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Inertial Optimization Based Two-Step Methods for Solving Equilibrium Problems with Applications in Variational Inequality Problems and Growth Control Equilibrium Models

Habib ur Rehman

Poom Kumam

^1,2,3,*

Meshal Shutaywi

⁴

Nasser Aedh Alreshidi

⁵

and

Wiyada Kumam

^6,*

KMUTTFixed Point Research Laboratory, KMUTT-Fixed Point Theory and Applications Research Group, SCL 802 Fixed Point Laboratory, Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand

Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Science Laboratory Building, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand

Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan

⁴

Department of Mathematics, College of Science & Arts, King Abdulaziz University, P. O. Box 344, Rabigh 21911, Saudi Arabia

⁵

Department of Mathematics, College of Science, Northern Border University, Arar 73222, Saudi Arabia

⁶

Program in Applied Statistics, Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi, Thanyaburi, Pathumthani 12110, Thailand

Authors to whom correspondence should be addressed.

Energies 2020, 13(12), 3292; https://doi.org/10.3390/en13123292

Submission received: 8 April 2020 / Revised: 1 June 2020 / Accepted: 15 June 2020 / Published: 26 June 2020

(This article belongs to the Section F: Electrical Engineering)

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Abstract

This manuscript aims to incorporate an inertial scheme with Popov’s subgradient extragradient method to solve equilibrium problems that involve two different classes of bifunction. The novelty of our paper is that methods can also be used to solve problems in many fields, such as economics, mathematical finance, image reconstruction, transport, elasticity, networking, and optimization. We have established a weak convergence result based on the assumption of the pseudomonotone property and a certain Lipschitz-type cost bifunctional condition. The stepsize, in this case, depends upon on the Lipschitz-type constants and the extrapolation factor. The bifunction is strongly pseudomonotone in the second method, but stepsize does not depend on the strongly pseudomonotone and Lipschitz-type constants. In contrast, the first convergence result, we set up strong convergence with the use of a variable stepsize sequence, which is decreasing and non-summable. As the application, the variational inequality problems that involve pseudomonotone and strongly pseudomonotone operator are considered. Finally, two well-known Nash–Cournot equilibrium models for the numerical experiment are reviewed to examine our convergence results and show the competitive advantage of our suggested methods.

Keywords:

energy production models; optimization problems; control parameters; Lipschitz-type conditions; variational inequality; Nash-Cournot oligopolistic equilibrium model

1. Introduction

An Equilibrium problem (EP) was originally started in the unifying feature by Blum and Oettli [1] in 1994 and provided a detailed investigation of their theoretical properties. This study contributes significantly to the advancement of applied and pure science. This problem is primarily related to Ky Fan Inequity due to his early contributions to this field [2]. It has been established that the equilibrium problem theory has set up an unique approach to investigate an immense range of topics that have appeared in social and physical science. For instance, it might involve physical or mechanical structures, chemical processes [3], the distribution of traffic over computer, and telecommunication networks or public roads [4,5,6,7]. In economics, it often refers to production competition [8] or the dynamics of offer and demand [9], exploiting the mathematical model of non-cooperative games and the analogous equilibrium concept by Nash [10,11]. The problem of equilibrium, as a particular case, includes many mathematical problems as a particular case, such as the variational inequality problems (VIP), problems of minimization, the fixed point problems, Nash equilibrium of non-cooperative games, complementarity problems, and saddle point problem (see e.g., [1,12]).

On the other hand, iterative methods are efficient techniques for determining the approximate solution of an equilibrium problem. In that case, two major approaches that are well-known i.e., the proximal point method [13] and auxiliary problem principle [14]. The proximal point method strategy was initially developed by Martinet [15] for the monotone variational inequality problems and later Rockafellar [16] extends this approach for monotone operators. Moudafi [13] proposed the proximal point method for monotone equilibrium problems. Konnov [17] also suggests a different interpretation of the proximal point method with weaker assumptions for equilibrium problems.

