|F(x)\|\leq L\|y-x\|+\|F(x)\|.$

Thus, considering that $y\in{\cal P}^{\gamma}_{\cal C}(x,x-\alpha F(x))$, we
can apply the item $(i)$ to obtain

$\|F(y)\|\leq\dfrac{\alpha
L}{1-\gamma}\|F(x)\|+\|F(x)\|=\dfrac{1-\gamma+\alpha L}{1-\gamma}\|F(x)\|.$

By combing the last inequality with item $(ii)$, the desired inequality
follows. ∎

##  4 Extragradient inexact method with constant step size

In this section, we describe the extragradient method with a feasible inexact
projection for solving problem (1). It should be noted that the proposed
method uses appropriate relative error criteria to compute inexact projections
on the constraint set, unlike the extragradient method which uses exact
projections on the constraint set.

The inexact version of the classical extragradient method is stated as
follows:

Algorithm 1 Extragradient inexact projection method-EInexPM

1:Take $\alpha>0$, $0<\bar{\gamma}<1/2$ and $(a_{k})_{k\in\mathbb{N}}$
satisfying $\sum_{k\in\mathbb{N}}a_{k}<+\infty$. Let $x^{1}\in{\cal C}$ and
set $k=1$.

2: Choose a error tolerance $\gamma_{k}$ such that
$0\leq\gamma_{k}\|F(x^{k})\|^{2}\leq a_{k},\qquad
0\leq\gamma_{k}<\bar{\gamma},$ (5) and compute the following feasible inexact
projections:  $\displaystyle y^{k}$ $\displaystyle\in{\cal
P}^{\gamma_{k}}_{\cal C}\big{(}x^{k},x^{k}-\alpha F(x^{k})\big{)};$ (6)
$\displaystyle x^{k+1}$ $\displaystyle\in{\cal P}^{\gamma_{k}}_{\cal
C}\big{(}x^{k},x^{k}-\alpha F(y^{k})\big{)}.$ (7)

3:If $x^{k}=y^{k}$ or $y^{k}=x^{k+1}$, stop, else set $k\leftarrow k+1$, and
go to step 2.

Let us examine the main features of the EInexPM. To begin, we select a
constant step size $\alpha>0$, an upper bound for error tolerances
${\bar{\gamma}}$ such that $0<\bar{\gamma}<1/2$ and select an exogenous
summable sequence $(a_{k})_{k\in\mathbb{N}}$ to control the error tolerance.
The stopping criterion $F(x^{k})=0$ is then evaluated in the current iteration
$x^{k}$. If this criterion is not satisfied, a non-negative error tolerance
$\gamma_{k}$ that fulfils the requirements (5) is selected. By using an inner
procedure compute $y^{k}$ as any feasible inexact projection $x^{k}-\alpha
F(x^{k})$ onto the feasible set ${\cal C}$ relative to $x^{k}$, i.e.
$y^{k}\in{\cal P}^{\gamma_{k}}_{\cal C}\big{(}x^{k},x^{k}-\alpha
F(x^{k})\big{)}$. Finally, using again an inner procedure, the next iterate
$x^{k+1}$ is computed as any feasible inexact projection of $x^{k}-\alpha
F(y^{k})$ onto the feasible set ${\cal C}$ relative to $x^{k}$, i.e.,
$x^{k+1}\in{\cal P}^{\gamma_{k}}_{\cal C}\big{(}x^{k},x^{k}-\alpha
F(y^{k})\big{)}$.

It is worth noting that if $\gamma_{k}\equiv 0$, then Remark 1 implies that
inexact projections are the exact ones. Hence, EInexPM corresponds to the
classical extragradient method introduced in [24]. It is important to note
that $\gamma_{k}$ in (5) can be selected as any nonnegative real number
fulfilling $0\leq\gamma_{k}\|F(x^{k})\|^{2}\leq a_{k}$, for a prefixed
sequence $(a_{k})_{k\in\mathbb{N}}$. In this case, we have

$\sum_{k\in\mathbb{N}}\big{(}\gamma_{k}\|F(x^{k})\|^{2}\big{)}<+\infty.$ (8)

Since $x^{1}\in{\cal C}$ and, for all $k\in\mathbb{N}$, $x^{k+1}$ is a
feasible inexact projection onto ${\cal C}$, we conclude
$(x^{k})_{k\in\mathbb{N}}\subset{\cal C}$. As a consequence of ${\cal C}$
being a closed set, any cluster point of $(x^{k})_{k\in\mathbb{N}}$, if any
exists, belongs to ${\cal C}$.

Next we present two examples of sequences $(a_{k})_{k\in\mathbb{N}}$
satisfying $\sum_{k\in\mathbb{N}}a_{k}<+\infty$.

######  Example 1.

Sequences $(a_{k})_{k\in\mathbb{N}}$ satisfying
$\sum_{k\in\mathbb{N}}a_{k}<+\infty$ are obtained by taking
$a_{k}:=b_{k-1}-b_{k}$ and $\bar{b}>0$ satisfying one the following
conditions: (i) $b_{0}=2\bar{b}$, $b_{k}=\bar{b}/k$, for all $k=1,2,\ldots$;
(ii)$b_{0}=2\bar{b}$, $b_{k}=\bar{b}/\ln(k+1)$, for all $k=1,2,\ldots$.

The convergence analysis of the sequence $(x^{k})_{k\in\mathbb{N}}$ produced
by EInexPM will be discussed in the following sections.

###  4.1 Convergence analysis

We will show in this section that the sequence $(x^{k})_{k\in\mathbb{N}}$
generated by EInexPM converges to a solution of VIP(F, ${\cal C}$) for
Lipschitz continuous pseudo-monotone operator $F$ with Lipschitz constant
$L\geq 0$. To state our first result, let us recall that the solution set of
VIP(F, ${\cal C}$) is denoted by ${\cal C}^{*}$ and define the following
constants:

${\bar{\eta}}:=1-\alpha^{2}L^{2}-2{\bar{\gamma}},\qquad\qquad{\bar{\nu}}:=\dfrac{\alpha^{2}(1-{\bar{\gamma}}+\alpha
L)^{2}}{(1-{\bar{\gamma}})^{4}}.$ (9)

######  Lemma 8.

Let ${\cal C}\subset{\mathbb{R}}^{n}$ be a nonempty and closed convex set and
$(x^{k})_{k\in{\mathbb{N}}}$ the sequence generated by Algorithm 1. Assume
that $F:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ is a pseudo monotone operator on
${\cal C}$ with respect to ${\cal C}^{*}$ and Lipschitz continuous on ${\cal
C}$ with constant $L>0$. Then, for any $x^{*}\in{\cal C}^{*}$, there holds:

$\|x^{k+1}-x^{*}\|^{2}\leq\|x^{k}-x^{*}\|^{2}-{\bar{\eta}}\|x^{k}-y^{k}\|^{2}+{\bar{\nu}}\gamma_{k}\|F(x^{k})\|^{2},\qquad
k=1,2,\ldots.$

###### Proof.

First note that

$\big{\langle}x^{k}-\alpha
F(y^{k})-y^{k},x^{k+1}-y^{k}\big{\rangle}=\big{\langle}x^{k}-\alpha
F(x^{k})-y^{k},x^{k+1}-y^{k}\big{\rangle}\\\
+\alpha\big{\langle}F(x^{k})-F(y^{k}),x^{k+1}-y^{k}\big{\rangle}.$

Since (6) implies that $y^{k}\in{\cal P}^{\gamma_{k}}_{\cal
C}\big{(}x^{k},x^{k}-\alpha F(x^{k})\big{)}$, we conclude that

$\big{\langle}x^{k}-\alpha
F(y^{k})-y^{k},x^{k+1}-y^{k}\big{\rangle}\leq\gamma_{k}\|y^{k}-x^{k}\|^{2}+\alpha\big{\langle}F(x^{k})-F(y^{k}),x^{k+1}-y^{k}\big{\rangle}.$
(10)

For simplicity set $z^{k}:=x^{k}-\alpha F(y^{k})$. As nothing more than a
result of some algebraic manipulations, we arrive to the conclusion that

$\displaystyle\|x^{k+1}-x^{*}\|^{2}$
$\displaystyle=\|x^{k+1}-z^{k}\|^{2}+\|z^{k}-x^{*}\|^{2}-2\langle
x^{k+1}-z^{k},x^{*}-z^{k}\rangle$
$\displaystyle=-\|x^{k+1}-z^{k}\|^{2}+\|z^{k}-x^{*}\|^{2}+2\langle
z^{k}-x^{k+1},x^{*}-x^{k+1}\rangle.$

Using that $x^{k+1}\in{\cal P}^{\gamma_{k}}_{\cal C}\big{(}x^{k},z^{k}\big{)}$
we conclude that $\langle
z^{k}-x^{k+1},x^{*}-x^{k+1}\rangle\leq\gamma_{k}\|x^{k+1}-x^{k}\|^{2}$, which
combined with the previous equality give us

$\|x^{k+1}-x^{*}\|^{2}\leq\|z^{k}-x^{*}\|^{2}-\|x^{k+1}-z^{k}\|^{2}+2\gamma_{k}\|x^{k+1}-x^{k}\|^{2}$
(11)

Taking into account that $z^{k}=x^{k}-\alpha F(y^{k})$, some calculations show
that

$\displaystyle\|z^{k}-x^{*}\|^{2}-\|x^{k+1}-z^{k}\|^{2}$
$\displaystyle=\|x^{k}-x^{*}-\alpha F(y^{k})\|^{2}-\|x^{k}-x^{k+1}-\alpha
F(y^{k})\|^{2}$
$\displaystyle=\|x^{k}-x^{*}\|^{2}-\|x^{k}-x^{k+1}\|^{2}+2\alpha\big{\langle}F(y^{k}),x^{*}-x^{k+1}\big{\rangle}.$
(12)

On the other hand, considering that $x^{*}\in{\cal C}^{*}$, $y^{k}\in{\cal C}$
and $F$ is pseudo monotone operator on ${\cal C}$ with respect to ${\cal
C}^{*}$ we have $\big{\langle}F(y^{k}),y^{k}-{x^{*}}\big{\rangle}\geq 0$.
Thus, we conclude that

$\big{\langle}F(y^{k}),{x^{*}}-x^{k+1}\big{\rangle}\leq\big{\langle}F(y^{k}),y^{k}-x^{k+1}\big{\rangle}.$

The last inequality, when combined with (4.1), implies that

$\|z^{k}-x^{*}\|^{2}-\|x^{k+1}-z^{k}\|^{2}\leq\|x^{k}-x^{*}\|^{2}-\|x^{k}-x^{k+1}\|^{2}+2\alpha\big{\langle}F(y^{k}),y^{k}-x^{k+1}\big{\rangle}.$

The previous inequality is now combined with (11) to provide the following
inequality

$\|x^{k+1}-x^{*}\|^{2}\leq\|x^{k}-x^{*}\|^{2}-\|x^{k}-x^{k+1}\|^{2}+\gamma_{k}\|x^{k+1}-x^{k}\|^{2}+2\alpha\big{\langle}F(y^{k}),y^{k}-x^{k+1}\big{\rangle}.$

Since
$\|x^{k}-x^{k+1}\|^{2}=\|x^{k}-y^{k}\|^{2}+\|y^{k}-x^{k+1}\|^{2}+2\langle
x^{k}-y^{k},y^{k}-x^{k+1}\rangle$, the last inequality is equivalent to