In addition, inertial-type methods are additionally significant, depending on the heavy-ball methods of the second-order time dynamic system. Polyak began by considering inertial extrapolation as an acceleration procedure to deal with the problem of smooth convex minimization. Inertial-type algorithms are two-step iterative schemes, and the next iteration is determined by using the previous two iterations and it can be viewed as an accelerating step of the iterative sequence. A large number of methods are the earliest, being set up for solving the problem (EP) in finite and infinite-dimensional spaces, such as the proximal point-like methods [13,18], the extragradient methods [19,20,21,22,23], the subgradient extragradient methods [24,25,26], the inertia methods [27,28,29,30,31,32] and others in [33,34].

In this work, our focus is on the proximal point method, in particular projection methods, which are well established and technically easy to implement due to their convenient numerical computation. This manuscript aims to suggest two modifications of the results that appeared in [21,35,36] by applying the inertial scheme that is useful for speeding up the iteration process. The first result includes the two-step inertial Popov’s extragradient method for determining a numerical solution to the pseudomonotone equilibrium problems and the weak convergence of the suggested method is achieved based on the standard assumptions. We also propose an alternative inertial-type method, the second variant of the first method. The second method does not need any information regarding the Lipschitz-type and strongly pseudomonotone constants of a bifunction. A practical explanation for the second method is that it uses a diminishing and non-summable sequence of non-negative real numbers, which are useful in achieving the strong convergence.

This manuscript is arranged, as follows: in Section 2, we provide some essential definitions and useful results. Section 3 and Section 4 include all of our main methods and corresponding convergence results. Section 5 provides the methods for variational inequality problems. Section 6 sets out the numerical tests to show the numerical efficiency of the proposed methods for the test problems based on the Nash–Cournot equilibrium model compare to other existing methods.

2. Background

Let K be a non-empty, convex, and closed subset of the Hilbert space

E

. Let

H : K \to E

be an operator and

S O L_{V I (H, K)}

is the solution set of a variational inequality problem relative to the operator H upon the set K. Likewise,

S O L_{E P (f, K)}

denotes the solution set of an equilibrium problem on the set K and

ξ^{*}

is any arbitrary element of the solution set

S O L_{E P (f, K)}

S O L_{V I (H, K)}

Definition 1.

[1] Let

f : E \times E \to R

be a bifunction with

f (\tilde{u}, \tilde{u}) = 0

, for each

\tilde{u} \in K

. The equilibrium problem for f upon K is defined, as follows:

Find ξ^{*} \in K such that f (ξ^{*}, \tilde{v}) \geq 0, \forall \tilde{v} \in K .

Definition 2.

[37] The metric projection

P_{K} (\tilde{u})

\tilde{u}

on a closed and convex subset K of

E

is determined, as follows:

P_{K} (\tilde{u}) = \underset{}{arg min} {∥ \tilde{v} - \tilde{u} ∥ : \tilde{v} \in K} .

Next, we take the concept of monotonicity of a bifunction into account (see [1,38] for details).

Definition 3.

Let

f : E \times E \to R

on K for

γ > 0

(1): strongly monotone if

$f (\tilde{u}, \tilde{v}) + f (\tilde{v}, \tilde{u}) \leq - γ {∥ \tilde{u} - \tilde{v} ∥}^{2}, \forall \tilde{u}, \tilde{v} \in K;$
(2): monotone if

$f (\tilde{u}, \tilde{v}) + f (\tilde{v}, \tilde{u}) \leq 0, \forall \tilde{u}, \tilde{v} \in K;$
(3): strongly pseudomonotone if

$f (\tilde{u}, \tilde{v}) \geq 0 ⟹ f (\tilde{v}, \tilde{u}) \leq - γ {∥ \tilde{u} - \tilde{v} ∥}^{2}, \forall \tilde{u}, \tilde{v} \in K;$
(4): pseudomonotone if

$f (\tilde{u}, \tilde{v}) \geq 0 ⟹ f (\tilde{v}, \tilde{u}) \leq 0, \forall \tilde{u}, \tilde{v} \in K;$
(5): satisfying the Lipschitz-type condition on K if there exist constants $L_{1}, L_{2} > 0,$ such that

$f (\tilde{u}, \tilde{w}) \leq f (\tilde{u}, \tilde{v}) + f (\tilde{v}, \tilde{w}) + L_{1} ∥ \tilde{u} - \tilde{v} ∥^{2} + L_{2} {∥ \tilde{v} - \tilde{w} ∥}^{2}, \forall \tilde{u}, \tilde{v}, \tilde{w} \in K,$

holds.