$\|x^{k+1}-x^{*}\|^{2}\leq\|x^{k}-x^{*}\|^{2}-\|x^{k}-y^{k}\|^{2}-\|y^{k}-x^{k+1}\|^{2}+\gamma_{k}\|x^{k+1}-x^{k}\|^{2}+2\big{\langle}x^{k}-\alpha
F(y^{k})-y^{k},x^{k+1}-y^{k}\big{\rangle}.$

The last inequality together with (10) yield

$\|x^{k+1}-x^{*}\|^{2}\leq\|x^{k}-x^{*}\|^{2}-\|x^{k}-y^{k}\|^{2}-\|y^{k}-x^{k+1}\|^{2}+\gamma_{k}\|x^{k+1}-x^{k}\|^{2}\\\
+2\gamma_{k}\|y^{k}-x^{k}\|^{2}+2\alpha\big{\langle}F(x^{k})-F(y^{k}),x^{k+1}-y^{k}\big{\rangle}.$

Considering that $F$ is Lipschitz continuous on ${\cal C}$ with constant
$L>0$, we have

$\langle F(x^{k})-F(y^{k}),x^{k+1}-y^{k}\rangle\leq
L\|x^{k}-y^{k}\|\|x^{k+1}-y^{k}\|,$

which combined with the last inequality yields

$\|x^{k+1}-x^{*}\|^{2}\leq\|x^{k}-x^{*}\|^{2}-\|x^{k}-y^{k}\|^{2}-\|y^{k}-x^{k+1}\|^{2}+\gamma_{k}\|x^{k+1}-x^{k}\|^{2}\\\
+2\gamma_{k}\|y^{k}-x^{k}\|^{2}+2\alpha L\|x^{k}-y^{k}\|\|x^{k+1}-y^{k}\|.$

or equivalently,

$\|x^{k+1}-x^{*}\|^{2}\leq\|x^{k}-x^{*}\|^{2}-(1-\alpha^{2}L^{2}-2\gamma_{k})\|x^{k}-y^{k}\|^{2}\\\
+\gamma_{k}\|x^{k+1}-x^{k}\|^{2}-(\alpha
L\|x^{k}-y^{k}\|-\|x^{k+1}-y^{k}\|)^{2}.$

Hence, we have

$\|x^{k+1}-x^{*}\|^{2}\leq\|x^{k}-x^{*}\|^{2}-(1-\alpha^{2}L^{2}-2\gamma_{k})\|x^{k}-y^{k}\|^{2}+\gamma_{k}\|x^{k+1}-x^{k}\|^{2}.$

Thus, taking into account that $x^{k+1}\in{\cal P}^{\gamma_{k}}_{\cal
C}\big{(}x^{k},x^{k}-\alpha F(y^{k})\big{)}$, by applying Corollary 7 with
$x^{+}=x^{k+1}$, $x=x^{k}$ and $\gamma=\gamma_{k}$ we obtain that

$\|x^{k+1}-x^{*}\|^{2}\leq\|x^{k}-x^{*}\|^{2}-(1-\alpha^{2}L^{2}-2\gamma_{k})\|x^{k}-y^{k}\|^{2}+\dfrac{\alpha^{2}(1-\gamma_{k}+\alpha
L)^{2}}{(1-\gamma_{k})^{4}}\gamma_{k}\|F(x^{k})\|^{2}.$

Therefore, using (9) and considering that $0\leq\gamma_{k}<\bar{\gamma}$, the
desired inequality follows. ∎

######  Theorem 9.

Let ${\cal C}\subset{\mathbb{R}}^{n}$ be a nonempty and closed convex set and
$(x^{k})_{k\in{\mathbb{N}}}$ the sequence generated by Algorithm 1. Assume
that $F:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ is a pseudo monotone operator on
${\cal C}$ with respect to ${\cal C}^{*}$ and Lipschitz continuous on ${\cal
C}$ with constant $L>0$. If

$0<\alpha<\frac{\sqrt{1-2{\bar{\gamma}}}}{L},$ (13)

then the sequence $(x^{k})_{k\in{\mathbb{N}}}$ converges to a solution of the
VIP(F,${\cal C}$).

###### Proof.

Let $x^{*}\in{\cal C}^{*}$ be a arbitrary solution of the VIP(F,${\cal C}$).
The condition (13) and $0<\bar{\gamma}<1/2$ imply that ${\bar{\eta}}>0$ and
${\bar{\nu}}>0$. Thus, it follows from Lemma 8 that

$\|x^{k+1}-x^{*}\|^{2}\leq\|x^{k}-x^{*}\|^{2}+{\bar{\nu}}\gamma_{k}\|F(x^{k})\|^{2},\qquad
k=1,2,\ldots.$

The last inequality together with (8) implies that
$(x^{k})_{k\in{\mathbb{N}}}$ is quasi-Fejér convergent to ${\cal C}^{*}$.
Considering that ${\cal C}^{*}$ is nonempty, the item (i) of Lemma 3 implies
that $(x^{k})_{k\in{\mathbb{N}}}$ is bounded. Let ${\bar{x}}$ be a cluster
point of $(x_{k})_{k\in\mathbb{N}}$ and $(x_{k_{j}})_{j\in\mathbb{N}}$ a
subsequence of $(x_{k})_{k\in\mathbb{N}}$ such that
$\lim_{j\to+\infty}x_{k_{j}}={\bar{x}}$. To continue the proof, keep in mind
that Lemma 8 also implies that

${\bar{\eta}}\|x^{k}-y^{k}\|^{2}\leq\|x^{k}-x^{*}\|^{2}-\|x^{k+1}-x^{*}\|^{2}+{\bar{\nu}}\gamma_{k}\|F(x^{k})\|^{2},\qquad
k=1,2,\ldots.$

By adding both sides of the previous inequality and using (8), we arrive at
the conclusion that

${\bar{\eta}}\sum_{k=0}^{+\infty}\|x^{k}-y^{k}\|^{2}\leq\|x^{1}-x^{*}\|^{2}+{\bar{\nu}}\sum_{k=0}^{+\infty}\gamma_{k}\|F(x^{k})\|^{2}<+\infty.$

Hence, we have $\lim_{k\to+\infty}\|x^{k}-y^{k}\|=0$. Thus, taking into
account that $\lim_{j\to+\infty}x_{k_{j}}={\bar{x}}$, we conclude that
$\lim_{j\to+\infty}y_{k_{j}}={\bar{x}}$. It follows from (6) that

$y^{k_{j}}\in{\cal P}^{\gamma_{k_{j}}}_{\cal
C}\big{(}x^{k_{j}},x^{k_{j}}-\alpha F(x^{k_{j}})\big{)}.$

Considering that $\gamma_{k_{j}}<\bar{\gamma}$, the last inclusion and
Definition 2 imply that

$\big{\langle}x^{k_{j}}-\alpha
F(x^{k_{j}})-y^{k_{j}},y-y^{k_{j}}\big{\rangle}\leq\gamma\|y^{k_{j}}-x^{k_{j}}\|^{2},\qquad\qquad~{}\forall
y\in{\cal C}.$

Since $\lim_{j\to+\infty}x_{k_{j}}={\bar{x}}$ and
$\lim_{j\to+\infty}y_{k_{j}}={\bar{x}}$, taking the limit in the previous
inequality as $j$ tending to infinity yields

$\big{\langle}{\bar{x}}-\alpha
F({\bar{x}})-{\bar{x}},y-{\bar{x}}\big{\rangle}\leq{\bar{\gamma}}\|{\bar{x}}-{\bar{x}}\|{},\qquad\qquad~{}\forall
y\in{\cal C}.$

which, by using that $\alpha>0$, is equivalent to
$\big{\langle}F({\bar{x}}),y-{\bar{x}}\big{\rangle}\geq 0$, for all $y\in{\cal
C}$. Hence, ${\bar{x}}\in{\cal C}^{*}$. Given that ${\bar{x}}$ is a cluster
point of $(x_{k})_{k\in\mathbb{N}}$, item (ii) of Lemma 3 implies that
$\lim_{k\to+\infty}x_{k}={\bar{x}}$, and the proof is complete. ∎

##  5 Extragradient inexact method with line search

In this section, we introduce an inexact variant of the extragradient method
for VIP($F,{\cal C}$) with $F$ pseudo monotone operator on ${\cal C}$ with
respect to the solution set ${\cal C}^{*}$ of problem 1, see for example [21,
3, 23]. Instead of a fixed step size, the method presented finds a suitable
step size in each iteration by performing an Armijo type line search. It is
worth noting that, like the classical extragradient method, the method does
just two projections into the feasible set in each iteration. A full
convergence analysis is provided, with no Lipschitz continuity assumption of
the operator defining the variational inequality problem.

The inexact version of the proposed version of the extragradient method is
stated as follows:

Algorithm 2 Extragradient inexact projection method with line search-EInexPMLS

1:Take $0<{\hat{\beta}}\leq{\bar{\beta}}$ and $(\beta_{k})_{k\in\mathbb{N}}$
satisfying $0<{\hat{\beta}}\leq\beta_{k}\leq{\bar{\beta}}$. Take also
$\sigma$, $\rho$ and $\alpha\in(0,1)$, and
$0<\bar{\gamma}<\min\big{\\{}1-\rho,2-\sqrt{3}\big{\\}}.$ (14) Let
$x^{1}\in{\cal C}$ and set $k=1$.

2: Choose a error tolerance $\gamma_{k}$ such that
$0\leq\gamma_{k}<\bar{\gamma}$ and compute the following feasible inexact
projection:  $y^{k}\in{\cal P}^{\gamma_{k}}_{\cal
C}\big{(}x^{k},x^{k}-\beta_{k}F(x^{k})\big{)}.$ (15)

3: If $y^{k}=x^{k}$, then stop; otherwise, compute
$i_{k}:=\min\Big{\\{}i\in{\mathbb{N}}:~{}\big{\langle}F{\big{(}}x^{k}+\sigma\alpha^{i}(y^{k}-x^{k}){\big{)}},y^{k}-x^{k}\big{\rangle}\leq\rho\big{\langle}F(x^{k}),y^{k}-x^{k}\big{\rangle}\Big{\\}},$
(16) and set  $z^{k}:=x^{k}+\sigma\alpha^{i_{k}}(y^{k}-x^{k}).$ (17)

4:Compute the next iteration as a feasible inexact projection:
$x^{k+1}\in{\cal P}^{\gamma_{k}}_{\cal
C}\big{(}x^{k},x^{k}-\lambda_{k}F(z^{k})\big{)},\qquad\quad\lambda_{k}:=-\frac{1}{\|F(z^{k})\|^{2}}\big{\langle}F(z^{k}),z^{k}-x^{k}\big{\rangle}.$
(18)

5: Update $k\leftarrow k+1$ and go to Step 2.

Let us go through the main features of the EInexPMLS. First, we must select
some parameters that will control the behaviour of the algorithm and will be
essential in your convergence analysis. The most important of these parameters
is the upper bound ${\bar{\gamma}}$ for error tolerance ${\gamma_{k}}$, which
is connected to the line search parameter $\rho$. In step 2 of the algorithm,
by using an inner procedure compute $y^{k}$ as any feasible inexact projection
of $x^{k}-\beta_{k}F(x^{k})$ onto the feasible set ${\cal C}$ relative to
$x^{k}$, i.e. (15). Then, in step 3, the conceptual stopping criterion
$y^{k}=x^{k}$ is then evaluated in the current iteration. If this stopping
criterion is not satisfied, a line search in the segment between the points
$x^{k}$ and $y^{k}$ is done in order to decrease the mapping

$(0,1)\ni t\mapsto\langle F(x^{k}+t(y^{k}-x^{k})),y^{k}-x^{k}\rangle.$

In step 4, the line search resultant point $z^{k}$ is utilized to define the
following half space

$H_{k}:=\Big{\\{}x\in{\mathbb{R}}^{n}:~{}\langle F(z^{k}),x-z^{k}\rangle\leq
0\Big{\\}},$ (19)

whose boundary separates the current iterate $x_{k}$ of the solution set
${\cal C}^{*}$. Then, by applying Proposition 2, is computed the projection of
$x^{k}$ onto the hyperplane $H_{k}$ as follows

${\cal{P}}_{H_{k}}(x^{k})=x^{k}-\lambda_{k}F(z^{k}),\qquad\quad\lambda_{k}:=-\frac{1}{\|F(z^{k})\|^{2}}\big{\langle}F(z^{k}),z^{k}-x^{k}\big{\rangle}.$
(20)

Finally, using again an inner procedure, the next iterate $x^{k+1}$ is
computed as any feasible inexact projection of ${\cal{P}}_{H_{k}}(x^{k})$ onto
the feasible set ${\cal C}$ relative to $x^{k}$. Thus, by using (20), we
conclude that (18) is equivalently stated as follows

$x^{k+1}\in{\cal P}^{\gamma_{k}}_{\cal
C}\big{(}x^{k},{\cal{P}}_{H_{k}}(x^{k})\big{)},$ (21)

It is noteworthy that if $\gamma_{k}\equiv 0$, then Remark 1 implies that
inexact projection is the exact one. Consequently, EInexPMLS corresponds to a
version of the extragradient method addressed in [21], see also [3, 8].

######  Remark 2.

The stopping criteria is well defined, i.e., if $x^{k}=y^{k}$, then $x^{k}$ is
a solution of the Problem 1. In fact, $y^{k}=x^{k}$ and (15) implies that
$x^{k}\in{\cal P}^{\gamma_{k}}_{\cal
C}\big{(}x^{k},x^{k}-\alpha_{k}F(x^{k})\big{)}.$ Thus, it follows from
Corollary 6 that $x^{k}\in{\cal C}^{*}$.

Next, we show that the Algorithm 2 is well defined, namely that there exists
$i_{k}$ fulfilling (16).

######  Proposition 10.

Step 3 is well-defined, i.e., there exists $i_{k}$ satisfying 16.

###### Proof.

Assume by contradiction that
$\big{\langle}F{\big{(}}x^{k}+\sigma\alpha^{i}(y^{k}-x^{k}){\big{)}},y^{k}-x^{k}\big{\rangle}>\big{\langle}F(x^{k}),y^{k}-x^{k}\big{\rangle}$,
for all $i\in{\mathbb{N}}$ and $y^{k}\neq x^{k}$. Since $F$ is continuous and
$0<\alpha<1$, taking the limit in the last inequality as $i$ tends to
infinity, we conclude that
$\big{\langle}F(x^{k}),y^{k}-x^{k}\big{\rangle}\geq\rho\big{\langle}F(x^{k}),y^{k}-x^{k}\big{\rangle}.$
Thus, taking into account that $0<\rho<1$, the last inequality that

$\big{\langle}F(x^{k}),y^{k}-x^{k}\big{\rangle}\geq 0.$ (22)

Considering that $y^{k}\in{\cal P}^{\gamma_{k}}_{\cal
C}(x^{k},x^{k}-\beta_{k}F(x^{k}))$ and $x^{k}\in{\cal C}$, it follows from
Definition 2 that

$\langle
x^{k}-\beta_{k}F(x^{k})-y^{k},x^{k}-y^{k}\rangle\leq\gamma_{k}\|y^{k}-x^{k}\|^{2}.$

Since $0<{\hat{\beta}}\leq\beta_{k}$ and $0<\bar{\gamma}<1$, we can deduce
from some algebraic manipulations in the preceding inequality that

$0\leq\big{\langle}F(x^{k}),y^{k}-x^{k}\big{\rangle}\leq\frac{\gamma_{k}-1}{\beta_{k}}\|y^{k}-x^{k}\|^{2}<0,$

which is a contradiction. Therefore, there exists $i_{k}$ satisfying (16). ∎

Let $(x_{k})_{k\in\mathbb{N}}$ be the sequence generated by Algorithm 2. Since
$x^{1}\in{\cal C}$ and, for any $k\in\mathbb{N}$, (18) implies that $x^{k+1}$
is a feasible inexact projection onto ${\cal C}$, we conclude that
$(x^{k})_{k\in\mathbb{N}}\subset{\cal C}$. Consequently, due to ${\cal C}$ be
a closed set, each clustering point of $(x^{k})_{k\in\mathbb{N}}$ that exists
belongs to ${\cal C}$.

###  5.1 Convergence analysis

In the previous section, we show that EInexPMLs generates a
$(x^{k})_{k\in\mathbb{N}}$ belonging to the set ${\cal C}$. In this section we
will show that $(x^{k})_{k\in\mathbb{N}}$ converges to a solution of VIP(F,
${\cal C}$). To this purpose, we will begin by establishing a few initial
results. First, we show that the boundary of the half space $H_{k}$ defined as
in (19) separates the current iterates $x_{k}$ of the solution set ${\cal
C}^{*}$.

######  Proposition 11.

Let $H_{k}$ be defined as in (19). Then, $x^{k}\in H_{k}$ if and only if
$x^{k}\in\mathcal{C}^{*}$.

###### Proof.

First, we assume that $x^{k}\in H(z^{k})$. Thus, taking into account (17) we
conclude that

$0\geq\langle F(z^{k}),x^{k}-z^{k}\rangle=\sigma\alpha^{i_{k}}\langle
F(z^{k}),x^{k}-y^{k}\rangle,$

which implies that $\langle F(z^{k}),y^{k}-x^{k}\rangle\geq 0$. Hence, by
using (16) and (17), we conclude that
$\big{\langle}F(x^{k}),y^{k}-x^{k}\big{\rangle}\geq 0$. Therefore, using (15)
together with Corollary 6 we conclude that $x^{k}\in\mathcal{C}^{*}$.

Conversely, we assume that $x^{k}\in\mathcal{C}^{*}$. Since $x^{k}$ and
$y^{k}$ belong to ${\cal C}$, it follows from (17) that $z^{k}\in\mathcal{C}$
by convexity of $\mathcal{C}$, for all $k\in\mathbb{N}$. Thus, due to
$x^{k}\in\mathcal{C}^{*}$, we have $\langle F(x^{k}),z^{k}-x^{k}\rangle\geq
0$. Since $F$ is a pseudo-monotone operator, $\langle
F({x}^{k}),z^{k}-x^{k}\rangle\geq 0$ implies $\langle
F(z^{k}),z^{k}-x^{k}\rangle\geq 0$, which means that $x^{k}\in H_{k}$. ∎

######  Proposition 12.

The following inequality holds:

$\big{\langle}F(x^{k}),x^{k}-y^{k}\big{\rangle}\geq\frac{\max\\{\rho,\sqrt{3}-1\\}}{\bar{\beta}}\|y^{k}-x^{k}\|^{2},\qquad
k=1,2,\ldots.$ (23)

###### Proof.

Keeping in mind that $y^{k}\in{\cal P}^{\gamma_{k}}_{\cal
C}(x^{k},x^{k}-\beta_{k}F(x^{k}))$ and $x^{k}\in{\cal C}$, from Definition 2
we have

$\langle
x^{k}-\beta_{k}F(x^{k})-y^{k},x^{k}-y^{k}\rangle\leq\gamma_{k}\|y^{k}-x^{k}\|^{2},$

which, after some algebraic manipulation, is rewritten as follows

$\langle
F(x^{k}),x^{k}-y^{k}\rangle\geq\frac{1-\gamma_{k}}{\beta_{k}}\|y^{k}-x^{k}\|^{2}.$
(24)

Considering that $0<\beta_{k}\leq{\bar{\beta}}$ and
$0\leq\gamma_{k}<\bar{\gamma}<\min\big{\\{}1-\rho,2-\sqrt{3}\big{\\}}$, we
conclude that

$\frac{1-\gamma_{k}}{\beta_{k}}\geq\frac{\max\\{\rho,\sqrt{3}-1\\}}{\bar{\beta}}.$

The combination of (24) with the previous inequality yields the desired
inequality. ∎

Next, we are going to establish two important inequalities to show the
convergence of Algorithm 1.

######  Lemma 13.

Let ${\cal C}\subset{\mathbb{R}}^{n}$ be a nonempty and closed convex set and
$(x^{k})_{k\in{\mathbb{N}}}$ the sequence generated by Algorithm 1. Assume
that $F:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ is a pseudo monotone operator on
${\cal C}$ with respect to the solution set ${\cal C}^{*}$ of problem 1 and
$x^{k}\notin{\cal C}^{*}$, for all $k=1,2,\ldots$ Then, for any $x^{*}\in{\cal
C}^{*}$, there holds:

$\|x^{k+1}-x^{*}\|^{2}\leq\|x^{k}-x^{*}\|^{2}-\frac{1}{(1-{\bar{\gamma}})^{2}}\left({\bar{\gamma}}^{2}-4{\bar{\gamma}}+1\right)\lambda_{k}^{2}\|F(z^{k})\|^{2},\qquad
k=1,2,\ldots.$ (25)

As a consequence, $(x^{k})_{k\in{\mathbb{N}}}$ is Fejér convergent to ${\cal
C}^{*}$, i.e., for any $x^{*}\in{\cal C}^{*}$ there holds

$\|x^{k+1}-x^{*}\|\leq\|x^{k}-x^{*}\|,\qquad k=1,2,\ldots.$ (26)

###### Proof.

Take $x^{*}\in\mathcal{C}^{*}$. Denotes the boundary of $H_{k}$ by $L_{k}$,
which is given by

$L_{k}:=\\{x\in{\mathbb{R}}^{n}:~{}\langle F(z^{k}),x-z^{k}\rangle=0\\}.$ (27)

Since $x^{k}$ and $y^{k}$ belong to the convex set ${\cal C}$, we obtain from
(17) that $z^{k}\in\mathcal{C}$, for all $k\in\mathbb{N}$. Thus, due to $F$ be
pseudo monotone on ${\cal C}$ with respect to the solution set ${\cal C}^{*}$
of problem 1, we have

$\big{\langle}F(z^{k}),x^{k}-{x^{*}}\big{\rangle}\geq 0.$

The last inequality and the definition of $H_{k}$ in (19) imply that $x^{*}\in
H_{k}$. Thus, we conclude that

${\cal{P}}_{H_{k}}(x^{*})=x^{*}.$ (28)

We know as well that applying Proposition 4 with
$v=x^{k}-\lambda_{k}F(z^{k})$, $u=x^{k}$, $\gamma=\gamma_{k}$, ${w}=x^{k+1}$
and ${\bar{w}}={\bar{v}}=x^{*}$ we obtain that