This section ends with a few essential lemmas that are useful for examining convergence.

Lemma 1.

[39] Assume that K is non-empty, convex, and closed subset of Hilbert space

E

and

g : K \to R

is a convex, subdifferentiable, and lower semi-continuous function on

K

. Furthermore,

\tilde{u} \in K

is a minimizer of g if and only if

0 \in \partial g (\tilde{u}) + N_{K} (\tilde{u})

where

\partial g (\tilde{u})

and

N_{K} (\tilde{u})

denotes the subdifferential of g at

\tilde{u}

and normal cone of K at

\tilde{u}

, respectively.

Lemma 2.

[40] Let

{p_{n}}, {q_{n}} \subset [0, + \infty)

be two sequences and

\sum_{n = 1}^{\infty} p_{n} = \infty

with

\sum_{n = 1}^{\infty} p_{n} q_{n} < \infty

, then

{lim inf}_{n \to \infty} q_{n} = 0

Lemma 3.

[41] For

\tilde{u}, \tilde{v} \in E

and

μ \in R

then the following relation is true:

∥ μ \tilde{u} + (1 - μ) \tilde{v} ∥^{2} = μ ∥ \tilde{u} ∥^{2} + (1 - μ) ∥ \tilde{v} ∥^{2} - μ (1 - μ) {∥ \tilde{u} - \tilde{v} ∥}^{2} .

Lemma 4.

[42] Assume that

{\tilde{a}}_{n}

{\tilde{b}}_{n}

and

{\tilde{c}}_{n}

are sequences in

[0, + \infty)

, such that

{\tilde{a}}_{n + 1} \leq {\tilde{a}}_{n} + {\tilde{b}}_{n} ({\tilde{a}}_{n} - {\tilde{a}}_{n - 1}) + {\tilde{c}}_{n}, \forall n \geq 1, with \sum_{n = 1}^{+ \infty} {\tilde{c}}_{n} < + \infty,

and also with

\tilde{b} > 0

, such that

0 \leq {\tilde{b}}_{n} \leq \tilde{b} < 1

for all

n \in N

. Subsequently, the following relations are hold.

(i): $\sum_{n = 1}^{+ \infty} {[{\tilde{a}}_{n} - {\tilde{a}}_{n - 1}]}_{+} < \infty,$ with ${[s]}_{+} : = max {s, 0};$
(ii): ${lim}_{n \to + \infty} {\tilde{a}}_{n} = a^{*} \in [0, + \infty) .$

Lemma 5.

[43] Let

{{\tilde{u}}_{n}}

be a sequence in

E

and

K \subset E

such that the following relations are true:

(i): For each $\tilde{u} \in K$ , ${lim}_{n \to \infty} ∥ {\tilde{u}}_{n} - \tilde{u} ∥$ exists;
(ii): Every sequentially weak cluster point of ${{\tilde{u}}_{n}}$ belongs to K;

Subsequently,

{{\tilde{u}}_{n}}

weakly converges to a point in

K

A normal cone of K at

\tilde{u} \in K

is defined as:

N_{K} (\tilde{u}) = {\tilde{z} \in E : 〈 \tilde{z}, \tilde{v} - \tilde{u} 〉 \leq 0, \forall \tilde{v} \in K} .

Let

g : K \to R

be a convex function with subdifferential of g at

\tilde{u} \in K

is defined as:

\partial g (\tilde{u}) = {\tilde{z} \in E : g (\tilde{v}) - g (\tilde{u}) \geq 〈 \tilde{z}, \tilde{v} - \tilde{u} 〉, \forall \tilde{v} \in K} .

3. Inertial Popov’s Two-Step Subgradient Extragradient Algorithm for Pseudomonotone EP

We present our first method to solve the pseudomonotone equilibrium problems involving the Lipschitz-type condition of a bifunction. It uses an inertial term to boost up the iterative sequence, so we referred it as an “Inertial Popov’s Two-step Subgradient Extragradient Algorithm” for a class pseudomonotone equilibrium problems. The detailed algorithm is given below.