$\|x^{k+1}-x^{*}\|^{2}\leq\|x^{k}-\lambda_{k}F(z^{k})-x^{*}\|^{2}-\|(x^{k}-\lambda_{k}F(z^{k})-x^{*})-({x^{k+1}}-x^{*})\|^{2}+2\gamma_{k}\|x^{k+1}-x^{k}\|^{2},$

which implies that

$\|x^{k+1}-x^{*}\|^{2}\leq\|x^{k}-\lambda_{k}F(z^{k})-x^{*}\|^{2}+2\gamma_{k}\|x^{k+1}-x^{k}\|^{2}.$
(29)

Using (28) and the item $(ii)$ of Lemma 1 we have

$\|{\cal{P}}_{H_{k}}(x^{k})-x^{*}\|^{2}\leq\|x^{k}-x^{*}\|^{2}-\|{\cal{P}}_{H_{k}}(x^{k})-x^{k}\|^{2}.$

Since $x^{k}\notin{\cal C}^{*}$, Proposition 11 implies that $x^{k}\notin
H_{k}$. Hence, the last inequality together with the first equality in (20)
yield

$\|x^{k}-\lambda_{k}F(z^{k})-x^{*}\|^{2}\leq\|x^{k}-x^{*}\|^{2}-\lambda_{k}^{2}\|F(z^{k})\|^{2}.$

As a result of combining the last inequality with (29), we arrive at the
conclusion that

$\|x^{k+1}-x^{*}\|^{2}\leq\|x^{k}-x^{*}\|^{2}-\lambda_{k}^{2}\|F(z^{k})\|^{2}+2\gamma_{k}\|x^{k+1}-x^{k}\|^{2}.$
(30)

Since $x^{k+1}\in{\cal P}^{\gamma_{k}}_{\cal
C}\big{(}x^{k},x^{k}-\lambda_{k}F(z^{k})\big{)}$, applying item (ii) of Lemma
5 with $\gamma=\gamma_{k}$, $x=x^{k}$, $z=z^{k}$, $\alpha=\lambda_{k}$ and
$w(\alpha)=x^{k+1}$ we obtain that

$\|x^{k+1}-x^{k}\|\leq\dfrac{\lambda_{k}}{1-\gamma_{k}}\|F(z^{k})\|,$

which combined with (30) yields

$\|x^{k+1}-x^{*}\|^{2}\leq\|x^{k}-x^{*}\|^{2}-\lambda_{k}^{2}\|F(z^{k})\|^{2}+2\gamma_{k}\dfrac{\lambda_{k}^{2}}{(1-\gamma_{k})^{2}}\|F(z^{k})\|^{2},$

or equivalently,

$\|x^{k+1}-x^{*}\|^{2}\leq\|x^{k}-x^{*}\|^{2}-\frac{1}{(1-\gamma_{k})^{2}}\left(\gamma_{k}^{2}-4\gamma_{k}+1\right)\lambda_{k}^{2}\|F(z^{k})\|^{2}.$
(31)

Since $0<\bar{\gamma}<2-\sqrt{3}$, the function
$(0,{\bar{\gamma}}]\mapsto\left(\gamma_{k}^{2}-4\gamma_{k}+1\right)/(1-\gamma_{k})^{2}$
is increasing and positive, the inequality (25) follows (31). As a
consequence, (26) follows from (25) and the proof is complete. ∎

######  Theorem 14.

Let ${\cal C}\subset{\mathbb{R}}^{n}$ be a nonempty and closed convex set and
$(x^{k})_{k\in{\mathbb{N}}}$ the sequence generated by Algorithm 1. Assume
that $F:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ is a pseudo monotone operator on
${\cal C}$ with respect to the solution set ${\cal C}^{*}$ of problem 1. If
${\cal C}^{*}\neq\varnothing$, then Algorithm 1 either ends at iteration $k$,
in which case $x^{k}\in{\cal C}^{*}$, or generates an infinite sequence
$(x^{k})_{k\in{\mathbb{N}}}$ that converges to a point belonging to ${\cal
C}^{*}$.

###### Proof.

First, we assume that Algorithm 1 ends at iteration $k$. In this case, we have
$y^{k}=x^{k}$ and Remark 2 implies that $x^{k}\in{\cal C}^{*}$. Now, we assume
that the sequence $(x^{k})_{k\in{\mathbb{N}}}$ is infinite. Hence, we have
$x^{k}\notin{\cal C}^{*}$, for all $k=1,2,\ldots$.

Since $(x^{k})_{k\in{\mathbb{N}}}$ satisfies (26) in Lemma 13, it also
satisfies Definition 1. Thus, due to ${\cal C}^{*}\neq\varnothing$, it follows
from item $(ii)$ of Lemma 3 that $(x^{k})_{k\in{\mathbb{N}}}$ is bounded.
Using (25) of Lemma 13 we have

$0<\frac{1}{(1-{\bar{\gamma}})^{2}}\left({\bar{\gamma}}^{2}-4{\bar{\gamma}}+1\right)\lambda_{k}^{2}\|F(z^{k})\|^{2}\leq\|x^{k}-x^{*}\|^{2}-\|x^{k+1}-x^{*}\|^{2},\qquad
k=1,2,\ldots.$ (32)

On the other hand, (26) implies that the sequence
$(\|x^{k}-x^{*}\|)_{k\in{\mathbb{N}}}$ is monotone non-increasing and bounded
from below. Thus, $(\|x^{k}-x^{*}\|)_{k\in{\mathbb{N}}}$ converges. Hence,
taking the limit in (32) as $k$ tends to infinity, we have
$\lim_{k\to+\infty}\lambda_{k}\|F(z^{k})\|=0.$ And, in view of (18) we
conclude that

$\lim_{k\to+\infty}\lambda_{k}\|F(z^{k})\|=\lim_{k\to+\infty}\frac{1}{\|F(z^{k})\|}\big{\langle}F(z^{k}),z^{k}-x^{k}\big{\rangle}=0.$
(33)

Since $y^{k}\in{\cal P}^{\gamma_{k}}_{\cal
C}\big{(}x^{k},x^{k}-\beta_{k}F(x^{k})\big{)}$, applying item $(i)$ of
Corollary 7 with $\gamma=\gamma_{k}$, $x=x^{k}$, $\alpha=\beta_{k}$ and
$y=y^{k}$ we obtain that

$\|y^{k}-x^{k}\|\leq\dfrac{\beta_{k}}{1-\beta_{k}}\|F(x^{k})\|.$

Because $0<{\hat{\beta}}<\beta_{k}<{\bar{\beta}}$,
$0\leq\gamma_{k}<\bar{\gamma}$ and $(x^{k})_{k\in{\mathbb{N}}}$ is bounded,
the latter inequality implies that $(y^{k})_{k\in{\mathbb{N}}}$ is bounded.
Hence, it follows from (17) that $(z^{k})_{k\in{\mathbb{N}}}$ is also bounded.
In addition, due to $F$ be continuous, we conclude that
$(F(z^{k}))_{k\in{\mathbb{N}}}$ is bounded. Thus, from (33) we have
$\lim_{k\to+\infty}\big{\langle}F(z^{k}),z^{k}-x^{k}\big{\rangle}=0.$
Therefore, it follows from the last equality and (17) that

$\lim_{k\to+\infty}\sigma\alpha^{i_{k}}\big{\langle}F(z^{k}),y^{k}-x^{k}\big{\rangle}=0.$
(34)

Since the sequences $(x^{k})_{k\in{\mathbb{N}}}\subset{\cal C}$,
$(y^{k})_{k\in{\mathbb{N}}}\subset{\cal C}$ and
$(z^{k})_{k\in{\mathbb{N}}}\subset{\cal C}$ are bounded, we can take
subsequences $(x^{k_{j}})_{j\in{\mathbb{N}}}$,
$(y^{k_{j}})_{j\in{\mathbb{N}}}$ and $(z^{k_{j}})_{j\in{\mathbb{N}}}$ of them,
respectively, and ${\bar{x}}\in{\cal C}$, ${\bar{y}}\in{\cal C}$ and
${\bar{z}}\in{\cal C}$ such that $\lim_{j\to+\infty}x^{k_{j}}={\bar{x}}$,
$\lim_{j\to+\infty}y^{k_{j}}={\bar{y}}$ and
$\lim_{j\to+\infty}z^{k_{j}}={\bar{z}}$. Furthermore, due to $0<\alpha<1$,
$0\leq\gamma_{k}<\bar{\gamma}$ and $0<{\hat{\beta}}<\beta_{k}<{\bar{\beta}}$
for all $k\in{\mathbb{N}}$, we can also assume without loss of generality that
$\lim_{j\to+\infty}\alpha^{i_{k_{j}}}=\bar{\alpha}\in[0,1]$,
$\lim_{j\to+\infty}\gamma_{k_{j}}={\hat{\gamma}}\leq{\bar{\gamma}}$ and
$\lim_{j\to+\infty}\beta_{k_{j}}={\tilde{\beta}}\geq{\hat{\beta}}$. We have
two possibilities for $\bar{\alpha}$: $\bar{\alpha}>0$ or $\bar{\alpha}=0$.

First we assume that $\bar{\alpha}>0$. In this case, it follows from (34) that

$0=\lim_{j\to+\infty}{\sigma\alpha^{i_{k_{j}}}}\big{\langle}F(z^{k_{j}}),y^{k_{j}}-x^{k_{j}}\big{\rangle}=\sigma{\bar{\alpha}}\big{\langle}F({\bar{z}}),{\bar{y}}-{\bar{x}}\big{\rangle}.$

Because we are assuming that ${\bar{\alpha}}>0$, we conclude that $\langle
F({\bar{z}}),{\bar{y}}-{\bar{x}}\rangle=0$. Using, (16) and (17) together with
Proposition 12 we conclude that

$\big{\langle}F(z^{k_{j}}),y^{k_{j}}-x^{k_{j}}\big{\rangle}\leq\rho\big{\langle}F(x^{k_{j}}),y^{k_{j}}-x^{k_{j}}\big{\rangle}\leq-\rho\frac{\max\\{\rho,\sqrt{3}-1\\}}{\bar{\beta}}\|y^{k_{j}}-x^{k_{j}}\|^{2}.$

Taking the limit in the previous inequality as $j$ tending to infinity and
taking into account $\lim_{j\to+\infty}x^{k_{j}}={\bar{x}}$,
$\lim_{j\to+\infty}y^{k_{j}}={\bar{y}}$ and $\langle
F({\bar{z}}),{\bar{y}}-{\bar{x}}\rangle=0$, we conclude that
${\bar{y}}={\bar{x}}$. Considering that $y^{k}\in{\cal P}^{\gamma_{k}}_{\cal
C}\big{(}x^{k},x^{k}-\beta_{k}F(x^{k})\big{)}$, it follows from Definition 2
that

$\big{\langle}x^{k_{j}}-\beta_{k_{j}}F(x^{k_{j}})-y^{k_{j}},y-y^{k_{j}}\big{\rangle}\leq\gamma_{k_{j}}\|y^{k_{j}}-x^{k_{j}}\|^{2},\qquad\quad~{}\forall
y\in{\cal C}.$

Thus, taking the limit in the previous inequality as $j$ tending to infinity,
using that ${\bar{y}}={\bar{x}}$ and
$\lim_{j\to+\infty}\beta_{k_{j}}={\tilde{\beta}}>0$, we obtain that

$\left\langle F({\bar{x}}),y-{\bar{x}}\right\rangle\geq
0,\qquad\forall~{}y\in{\cal C},$

which implies that ${\bar{x}}\in{\cal C}^{*}$. Since ${\bar{x}}$ is a cluster
point of $(x_{k})_{k\in\mathbb{N}}$ and the sequence
$(x_{k})_{k\in\mathbb{N}}$ is Fejér convergent to ${\cal C}^{*}$, item (ii) of
Lemma 3 implies that $\lim_{k\to+\infty}x_{k}={\bar{x}}$.

Now, let us assume that $\bar{\alpha}=0$. We proceed to prove that in this
case, $(x^{k})_{k\in{\mathbb{N}}}$ likewise converges to some point belonging
to the set ${\cal C}^{*}$. For that, we consider the auxiliary sequence
$({\hat{z}}_{k})_{k\in\mathbb{N}}$ defined by

${\hat{z}}^{k}:=x^{k}+\sigma\frac{\alpha^{i_{k}}}{\alpha}(y^{k}-x^{k}),\qquad\qquad
k=1,2,\ldots,$ (35)

where $i_{k}$ is defined in (16). Since
$\lim_{j\to+\infty}x^{k_{j}}={\bar{x}}$,
$\lim_{j\to+\infty}y^{k_{j}}={\bar{y}}$ and
$\lim_{j\to+\infty}\alpha_{k_{j}}=\bar{\alpha}=0$, it follows from (35) that
$\lim_{j\to+\infty}{\hat{z}}^{k_{j}}={\bar{x}}$. The definition of
${i_{k_{j}}}$ in (16) implies that

$\big{\langle}F{\big{(}}{\hat{z}}^{k_{j}}{\big{)}},y^{k_{j}}-x^{k_{j}}\big{\rangle}>\rho\big{\langle}F(x^{k_{j}}),y^{k_{j}}-x^{k_{j}}\big{\rangle}.$
(36)

Thus, (36) implies that
$\big{\langle}F({\bar{x}}),{\bar{y}}-{\bar{x}}\big{\rangle}\geq\rho\big{\langle}F({\bar{x}}),{\bar{y}}-{\bar{x}}\big{\rangle}$,
and since $\rho<1$ we conclude that

$\big{\langle}F({\bar{x}}),{\bar{y}}-{\bar{x}}\big{\rangle}\geq 0.$ (37)

Given that $y^{k}\in{\cal P}^{\gamma_{k}}_{\cal
C}\big{(}x^{k},x^{k}-\alpha_{k}F(x^{k})\big{)}$, it follows from Definition 2
that

$\big{\langle}x^{k_{j}}-\beta_{k_{j}}F(x^{k_{j}})-y^{k_{j}},y-y^{k_{j}}\big{\rangle}\leq\gamma_{k_{j}}\|y^{k_{j}}-x^{k_{j}}\|^{2},\qquad\quad~{}\forall
y\in{\cal C}.$

Taking the limit in the previous inequality as $j$ going to infinity, and
using that ${\hat{\gamma}}\leq{\bar{\gamma}}$, we have

$\langle{\bar{x}}-{\bar{y}},y-{\bar{y}}\rangle-{\tilde{\beta}}\big{\langle}F({\bar{x}}),y-{\bar{y}}\big{\rangle}\leq{\bar{\gamma}}\|{\bar{y}}-{\bar{x}}\|^{2},\qquad\quad~{}\forall
y\in{\cal C}.$ (38)

Substituting $y\in{\cal C}$ for ${\bar{x}}\in{\cal C}$ in the last inequality,
after some algebraic manipulations yields

${\tilde{\beta}}\big{\langle}F({\bar{x}}),{\bar{y}}-{\bar{x}}\big{\rangle}\leq({\bar{\gamma}}-1)\|{\bar{y}}-{\bar{x}}\|^{2}.$
(39)

Combining (37) with the latter inequality we obtain that
$(1-{\bar{\gamma}})\|{\bar{y}}-{\bar{x}}\|^{2}\leq 0$. Hence, because (14)
implies that $1-{\bar{\gamma}}>0$, we conclude that ${\bar{y}}={\bar{x}}$.
Therefore, due to ${\tilde{\beta}}>0$ and ${\bar{y}}={\bar{x}}$, it follows
from (38) that

$\big{\langle}F({\bar{x}}),y-{\bar{x}}\big{\rangle}\geq
0,\qquad\quad~{}\forall y\in{\cal C},$

which also implies that ${\bar{x}}\in{\cal C}^{*}$. Again, because ${\bar{x}}$
is a cluster point of $(x_{k})_{k\in\mathbb{N}}$ and the sequence
$(x_{k})_{k\in\mathbb{N}}$ is Fejér convergent to ${\cal C}^{*}$, item (ii) of
Lemma 3 implies that $\lim_{k\to+\infty}x_{k}={\bar{x}}$ and the proof is
concluded. ∎

##  6 Numerical Results

In order to demonstrate the behaviour of the proposed algorithm we now present
the results of numerical experiments. We implemented Algorithms 1 and 2 and
applied them on two test problems adapted from [8]. In both cases we slightly
change the feasible set $C$ to be the norm 10 unit ball defined by
$C=\\{x\in\mathbb{R}^{2}:{x_{1}}^{10}+{x_{2}}^{10}\leq 1\\}$. Projections on
this set are more challenging than the projection over the feasible sets (the
Euclidean ball and the set $[0,1]\times[0,1]$ in the original problems. The
algorithms were implemented in the Julia language. The code can be obtained
from https://github.com/ugonj/extragradient.

###  6.1 Lipschitz operator

In this section, we show the convergence of Algorithm 1 on a Lipschitz
continuous operator.

Consider the operator

$T_{1}(x)=\begin{bmatrix}-1&-1\\\ 1&-1\end{bmatrix}x+\begin{bmatrix}3/2\\\
1/2\end{bmatrix}$

We applied Algorithm 1 on VIP($T_{1}$,$C$). The iterates are depicted in
Figure 1.

$-1.0$$-0.8$$-0.6$$-0.4$$-0.2$$0.0$$0.90$$0.92$$0.94$$0.96$$0.98$Results for
exact and inexact projections for $\alpha$=0.31exactapproximate Figure 1:
Convergence of Algorithm 1 with exact and inexact projections. On this problem
the number of steps is the same in both cases.

Table 1 shows how the algorithm performed for various values of $\alpha$ and
$\gamma$. The third columns represent the number of steps taken by the
extragradient algorithm to reach the solution, and the last column represents
the total number of linear searches applied by the Frank-Wolfe method. It is
interesting to compare the case when $\gamma$ is small (say, $\gamma=0.01$) to
the cases when $\gamma$ is larger: it can be seen that although the algorithm
takes the same number of steps to reach the solution, when $gamma\approx 0$
(in the case when the projection is almost exact), significantly more Frank-
Wolfe iterations were performed. In other words, since performing approximate
projections does not increase the total number of steps in the extragradient
method, it is beneficial to use them, as each step of this method requires
less work.

$\mathbf{\alpha}$ | $\mathbf{\gamma}$ | N. steps | N. linear searches
---|---|---|---
0.01 | 0.01 | 109 | 1.0317e7
0.01 | 0.106 | 109 | 6.62431e6
0.01 | 0.49 | 109 | 6.57129e6
0.11 | 0.01 | 16 | 12997
0.11 | 0.106 | 16 | 1085
0.11 | 0.394 | 16 | 966
0.21 | 0.01 | 11 | 2444
0.21 | 0.106 | 11 | 237
0.21 | 0.394 | 11 | 239
0.31 | 0.01 | 9 | 935
0.31 | 0.106 | 9 | 129
0.31 | 0.298 | 9 | 126
0.41 | 0.01 | 9 | 476
0.41 | 0.106 | 9 | 113
Table 1: Behaviour of the algorithm for various values of $\alpha$ and
$\gamma$.

###  6.2 Non-Lipschitz operator

Let $t(x_{1},x_{2})=(x_{1}+\sqrt{x_{1}^{2}+4x_{2}})/2$ and define
$T_{2}(x_{1},x_{2})=-t/(1+t)(1,1)$. The operator $T$ is quasimonotone, and it
is pseudomonotone with respect to the solution set, as it has a unique
solution $(1,1)$ (see [8]). However it is not Lipschitz. We applied Algorithm
2 on VIP($T_{2}$,$C$), with initial point at $(0,1)$. The iterates are
depicted in Figure 1.

$0.0$$0.2$$0.4$$0.6$$0.8$$1.0$$0.90$$0.92$$0.94$$0.96$$0.98$Results for
$\alpha$=0.9, $\rho$$\approx$ 0.73, $\beta$=0.9, $\sigma$=0.99iterates Figure
2: Convergence of Algorithm 2.

##  7 Conclusions

We investigate the Extragradient method for solving variational inequality
problems using inexact projections onto the feasible set in this paper. We
expect that our study can contribute to further research on the subject,
particularly in solving large-scale problems where the computing effort of
each iteration is connected with projections into the feasible set. Indeed,
the idea of employing inexactness in the projection rather than exactness is
very attractive from a computational perspective. It is worth noting that the
Frank-Wolfe approach has a low computing cost for each iteration, resulting in
great computational performance in various types of compact sets, as cited in
[18, 22]. Searching for new efficient methods, such as the Frank-Wolfe method
which generates inexact projections, is a subject that calls for attention.

##  8 Acknowledgements

The first and last authors were supported by the Australian Research Council
(ARC), Solving hard Chebyshev approximation problems through nonsmooth
analysis (Discovery Project DP180100602).

The second author was supported in part by CNPq grant 304666/2021-1.

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11institutetext: Departamento de Física, Universidad de Murcia, 30071 Murcia,
Spain 22institutetext: Department of Physics, Hebei Normal University,
Shijiazhuang 050024, China

# Compositeness and several applications to exotic hadronic states with heavy
quarks

José Antonio Oller 11<EMAIL_ADDRESS>Zhi-Hui Guo 22<EMAIL_ADDRESS>
###### Abstract

Several methods for studying the nature of a resonance are applied to
resonances recently discovered in the bottonomium and charmonium sectors. We
employ the effective-range expansion, the saturation of the width and
compositeness of a resonance, as well as direct fits to data. The latter stem
from generic $S$-matrix parameterization that account for relevant dynamical
features associated to channels that couple strongly in an energy region
around the resonance masses, in which their thresholds also lie. We report on
results obtained with these methods for the resonances $Z_{b}(10610)$,
$Z_{b}(10650)$, $Z_{cs}(3985)$, $Z_{c}(3900)$, $X(4020)$, $X(6900)$,
$X(6825)$, and $P_{cs}(4459)$.

##  1 Introduction

We use the methods reviewed in Ref. oller2.theseproceedings , and more details
can be found there. In the subsequent we denote this reference by [I], and we
just write here the basic formulas that we are going to use. The compositeness
$X$ from the effective-range expansion (ERE) is calculated as
$X={k_{i}}/{k_{r}}$, with the resonance momentum
$k_{R}=\sqrt{2\mu(E_{R}-m_{\rm th})}=k_{r}-ik_{i}$ calculated in the second
Riemann sheet (RS), so that $k_{r}\,,k_{i}\geq 0$. $E_{R}=M_{R}-i\Gamma/2$ is
the resonance pole position with mass $M_{R}>m_{\rm th}$ ($m_{\rm th}$ is the
threshold of the active channel) and width $\Gamma$.