Algorithm 1 (Two-step Subgradient Extragradient Algorithm for Pseudomonotone EP)

Initialization: Choose $u_{- 1}, u_{0}, v_{0} \in E$ , $0 \leq ϑ_{n} \leq ϑ < \sqrt{5} - 2$ and $λ (ϑ, L_{1}, L_{2}) > 0$ . Set

$u_{1} = \underset{y \in K}{arg min} {λ f (v_{0}, y) + \frac{1}{2} ∥ ρ_{0} - y ∥^{2}},$

$v_{1} = \underset{y \in K}{arg min} {λ f (v_{0}, y) + \frac{1}{2} ∥ ρ_{1} - y ∥^{2}},$

where $ρ_{0} = u_{0} + ϑ_{0} (u_{0} - u_{- 1})$ and $ρ_{1} = u_{1} + ϑ_{1} (u_{1} - u_{0})$ .
Iterative steps: Given $u_{n - 1}, u_{n}$ , $v_{n - 1}$ , $v_{n}$ for $n \geq 1$ and construct a half space
$H_{n} = {z \in E : 〈 ρ_{n} - λ ω_{n - 1} - v_{n}, z - v_{n} 〉 \leq 0},$

where $ω_{n - 1} \in \partial f (v_{n - 1}, v_{n})$ .
Step 1: Compute

$u_{n + 1} = \underset{y \in H_{n}}{arg min} {λ f (v_{n}, y) + \frac{1}{2} ∥ ρ_{n} - y ∥^{2}},$

where $ρ_{n} = u_{n} + ϑ_{n} (u_{n} - u_{n - 1})$ .
Step 2: Compute

$v_{n + 1} = \underset{y \in K}{arg min} {λ f (v_{n}, y) + \frac{1}{2} ∥ ρ_{n + 1} - y ∥^{2}},$

where $ρ_{n + 1} = u_{n + 1} + ϑ_{n + 1} (u_{n + 1} - u_{n})$ .
Step 3: If $u_{n + 1} = ρ_{n}$ and $v_{n} = v_{n - 1}$ , then STOP. Otherwise, set $n : = n + 1$ and go back to Step 1.

Assumption 1.

Assume that

f : E \times E \to R

satisfy the following conditions:

(A1): $f (\tilde{v}, \tilde{v}) = 0$ for all $\tilde{v} \in K$ and f is pseudomonotone on K;
(A2): f satisfy the Lipschitz-type condition on $E$ through two positive constants $L_{1}$ and $L_{2}$ ;
(A3): $\underset{n \to \infty}{lim sup} f ({\tilde{u}}_{n}, \tilde{v}) \leq f ({\tilde{u}}^{*}, \tilde{v})$ for all $\tilde{v} \in K$ and ${{\tilde{u}}_{n}} \subset K$ satisfy ${\tilde{u}}_{n} ⇀ {\tilde{u}}^{*}$ ;
(A4): $f (\tilde{u}, .)$ is convex and subdifferentiable on $E$ for each $\tilde{u} \in E .$

Lemma 6.

We have the following crucial inequality that results from the Algorithm 1.

λ f (v_{n}, y) - λ f (v_{n}, u_{n + 1}) \geq 〈 ρ_{n} - u_{n + 1}, y - u_{n + 1} 〉, \forall y \in H_{n} .

Proof.

By the value

u_{n + 1}

through Lemma 1, we have

0 \in \partial_{2} \{λ f (v_{n}, y) + \frac{1}{2} {∥ ρ_{n} - y ∥}^{2}\} (u_{n + 1}) + N_{H_{n}} (u_{n + 1}) .

For

ω \in \partial f (v_{n}, u_{n + 1})

, there exists

\bar{ω} \in N_{H_{n}} (u_{n + 1})

, such that

λ ω + u_{n + 1} - ρ_{n} + \bar{ω} = 0 .

The above implies that

〈 ρ_{n} - u_{n + 1}, y - u_{n + 1} 〉 = λ 〈 ω, y - u_{n + 1} 〉 + 〈 \bar{ω}, y - u_{n + 1} 〉, \forall y \in H_{n} .

Because

\bar{ω} \in N_{H_{n}} (u_{n + 1})

then

〈 \bar{ω}, y - u_{n + 1} 〉 \leq 0

\forall y \in H_{n} .

It implies that

λ 〈 ω, y - u_{n + 1} 〉 \geq 〈 ρ_{n} - u_{n + 1}, y - u_{n + 1} 〉, \forall y \in H_{n} .