Beyond the ERE we also use the following formula for calculating the partial
compositeness coefficient $X_{i}$ of the $i_{\rm th}$ channel with its
threshold above the resonance mass,

$\displaystyle X_{i}$
$\displaystyle=\left|g_{i}^{2}\frac{dG_{i}^{II}(s_{R})}{ds}\right|\leavevmode\nobreak\
.$ (1)

The superscript ${II}$ on the unitarity loop function $G_{i}(s)$ indicates
that it is calculated in the second RS for those channels in which the transit
to this sheet is involved to find the pole at $s_{R}$, otherwise it remains in
the first RS ($s$ is the standard the Mandelstam variable). The function
$G_{i}(s)$ is given in [I]. The couplings $g_{i}$ are calculated from the
residues of the partial-wave amplitudes (PWAs) at the resonance pole position.
In terms of them the partial decay width $\Gamma_{i}$ is calculated from two
equations, depending on whether the threshold of the $i_{\rm th}$ channel is
relatively far or close to $M_{R}$ compared to its width $\Gamma$,
respectively, as

$\displaystyle\Gamma_{i}$ $\displaystyle=\frac{|g_{i}|^{2}}{8\pi
M_{R}^{2}}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \text{ or
}\leavevmode\nobreak\ \Gamma_{i}=|g_{i}|^{2}\int_{m_{\rm
th,i}}^{M_{R}+n\Gamma_{R}}dw\,\frac{q_{i}(w^{2})}{16\pi^{2}\,w^{2}}\frac{\Gamma}{(M_{R}-w)^{2}+\Gamma^{2}/4}\leavevmode\nobreak\
,\leavevmode\nobreak\ n\gg 1\leavevmode\nobreak\ ,$ (2)

with $q_{i}$ the center of mass momentum of the $i_{\rm th}$ channel.

##  2 Resonances $Z_{b}(10610)$ and $Z_{b}(10650)$

We review the application performed in Ref. Kang:2016ezb  of calculating $X$
within the ERE to study the nature of the bottonomium resonances
$Z_{b}(10610)$ and $Z_{b}(10650)$ (also called $Z_{b}$ and $Z_{b}^{\prime}$,
in this order). These are, respectively, $B^{(*)}\bar{B}^{*}$ systems with
$I^{G}(J^{P})=1^{+}(1^{+})$ discovered in Ref. Belle:2011aa  by the Belle
Collaboration. The mass and width of the resonances determined in this
reference are

$\displaystyle E_{Z_{b}}$ $\displaystyle=10607.2\pm 2.0-i(9.2\pm
1.2)\leavevmode\nobreak\ {\rm MeV}\leavevmode\nobreak\ ,$ (3) $\displaystyle
E_{Z^{\prime}_{b}}$ $\displaystyle=10652.2\pm 1.5-i(5.5\pm
1.1)\leavevmode\nobreak\ {\rm MeV}\leavevmode\nobreak\ ,$

with the resonance mass below the corresponding $B^{(*)}\bar{B}^{*}$ threshold
by around 3 MeV. The resulting values for the scattering length $a$, effective
range $r$, and the compositeness $X$ are given in Table 1.

Table 1: Summary of scattering and compositeness properties of the $Z_{b}$ and $Z_{b}^{\prime}$.  | $Z_{b}(10610)$ | $Z_{b}(10650)$
---|---|---
$a\,(\rm{fm})$ | $-1.03\pm 0.17$ | $-1.18\pm 0.26$
$r\,(\rm{fm})$ | $-1.49\pm 0.20$ | $-2.03\pm 0.38$
$X$ | $0.75\pm 0.15$ | $0.67\pm 0.16$

For the $Z_{b}$, $Z_{b}^{\prime}$ the method based on saturating the total
decay width gives results that are almost identical to those in Table 1, which
is not surprising since the ERE method assumes only one channel. However, when
saturating the partial decay width of the $Z_{b}^{(^{\prime})}$ into
$B^{(*)}\bar{B}^{*}$, measured too in Ref. Belle:2011aa , one can account for
coupled channels and the resulting compositeness coefficients for the $Z_{b}$
and $Z_{b}^{\prime}$ are, respectively,

$\displaystyle X_{Z_{b}}$ $\displaystyle=0.66\pm 0.11\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ X_{Z_{b}^{\prime}}=0.51\pm
0.10\leavevmode\nobreak\ .$ (4)

The numbers are still compatible with Table 1, but they are on the lower side
due to the saturation of the partial decay widths only.

##  3 Study of the nature of the $Z_{cs}(3985)$, $Z_{c}(3900)$, and $X(4020)$

We now report here on the unified study of Ref. Guo:2020vmu  on the nature of
the charmonium resonances $Z_{cs}(3985)$, $Z_{c}(3900)$, and $X(4020)$. The
lighter and heavier channels (the one with its threshold close to the
resonance mass) used by us for the study of every resonance are

$\displaystyle
Z_{c}(3900):J/\psi\pi,\,\bar{D}D^{*}/D\bar{D}^{*}(3875.5)\,;\,X(4020):h_{c}\pi,\,D^{*}\bar{D^{*}}(4017.1)\,;$
(5) $\displaystyle Z_{cs}(3985):J/\psi K^{-},\leavevmode\nobreak\
D_{s}^{-}D^{*0}(3975.2)/D_{s}^{*-}D^{0}(3977.0)\,.$

The thresholds for the heavier channels are given between parentheses in MeV.

Table 2: Values of $a$, $r$ and $X$ from the elastic ERE study. For the $Z_{cs}(3985)$ the analysis is done twice by taking either the threshold of $D_{s}^{-}D^{*0}$ or $D_{s}^{*-}{D}^{0}$.  Charmonium | Mass | Width | $a$ | $r$ | $X$
---|---|---|---|---|---
Resonance | (MeV) | (MeV) | (fm) | (fm) |
$Z_{c}(3900)$ | $3888.4\pm 2.5$ | $28.3\pm 2.5$ | $-0.84\pm 0.13$ | $-2.52\pm 0.25$ | $0.45\pm 0.06$
$X(4020)$ | $4024.1\pm 1.9$ | $13\pm 5$ | $-1.04\pm 0.30$ | $-3.90\pm 1.35$ | $0.39\pm 0.14$
$Z_{cs}(3985)$ | $3982.5\pm 3.3$ | $12.8\pm 6.1$ | $-1.00\pm 0.47$ | $-4.04\pm 1.82$ | $0.38\pm 0.18$
|  |  | $-1.28\pm 0.60$ | $-3.65\pm 1.60$ | $0.46\pm 0.19$

The elastic ERE study, which considers only each channel with its threshold
around the resonance mass, yields the results shown in Table 2. One can
observe from Table 2 several interesting points. Firstly, it is notorious that
the values for $a$, $r$ and $X$ are quite similar among all the three
resonances. This is a clear hint towards a remarkable similar dynamics in the
constitution of all these resonances (a suspicion that one could already have
considering the heavier channels shown in Eq. (5), being related by heavy-
quark spin symmetry and $SU(3)$). Secondly, we note the rather large and
negative values of $r$ and that the compositeness is less than 0.5 for all the
resonances. These two aspects are related as discussed in Sec. 7 of [I].

Next, the method based on the saturation of the width and compositeness of
every resonance is also considered. This method fixes the two couplings in the
case of the resonance $Z_{c}(3900)$, because its separate decay widths into
the two channels in Eq. (5) have been measured BESIII:2013qmu . This analysis
gives

$\displaystyle Z_{c}(3900):$ $\displaystyle\leavevmode\nobreak\
\frac{\Gamma_{D\bar{D}^{*}}}{\Gamma_{J/\psi\pi}}=6.2\pm
2.9\,,\,|g_{1}|=1.46_{-0.23}^{+0.43}\,,|g_{2}|=7.89_{-0.44}^{+0.18}\,,\,|g_{1}|\ll|g_{2}|\leavevmode\nobreak\
,$ (6) $\displaystyle X_{1}=0.002\pm
0.001\,,\,X_{2}=0.436_{-0.047}^{+0.021}\,,\,X=X_{1}+X_{2}=0.438_{-0.047}^{+0.021}\,.$

It is worth stressing that $X$ is almost identical to the one determined in
Table 2 from the ERE method. Then, given the analogous dynamics of the
constituents of these resonances, as reflected in Table 2, we then take as the
compositeness $X$ for the $Z_{cs}(3985)$ and $X(4020)$ the one determined from
ERE study. In this way we can solve for the couplings and predict the partial-
decay widths of these resonances. The results of the analysis are given in
Table 3.

Table 3: The coupled-channel solutions for the $X(4020)$ and $Z_{cs}(3985)$.  Charmonium | $|g_{1}|$ | $|g_{2}|$ | $\Gamma_{1}$ | $\Gamma_{2}$ | $X_{1}\times 10^{3}$ | $X_{2}$
---|---|---|---|---|---|---
Resonance | (GeV) | (GeV) | (MeV) | (MeV) |  |
$X(4020)$ |  |  |  |  |  |
$X_{\text{ERE}}=0.39\pm 0.14$ | $1.1\pm 0.2$ | $6.5\pm 1.3$ | $1.4\pm 0.5$ | $11.6\pm 4.5$ | $1\pm 1$ | $0.39\pm 0.14$
$Z_{cs}(3985)$ |  |  |  |  |  |
Threshold$({D_{s}^{-}D^{*0}})$ |  |  |  |  |  |
$X_{\text{ERE}}=0.38\pm 0.18$ | $0.8\pm 0.2$ | $6.4\pm 1.7$ | $1.2\pm 0.6$ | $11.6\pm 5.3$ | $0.8\pm 0.4$ | $0.38\pm 0.18$
Threshold$({D_{s}^{*-}{D}^{0}})$ |  |  |  |  |  |
$X_{\text{ERE}}=0.46\pm 0.19$ | $0.9\pm 0.2$ | $6.8\pm 1.7$ | $1.2\pm 0.6$ | $11.6\pm 5.6$ | $0.8\pm 0.4$ | $0.46\pm 0.19$

##  4 $X(6900)$ and $X(6825)$

The LHCb Collaboration LHCb:2020bwg  discovered the fully charmed tetraquark
resonance $X(6900)$ in the di-$J/\psi$ mass distributions. The fitted mass and
width for this resonance are,

$\displaystyle\text{Model I:}\leavevmode\nobreak\ M$ $\displaystyle=6905\pm
11\pm 7\leavevmode\nobreak\ \text{MeV}\,,\leavevmode\nobreak\ \Gamma=80\pm
19\pm 33\leavevmode\nobreak\ \text{MeV}$ (7) $\displaystyle\text{Model
II:}\leavevmode\nobreak\ M$ $\displaystyle=6886\pm 11\pm
11\leavevmode\nobreak\ \text{MeV}\,,\leavevmode\nobreak\ \Gamma=168\pm 33\pm
69\leavevmode\nobreak\ \text{MeV}\leavevmode\nobreak\ ,$

depending on the model used for the non-resonant signal. Here we review the
approach and results of Ref. Guo:2020pvt , which studied the $S$-wave
scattering in coupled channels of $J/\psi J/\psi$($\eta_{c}\eta_{c}$),
$\chi_{c0}\chi_{c0}(6829.4)$, and $\chi_{c1}\chi_{c1}(7021.3)$ around the
thresholds (given between brackets in MeV) of the latter two channels. The
symbol $J/\psi J/\psi$($\eta_{c}\eta_{c}$) refers to the fact that by studying
explicitly the $J/\psi J/\psi$ channel one is also accounting for other light
channels like the $\eta_{c}\eta_{c}$, whose thresholds are far away from the
mass of the $X(6900)$. The $T$ matrix is calculated with a unitarization
formula Oller:2019opk ; Oller:2019rej  that considers linear interactions in
$s$ between the channels $\chi_{c0}\chi_{c0}$ (2) and $\chi_{c1}\chi_{c1}$(3),
allowing for a CDD pole at $M_{CDD}^{2}$. The relevant formulas are