(1)

Due to

ω \in \partial f (v_{n}, u_{n + 1})

and by definition of subdifferentiable, we obtain

f (v_{n}, y) - f (v_{n}, u_{n + 1}) \geq 〈 ω, y - u_{n + 1} 〉, \forall y \in E .

(2)

From expressions (1) and (2), we have the required result. □

Lemma 7.

We also have the following inequality from Algorithm 1.

λ f (v_{n}, y) - λ f (v_{n}, v_{n + 1}) \geq 〈 ρ_{n + 1} - v_{n + 1}, y - v_{n + 1} 〉, \forall y \in K .

Proof.

The proof is the same as that of Lemma 6. □

Lemma 8.

We have the following inequality from Algorithm 1.

λ \{f (v_{n - 1}, u_{n + 1}) - f (v_{n - 1}, v_{n})\} \geq 〈 ρ_{n} - v_{n}, u_{n + 1} - v_{n} 〉 .

Proof.

Because

u_{n + 1} \in H_{n}

then the definition of

H_{n}

implies that

〈 ρ_{n} - λ ω_{n - 1} - v_{n}, u_{n + 1} - v_{n} 〉 \leq 0 .

The above implies that

λ 〈 ω_{n - 1}, u_{n + 1} - v_{n} 〉 \geq 〈 ρ_{n} - v_{n}, u_{n + 1} - v_{n} 〉 .

(3)

From

ω_{n - 1} \in \partial f (v_{n - 1}, v_{n})

and due to subdifferential definition, we have

f (v_{n - 1}, y) - f (v_{n - 1}, v_{n}) \geq 〈 ω_{n - 1}, y - v_{n} 〉, \forall y \in E .

Set

y = u_{n + 1}

in the above expression

f (v_{n - 1}, u_{n + 1}) - f (v_{n - 1}, v_{n}) \geq 〈 ω_{n - 1}, u_{n + 1} - v_{n} 〉, \forall y \in E .

(4)

From expression (3) and (4), we obtain the desired result. □

Now, we are proving the validity of the stopping criterion for Algorithm 1.

Lemma 9.

u_{n + 1} = ρ_{n}

and

v_{n} = v_{n - 1}

in Algorithm 1, then

v_{n} \in S O L_{E P (f, K)}

Proof.

By substituting

u_{n + 1} = ρ_{n}

in Lemma 6, we have

λ f (v_{n}, y) - λ f (v_{n}, u_{n + 1}) \geq 0, \forall y \in H_{n} .

(5)

Because

u_{n + 1} \in H_{n}

and

v_{n} = v_{n - 1}

u_{n + 1} = ρ_{n}

, then from Lemma 8, we have

λ f (v_{n}, u_{n + 1}) \geq {∥ ρ_{n} - v_{n} ∥}^{2} \geq 0 .

(6)

The expression (5) and (6) implies that

v_{n} \in S O L_{E P (f, K)}

. □

Remark 1.

Two more conditions for stopping criterion are

u_{n + 1} = v_{n} = ρ_{n}

and

ρ_{n + 1} = v_{n + 1} = v_{n}

for Algorithm 1. The validity of these stopping criterion can be shown easily by Lemma 6 and Lemma 7, respectively.

Lemma 10.

Let

f : E \times E \to R

satisfying the Assumption 1. Assume that

S O L_{E P (f, K)}

is nonempty. Afterwards, for each

ξ^{*} \in S O L_{E P (f, K)}

, we have

\begin{matrix} ∥ u_{n + 1} - ξ^{*} ∥^{2} \\ \leq ∥ ρ_{n} - ξ^{*} ∥^{2} - (1 - 4 λ L_{1}) ∥ ρ_{n} - v_{n} ∥^{2} - (1 - 2 λ L_{2}) ∥ u_{n + 1} - v_{n} ∥^{2} + 4 λ L_{1} {∥ ρ_{n} - v_{n - 1} ∥}^{2} . \end{matrix}

(7)

Proof.

Substituting

y = ξ^{*}

into Lemma 6, we obtain

λ f (v_{n}, ξ^{*}) - λ f (v_{n}, u_{n + 1}) \geq 〈 ρ_{n} - u_{n + 1}, ξ^{*} - u_{n + 1} 〉, \forall y \in H_{n} .