$\displaystyle\mathcal{T}(s)=[1-\mathcal{V}(s)\cdot
G(s)]^{-1}\cdot\mathcal{V}(s)\,;\,\mathcal{V}(s)=\left(\begin{matrix}0&b_{12}&b_{13}\\\
b_{12}&\frac{b_{22}}{M_{J/\psi}^{2}}(s-M_{CDD}^{2})&\frac{b_{23}}{M_{J/\psi}^{2}}(s-M_{CDD}^{2})\\\
b_{13}&\frac{b_{23}}{M_{J/\psi}^{2}}(s-M_{CDD}^{2})&\frac{b_{33}}{M_{J/\psi}^{2}}(s-M_{CDD}^{2})\\\
\end{matrix}\right)\,.$ (8)

Because of the heavy-quark spin symmetry (HQSS) one has that
$b_{13}=\frac{b_{12}}{\sqrt{3}}\,$, $b_{23}=\frac{b_{22}}{\sqrt{3}}\,,$ and
$b_{33}=\frac{b_{22}}{3}$ . The $G(s)$ matrix is diagonal and it contains the
unitarity loop functions $G_{i}(s)$ [I]. The subtraction constant $a(\mu)$ is
estimated by matching at threshold $G_{i}(s)$ with its value when calculated
with a three-momentum cutoff $\mu$ around 1 GeV. In order to reproduce the
di-$J/\psi$ event distribution we take into account the final-state
interactions due to the interaction among the three channels with the generic
formula Oller:2019opk ; Oller:2000vu ; Oller:1999ag

$\displaystyle B(s)$ $\displaystyle=[1-\mathcal{V}(s)\cdot
G(s)]^{-1}\cdot\mathcal{P}\leavevmode\nobreak\ \leavevmode\nobreak\
;\leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{P}=\left(\begin{matrix}0\\\ d_{2}\\\
d_{2}/\sqrt{3}\end{matrix}\right)\leavevmode\nobreak\ ,$ (9)

where $\mathcal{P}$ is a vector containing the production vertices $d_{i}$,
with $d_{1}=0$ due to the assumed weak coupling of the $J/\psi J/\psi$. We
have also checked that fits are stable if releasing $d_{1}$. Note also that in
Eq. (9) we have employed HQSS to set $d_{3}=d_{2}/\sqrt{3}$. The final formula
for calculating the di-$J/\psi$ event distribution is

$\displaystyle\frac{d\mathcal{N}(s)}{d\sqrt{s}}=|B_{1}(s)|^{2}\frac{q_{{J/\psi}{J/\psi}}(s)}{M_{J/\psi}^{2}}\leavevmode\nobreak\
,$ (10)

in obvious notation. The free parameters in our approach are then $b_{12}$,
$b_{22}$, $M_{CDD}^{2}$, and $d_{2}$, which are fitted to data. We distinguish
between the Fits I and II corresponding to the non-resonant background being
treated as in the Models I and II of Ref. LHCb:2020bwg , respectively. The
fitted values for the free parameters are given in Table 4. The reproduction
of the data (crosses) by the fits around the resonance is shown in Fig. 1 by
the solid (Fit-I) and dashed (Fit-II) lines. The dotted lines show the non-
resonant backgrounds taken from Ref. LHCb:2020bwg , and the green histogram is
the average of the Fit-I over the experimental bin width of 27 MeV. The
resulting $|g_{i}|$ and $X_{i}$ are given in Table 5. We notice that the mass
and width of the resonance are well compatible with those determined by the
LHCb LHCb:2020bwg , cf. Eq. (7). The resulting total compositeness is quite
small, $X<0.2$, which is a clear hint of a large bare “elementary” component
of the $X(6900)$. This is in agreement with the fact of having the CDD pole so
close to the resonance mass [I], $M_{\rm CDD}\approx M_{R}$, which is also
tightly connected with the Morgan’s pole-counting criterion Morgan:1990ct ;
Morgan:1992ge . We have checked that the pole also appears in the 5th RS
$(-,-,-)$ (for the definition of RSs see [I]). We show in Ref. Guo:2020pvt
that a way to distinguish between Fits I and II could be measuring the
invariant mass distributions of $\chi_{c0}\chi_{c0}$.

Table 4: Fits I and II. The entries marked with asterisks are fixed during the fits.  | $\chi^{2}/{\rm d.o.f}$ | $a(\mu)$ | $M_{CDD}(\text{MeV})$ | $b_{22}$ | $b_{12}$ | $d_{2}$
---|---|---|---|---|---|---
Fit-I | $1.6/(12-3)$ | $-3.0^{*}$ | $6910^{*}$ | $10817_{-2096}^{+8378}$ | $151_{-99}^{+153}$ | $2213_{-316}^{+2106}$
Fit-II | $4.9/(12-3)$ | $-3.0^{*}$ | $6885^{*}$ | $21085_{-7359}^{+15141}$ | $484_{-112}^{+239}$ | $3646_{-714}^{+1325}$

Figure 1: $J/\psi J/\psi$ event distribution around the resonant region of the $X(6900)$. Table 5: Resonance poles of the $X(6900)$ and its properties.  | Mass | Width/2 | $|g_{1}|$ | $|g_{2}|$ | $|g_{3}|$ | $X=\sum_{i=1}^{3}X_{i}$
---|---|---|---|---|---|---
| (MeV) | (MeV) | (GeV) | (GeV) | (GeV) |
Fit-I | $6907_{-3}^{+5}$ | $33_{-10}^{+14}$ | $4.6_{-2.8}^{+2.5}$ | $9.7_{-2.6}^{+1.4}$ | $5.6_{-1.5}^{+0.8}$ | $0.17_{-0.07}^{+0.04}$
Fit-II | $6892_{-2}^{+2}$ | $80_{-17}^{+24}$ | $10.3_{-1.4}^{+1.8}$ | $6.9_{-1.9}^{+1.4}$ | $4.0_{-1.1}^{+0.8}$ | $0.13_{-0.03}^{+0.03}$

The solid line in Fig. 1 shows a clear peak around 6825 MeV, which is washed
out when taking the average over the bin width. This peak corresponds to a
pole in the 4th RS ($+,-,+)$ [I], a new resonance that we call the $X(6825)$.
Its properties are summarized in Table 6. Comparing this table with Table 5
for the $X(6900)$, we observe that the coupling to $J/\psi J/\psi$ of the
former is much smaller, driving a much smaller width. At the same time, the
couplings $|g^{\prime}_{2}|$ and $|g^{\prime}_{3}|$ are much larger than those
of the $X(6900)$. The new resonance is a $\chi_{c0}\chi_{c0}$ virtual state
present only at the 4th RS [I]. According to Morgan’s pole counting rule
Morgan:1990ct ; Morgan:1992ge  it should be of molecular nature.

Table 6: Pole position and couplings of the $X(6825)$.  Fit |  $E^{\prime}_{R}$ MeV | $|g^{\prime}_{1}|$ | $|g^{\prime}_{2}|$ | $|g^{\prime}_{3}|$
---|---|---|---|---
I | $6827.0^{+1.6}_{-4.8}-i\,1.1^{+1.3}_{-1.0}$ | $1.4^{+0.6}_{-0.9}$ | $11.9^{+3.2}_{-3.1}$ | $6.8^{+1.8}_{-1.8}$
II | $6820.6^{+3.0}_{-2.7}-i\,4.0^{+1.7}_{-1.6}$ | $2.5^{+0.5}_{-0.6}$ | $15.8^{+0.7}_{-0.6}$ | $9.1^{+0.4}_{-0.4}$

We also investigated in Guo:2020pvt  the importance of the channel
$\psi(3770)J/\psi$, which threshold is at $6870.6$ MeV. By taking the channels
$J/\psi J/\psi$ (1) and $\psi(3770)J/\psi$ (2) the resulting fits are not well
determined because the fitted parameters are affected by huge errors. In
addition, the coupling to $\psi(3770)J/\psi$ is much smaller than those to
$\chi_{c0}\chi_{c0}$ and $\chi_{c1}\chi_{c1}$, a fact that clearly indicates a
much less important role of the $\psi(3770)J/\psi$ channel. Related to these
observations, Ref. Guo:2020pvt  also considered a perturbative treatment of
the $\psi(3770)J/\psi$ as the fourth channel, so that this channel interacts
only through its couplings to $\chi_{c0}\chi_{c0}$ and $\chi_{c1}\chi_{c1}$.
It comes out that the Fits I and II obtained with only the first three
channels are stable and the conclusions remain unchanged.

##  5 Study of the $P_{cs}(4459)$

Here we review the work of Ref. Du:2021bgb  on the nature of the
$P_{cs}(4459)$ Wu:2010jy , a charmonium pentaquark resonance with strangeness
recently discovered by the LHCb Collaboration LHCb:2020jpq  in the
$J/\psi\Lambda$ mass distribution. The mass and width of the resonance from
this reference are given in Table 7. In Ref. LHCb:2020jpq  the question
whether the peak structure unveiled was due to one or two resonances with
$J=1/2$ or $3/2$ was also raised. Reference Du:2021bgb  applied three methods
to study the nature of this resonance, and all of them clearly signal it is a
$\Xi_{c}\bar{D}^{*}$ molecular resonance. We skip the discussions on the
hidden-charm pentaquark states $P_{c}(4312)$, $P_{c}(4440)$ and $P_{c}(4457)$
in Ref. Guo:2019kdc  due to length limitations.

Table 7: Results of the $P_{cs}(4459)$ for $a$, $r$ and $X$ from the ERE method. The relevant channel is indicated in the third column.  Mass | Width | Threshold | $a$ | $r$ | $X$
---|---|---|---|---|---
(MeV) | (MeV) | (MeV) | (fm) | (fm) |
$4458.8\pm 5.5$ | $17.3\pm 10.3$ |  ${\Xi^{\prime}}_{c}\bar{D}$ (4446.0) | $-0.63\pm 0.38$ | $-3.68\pm 2.11$ | $0.31\pm 0.19$
$4458.8\pm 5.5$ | $17.3\pm 10.3$ |  $\Xi_{c}\bar{D}^{*}$ (4478.0) | $-1.79\pm 0.23$ | $-0.94\pm 0.13$ | $--$

First, the elastic ERE method is separately applied to the channels
${\Xi^{\prime}}_{c}\bar{D}$ and $\Xi_{c}\bar{D}^{*}$ whose thresholds are
4446.0 MeV and 4478.0 MeV, respectively. The results for $a$, $r$ and $X$ are
given in Table 7. For the case of the ${\Xi^{\prime}}_{c}\bar{D}$ channel the
value of $X$ is rather small, which is also consistent with having a large and
negative $r$. That is, the resonance $P_{cs}(4459)$ has other more important
components than the ${\Xi^{\prime}}_{c}\bar{D}$. On the other hand, when
considering the case with the $\Xi_{c}D^{*}$ the effective range has a natural
value for strong interactions, which is a hint towards a $\Xi_{c}\bar{D}^{*}$
molecular nature of the resonance.