(8)

Since

ξ^{*} \in S O L_{E P (f, K)}

then

f (ξ^{*}, v_{n}) \geq 0 .

Thus, from (A1) the above expression becomes

〈 ρ_{n} - u_{n + 1}, u_{n + 1} - ξ^{*} 〉 \geq λ f (v_{n}, u_{n + 1}) .

(9)

Because of the Lipschitz-type condition, we have

f (v_{n - 1}, u_{n + 1}) \leq f (v_{n - 1}, v_{n}) + f (v_{n}, u_{n + 1}) + L_{1} ∥ v_{n - 1} - v_{n} ∥^{2} + L_{2} {∥ v_{n} - u_{n + 1} ∥}^{2} .

(10)

The expression (9) and (10) implies that

\begin{matrix} 〈 ρ_{n} - u_{n + 1}, u_{n + 1} - ξ^{*} 〉 \\ \geq λ \{f (v_{n - 1}, u_{n + 1}) - f (v_{n - 1}, v_{n})\} - λ L_{1} ∥ v_{n - 1} - v_{n} ∥^{2} - λ L_{2} {∥ v_{n} - u_{n + 1} ∥}^{2} . \end{matrix}

(11)

From expression (11) and Lemma 8, we obtain

\begin{matrix} 〈 ρ_{n} - u_{n + 1}, u_{n + 1} - ξ^{*} 〉 \\ \geq 〈 ρ_{n} - v_{n}, u_{n + 1} - v_{n} 〉 - λ L_{1} ∥ v_{n - 1} - v_{n} ∥^{2} - λ L_{2} {∥ v_{n} - u_{n + 1} ∥}^{2} . \end{matrix}

(12)

We have the following facts:

- 2 〈 ρ_{n} - u_{n + 1}, u_{n + 1} - ξ^{*} 〉 = - ∥ ρ_{n} - ξ^{*} ∥^{2} + ∥ u_{n + 1} - ρ_{n} ∥^{2} + {∥ u_{n + 1} - ξ^{*} ∥}^{2} .

2 〈 ρ_{n} - v_{n}, u_{n + 1} - v_{n} 〉 = ∥ ρ_{n} - v_{n} ∥^{2} + ∥ u_{n + 1} - v_{n} ∥^{2} - {∥ ρ_{n} - u_{n + 1} ∥}^{2} .

We also have the following inequality

∥ v_{n - 1} - v_{n} ∥^{2} \leq {(∥ v_{n - 1} - ρ_{n} ∥ + ∥ ρ_{n} - v_{n} ∥)}^{2} \leq 2 ∥ v_{n - 1} - ρ_{n} ∥^{2} + 2 {∥ ρ_{n} - v_{n} ∥}^{2} .

From the above two facts and last inequality with (12) provides the required result. □

Now, we are in a position to provide our first convergence result of this work.

Theorem 1.

Assume that

{u_{n}}

{v_{n}}

and

{ρ_{n}}

sequences in

E

generated by Algorithm 1, where the sequence

ϑ_{n}

is non-decreasing and λ is a positive real number, such that

0 < λ < \frac{\frac{1}{2} - 2 ϑ - \frac{1}{2} ϑ^{2}}{L_{2} {(1 - ϑ)}^{2} + 2 L_{1} (1 + ϑ + ϑ^{2} + ϑ^{3})} and 0 \leq ϑ_{n} \leq ϑ < \sqrt{5} - 2 .

Subsequently, the sequences

{u_{n}}

{v_{n}}

and

{ρ_{n}}

are converges weakly to an element

ξ^{*}

S O L_{E P (f, K)}

Proof.

From Lemma 10, we have

\begin{matrix} ∥ u_{n + 1} - ξ^{*} ∥^{2} + 4 λ L_{1} {∥ ρ_{n + 1} - v_{n} ∥}^{2} \\ \leq ∥ ρ_{n} - ξ^{*} ∥^{2} - (1 - 4 λ L_{1}) ∥ ρ_{n} - v_{n} ∥^{2} - (1 - 2 λ L_{2}) {∥ u_{n + 1} - v_{n} ∥}^{2} \\ + 4 λ L_{1} ∥ ρ_{n} - v_{n - 1} ∥^{2} + 4 λ L_{1} {∥ ρ_{n + 1} - v_{n} ∥}^{2} . \end{matrix}

(13)

By the definition of

ρ_{n}

in Algorithm 1, we have

\begin{matrix} ∥ ρ_{n} - ξ^{*} ∥^{2} & = ∥ (1 + ϑ_{n}) (u_{n} - ξ^{*}) - ϑ_{n} (u_{n - 1} - ξ^{*}) ∥^{2} \\ = (1 + ϑ_{n}) ∥ u_{n} - ξ^{*} ∥^{2} - ϑ_{n} ∥ u_{n - 1} - ξ^{*} ∥^{2} + ϑ_{n} (1 + ϑ_{n}) {∥ u_{n} - u_{n - 1} ∥}^{2} . \end{matrix}

(14)

By the definition of

ρ_{n + 1}

in Algorithm 1, we also have

\begin{matrix} ∥ ρ_{n + 1} - v_{n} ∥^{2} & = ∥ u_{n + 1} + ϑ_{n + 1} (u_{n + 1} - u_{n}) - v_{n} ∥^{2} \\ = ∥ (1 + ϑ_{n + 1}) (u_{n + 1} - v_{n}) - ϑ_{n + 1} (u_{n} - v_{n}) ∥^{2} \\ = (1 + ϑ_{n + 1}) ∥ u_{n + 1} - v_{n} ∥^{2} - ϑ_{n + 1} ∥ u_{n} - v_{n} ∥^{2} + ϑ_{n + 1} (1 + ϑ_{n + 1}) {∥ u_{n + 1} - u_{n} ∥}^{2} \\ \leq (1 + ϑ_{n}) ∥ u_{n + 1} - v_{n} ∥^{2} + ϑ_{n} (1 + ϑ_{n}) {∥ u_{n + 1} - u_{n} ∥}^{2} . \end{matrix}

(15)

Combining the expression (13)–(15), we obtain

\begin{matrix} ∥ u_{n + 1} - ξ^{*} ∥^{2} + 4 λ L_{1} {∥ ρ_{n + 1} - v_{n} ∥}^{2} \\ \leq (1 + ϑ_{n}) ∥ u_{n} - ξ^{*} ∥^{2} - ϑ_{n} ∥ u_{n - 1} - ξ^{*} ∥^{2} + ϑ_{n} (1 + ϑ_{n}) {∥ u_{n} - u_{n - 1} ∥}^{2} \\ + 4 λ L_{1} ∥ ρ_{n} - v_{n - 1} ∥^{2} - (1 - 4 λ L_{1}) ∥ ρ_{n} - v_{n} ∥^{2} - (1 - 2 λ L_{2}) {∥ u_{n + 1} - v_{n} ∥}^{2} \\ + 4 λ L_{1} (1 + ϑ_{n}) ∥ u_{n + 1} - v_{n} ∥^{2} + 4 λ L_{1} ϑ_{n} (1 + ϑ_{n}) {∥ u_{n + 1} - u_{n} ∥}^{2} \end{matrix}

(16)

\begin{matrix} \leq (1 + ϑ_{n}) ∥ u_{n} - ξ^{*} ∥^{2} - ϑ_{n} ∥ u_{n - 1} - ξ^{*} ∥^{2} + ϑ_{n} (1 + ϑ_{n}) {∥ u_{n} - u_{n - 1} ∥}^{2} \\ + 4 λ L_{1} ∥ ρ_{n} - v_{n - 1} ∥^{2} + 4 λ L_{1} ϑ_{n} (1 + ϑ_{n}) {∥ u_{n + 1} - u_{n} ∥}^{2} \\ - (1 - 4 λ L_{1}) ∥ ρ_{n} - v_{n} ∥^{2} - (1 - 2 λ L_{2} - 4 λ L_{1} (1 + ϑ_{n})) {∥ u_{n + 1} - v_{n} ∥}^{2} \end{matrix}