The consideration by Ref. Du:2021bgb  of the method based on saturating $X$
and $\Gamma$ of the $P_{cs}(4459)$ for the coupled channels
$J/\psi\Lambda$-$\Xi_{c}\bar{D}^{*}$, on the one hand, and
$J/\psi\Lambda$-$\Xi_{c}^{\prime}\bar{D}$, on the other, implies that no
solution for $X\gtrsim 0.3$ is obtained for the latter case. Thus, the
$P_{cs}(4459)$ cannot be a $\Xi_{c}^{\prime}\bar{D}$ molecular-type resonance.
However, it could be a $\Xi_{c}\bar{D}^{*}$ one since values of $X$ as large
as 1 can be reproduced.

The last method that Ref. Du:2021bgb  applied is to directly fit the data on
the $J/\psi\Lambda$ event distribution provided by Ref. LHCb:2020jpq . Here,
we consider a two coupled-channel problem involving the $J/\psi\Lambda$ and
$\Xi_{c}\bar{D}^{*}$ channels interacting in $S$ wave. The inclusion of the
channel $\eta_{c}\Lambda$ for $J=1/2$ would not add any new aspect Du:2021bgb
. We restrict to $S$-wave scattering because the $P_{cs}(4459)$ is very close
to the thresholds of $\Xi_{c}^{\prime}\bar{D}$ and $\Xi_{c}\bar{D}^{*}$. It is
also the case that in the HQSS $J/\psi\Lambda$ ($\eta_{c}\Lambda$) cannot
couple to the heavier channels considered in $D$ and higher partial waves. The
basic formulas to calculate the PWAs in $S$ wave are similar to Eq. (8) with
the interaction kernel ${\cal V}_{J}$, $J=1/2$ and $3/2$, given by

$\displaystyle\mathcal{V}_{\frac{1}{2}}$
$\displaystyle=\left(\begin{array}[]{ll}0&g\\\
g&C_{\frac{1}{2}}\end{array}\right)\,\leavevmode\nobreak\ \leavevmode\nobreak\
;\leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{V}_{\frac{3}{2}}=\left(\begin{array}[]{lll}0&g\\\
g&C_{\frac{3}{2}}\end{array}\right)\leavevmode\nobreak\ .$ (15)

The HQSS requires that $C_{\frac{1}{2}}=C_{\frac{1}{3}}$, but we have let them
to float in Eq. (15) as a check of the completeness of the model. The
$J/\psi\Lambda$ production amplitudes $F_{J}(s)$ and event distribution are
given by

$\displaystyle F_{J}(s)$
$\displaystyle=\frac{d_{J}}{\Delta_{J}(s)}=\frac{d_{J}}{1-(C_{J}+G_{1}(s)g^{2})G_{3}(s)}\leavevmode\nobreak\
\leavevmode\nobreak\ ;\leavevmode\nobreak\ \leavevmode\nobreak\
\Delta_{J}(s)={\rm det}\left[1-\mathcal{V}_{J}\cdot
G(s)\right]\leavevmode\nobreak\ ,$ (16) $\displaystyle\frac{dN(s)}{d\sqrt{s}}$
$\displaystyle=\frac{1}{128\pi^{3}M^{3}_{\Xi_{b}}}\frac{\sqrt{\lambda(M^{2}_{\Xi_{b}},s,M_{K}^{2})\lambda(s,M^{2}_{J/\psi},M_{\Lambda}^{2})}}{\sqrt{s}}\sum_{J}|F_{J}|^{2}\leavevmode\nobreak\
.$

In addition, a convolution to take into account energy resolution of the
experimental data is also made. The results of the fits with this scheme based
on using the two channels $J/\psi\Lambda$, $\Xi_{c}\bar{D}^{*}$ are given by
the first two lines of Table 8, and shown in Fig. 2. We designate the fits by
the tuple $(abc)$, where $a$ gives the number of channels, $b$ the number of
partial wave amplitudes and $c$ the maximum degree in powers of $s$ of the
matrix elements of ${\cal V}$. In the first line only the $J=1/2$ PWA is
included (this is why $C_{3/2}=d_{3/2}=0$) and the fit is not good. In the
second line both waves are included, the fit is good and has a much lower
$\chi^{2}$, but the couplings $C_{1/2}$ and $C_{3/2}$ are very different,
which indicates a clear violation of the HQSS expectation. Other fits were
also explored attending to the possible presence of CDD poles by substituting
$C_{1/2}$ with $C_{1/2}(s/M_{\rm CDD}^{2}-1)$. However, the resulting fits are
ruled out because they drive to poles in the first RS. As an informative
remark, Ref. Du:2021bgb  also studied in a similar way the coupled-channel
system $J/\psi\Lambda$-$\Xi_{c}^{\prime}\bar{D}$, which only interacts in
$J=1/2$. This fit is denoted by $(210)^{\prime}$ in Fig. 2 and is very poor.

|
---|---
|
Figure 2: Reproduction of the experimental data by the different fits around the $P_{cs}(4459)$ resonance. The actual fitted area has a green shadow. The dot-dashed line curve in (210) results after imposing that $\Delta_{1/2}=0$ at the mass and width values of Ref. LHCb:2020jpq .  Table 8: Fit parameters to data in the region of the $P_{cs}(4459)$.  Fit | $\chi^{2}$ | $g$ | $C_{\frac{1}{2}}$ | $d_{\frac{1}{2}}$ | $C_{\frac{3}{2}}$ | $d_{\frac{3}{2}}$
---|---|---|---|---|---|---
$(210)$ | 6.08 | $316.0^{+86.2}_{-69.3}$ | $1126.6^{+327.8}_{-214.9}$ | $290.5^{+71.1}_{-77.2}$ | $\times$ | $\times$
$(220)$ | 2.95 | $217.8^{+80.6}_{-79.3}$ | $1125.5^{+190.6}_{-185.9}$ | $174.7^{+86.5}_{-77.3}$ | $3862.7^{+1466.1}_{-1003.3}$ | $97.6^{+37.9}_{-35.5}$
$(320)$ | 3.06 | $124.5^{+130.4}_{-164.7}$ | $1105.8^{+191.9}_{-132.5}$ | $250.8^{+62.0}_{-40.2}$ | $C_{\frac{3}{2}}=C_{\frac{1}{2}}$ | $82.1^{+292.1}_{-128.0}$

Reference Du:2021bgb  investigated the dynamical impact of the
$\Xi_{c}^{\prime}\bar{D}$ (which in $S$ wave only interacts with $J=1/2$) on
top of $J/\psi\Lambda$ and $\Xi_{c}\bar{D}^{*}$. We number the channels
according to their increasing thresholds, such that $J/\psi\Lambda$ (1),
$\Xi_{c}^{\prime}\bar{D}$ (2), and $\Xi_{c}\bar{D}^{*}$ (3). However, in order
to avoid having too many free parameters this extra channel is introduced
perturbatively, and its interactions would take place only through its
coupling to $\Xi_{c}\bar{D}^{*}$. The resulting fit is (320) in the last panel
of Fig. 2. Its quality is good and it is also found that the HQSS equality
$C_{1/2}=C_{3/2}$ can be fulfilled. Then we conclude that the previously
commented violation of this HQSS expectation in the fit (220) is due to having
neglected a nearby threshold.

We give the pole content of the fit (320) in the Table 9, with the different
RSs denoted as RSII $(-,+,+)$, and RSIII $(-,-,+)$ (and its reduction for the
two-channel case).

Table 9: The poles for the fit $(320)$ with three channels.  Type | $J$ | RS | $\sqrt{s_{R}}$ | $|g_{1}|$ | $|g_{2}|$ | $|g_{3}|$
---|---|---|---|---|---|---
|  |  | (MeV) | (MeV) | (MeV) | (MeV)
$(320)$ | 3/2 | RSII $(-+)$ | $4466.6^{+1.9}_{-2.7}-i\,1.3^{+1.3}_{-3.7}$ | $1.4^{+1.4}_{-1.4}$ | $\times$ | $12.6^{+0.8}_{-0.6}$
$(320)$ | 1/2 | RSIII $(--+)$ | $4453.8^{+2.4}_{-3.3}-i\,2.8^{+0.9}_{-0.8}$ | $0.6^{+0.6}_{-0.6}$ | $4.2^{+0.2}_{-0.4}$ | $15.0^{+0.5}_{-0.3}$

Having the residue and the pole positions we can calculate the partial-decay
widths and partial compositeness coefficients,

$\displaystyle J=1/2:\leavevmode\nobreak\ $
$\displaystyle\Gamma_{1}=0.5_{-0.5}^{+1.9}\leavevmode\nobreak\ {\rm
MeV}\,,\leavevmode\nobreak\ \Gamma_{2}=4.3_{-1.4}^{+1.2}\leavevmode\nobreak\
{\rm MeV}\,,\leavevmode\nobreak\
\Gamma_{3}=0.9_{-0.6}^{+1.2}\leavevmode\nobreak\ {\rm MeV}\leavevmode\nobreak\
,$ (17) $\displaystyle X_{1}=0.0\pm 0.0\,,\leavevmode\nobreak\ X_{2}=0.15\pm
0.05\leavevmode\nobreak\ ,$ $\displaystyle J=3/2:\leavevmode\nobreak\ $
$\displaystyle\Gamma_{1}=2.6^{+8.2}_{-2.6}\leavevmode\nobreak\ {\rm
MeV}\,,\leavevmode\nobreak\ \Gamma_{3}=0.4_{-0.4}^{+2.5}\leavevmode\nobreak\
{\rm MeV}\leavevmode\nobreak\ ,\leavevmode\nobreak\ X_{1}=0.0\pm
0.0\,,\leavevmode\nobreak\ X_{3}=1.0^{+0.2}_{-0.2}\leavevmode\nobreak\ .$

From here one can conclude that the $P_{cs}$ is mostly a $\Xi_{c}\bar{D}^{*}$
resonance, having very large couplings to this channel driving to large
$X_{3}$. The latter was not calculated for the $J=1/2$ pole since the
threshold for the $\Xi_{c}\bar{D}^{*}$ channel lies clearly above its mass.
Nonetheless, the naively calculated $X_{3}=0.94$ in this case is very large,
being a clear indication of the preponderance of the (3) channel in this pole.
A similar comment is in order also for the $J=3/2$ pole, with $X_{3}$ having a
central value of 1. This composite nature of the $P_{cs}(4459)$ is analogous
to the one already unveiled for its non-strange pentaquark partners
$P_{c}(4312)$, $P_{c}(4380)$, $P_{c}(4440)$, and $P_{c}(4457)$ Guo:2019kdc ;
Du:2019pij ; Du:2021fmf .

## Acknowledgements

This work has been supported in part by the MICINN AEI (Spain) Grant No.
PID2019â106080GB-C22/AEI/10.13039/501100011033 , and the Natural Science
Foundation of China (NSFC) under Grant No. 11975090.

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