(17)

\begin{matrix} \leq (1 + ϑ_{n + 1}) ∥ u_{n} - ξ^{*} ∥^{2} - ϑ_{n} ∥ u_{n - 1} - ξ^{*} ∥^{2} + ϑ_{n} (1 + ϑ_{n}) {∥ u_{n} - u_{n - 1} ∥}^{2} \\ + 4 λ L_{1} ∥ ρ_{n} - v_{n - 1} ∥^{2} + 4 λ L_{1} ϑ_{n} (1 + ϑ_{n}) {∥ u_{n + 1} - u_{n} ∥}^{2} \\ - \frac{(1 - 2 λ L_{2} - 4 λ L_{1} (1 + ϑ_{n}))}{2} [2 (∥ u_{n + 1} - v_{n} ∥^{2} + ∥ ρ_{n} - v_{n} ∥^{2})] . \end{matrix}

(18)

By substituting

σ_{n} = \frac{1 - 2 λ L_{2} - 4 λ L_{1} (1 + ϑ_{n})}{2},

and due to the inequality

2 ∥ u_{n + 1} - v_{n} ∥^{2} + 2 ∥ ρ_{n} - v_{n} ∥^{2} \geq {∥ u_{n + 1} - ρ_{n} ∥}^{2} .

From this discussion, the expression (18) turns into following:

\begin{matrix} Λ_{n + 1} & \leq Λ_{n} + ϑ_{n} (1 + ϑ_{n}) ∥ u_{n} - u_{n - 1} ∥^{2} + 4 λ L_{1} ϑ_{n} (1 + ϑ_{n}) ∥ u_{n + 1} - u_{n} ∥^{2} - σ_{n} {∥ u_{n + 1} - ρ_{n} ∥}^{2}, \end{matrix}

(19)

where

Λ_{n} = ∥ u_{n} - ξ^{*} ∥^{2} - ϑ_{n} ∥ u_{n - 1} - ξ^{*} ∥^{2} + 4 λ L_{1} {∥ ρ_{n} - v_{n - 1} ∥}^{2} .

By the value

ρ_{n + 1}

, we have

\begin{matrix} ∥ u_{n + 1} - ρ_{n} ∥^{2} & = ∥ u_{n + 1} - u_{n} - ϑ_{n} (u_{n} - u_{n - 1}) ∥^{2} \\ = ∥ u_{n + 1} - u_{n} ∥^{2} + ϑ_{n}^{2} {∥ u_{n} - u_{n - 1} ∥}^{2} - 2 ϑ_{n} 〈 u_{n + 1} - u_{n}, u_{n} - u_{n - 1} 〉 \end{matrix}

(20)

\begin{matrix} \geq ∥ u_{n + 1} - u_{n} ∥^{2} + ϑ_{n}^{2} ∥ u_{n} - u_{n - 1} ∥^{2} - 2 ϑ_{n} ∥ u_{n + 1} - u_{n} ∥ ∥ u_{n} - u_{n - 1} ∥ \\ \geq (1 - ϑ_{n}) ∥ u_{n + 1} - u_{n} ∥^{2} + (ϑ_{n}^{2} - ϑ_{n}) {∥ u_{n} - u_{n - 1} ∥}^{2} . \end{matrix}

(21)

Combining the expression (19) and (21) implies that

\begin{matrix} Λ_{n + 1} & \leq Λ_{n} + ϑ_{n} (1 + ϑ_{n}) ∥ u_{n} - u_{n - 1} ∥^{2} + 4 λ L_{1} ϑ_{n} (1 + ϑ_{n}) {∥ u_{n + 1} - u_{n} ∥}^{2} \\ - σ_{n} (1 - ϑ_{n}) ∥ u_{n + 1} - u_{n} ∥^{2} - σ_{n} (ϑ_{n}^{2} - ϑ_{n}) {∥ u_{n} - u_{n - 1} ∥}^{2} \\ \leq Λ_{n} + r_{n} ∥ u_{n} - u_{n - 1} ∥^{2} - q_{n} {∥ u_{n + 1} - u_{n} ∥}^{2}, \end{matrix}

(22)

where

r_{n} : = ϑ_{n} (1 + ϑ_{n}) + σ_{n} ϑ_{n} (1 - ϑ_{n})

and

q_{n} : = σ_{n} (1 - ϑ_{n}) - 4 λ L_{1} ϑ_{n} (1 + ϑ_{n}) .

Further, we take

Γ_{n} = Λ_{n} + r_{n} {∥ u_{n} - u_{n - 1} ∥}^{2} .

It follows from (22) that

\begin{matrix} Γ_{n + 1} - Γ_{n} & = ∥ u_{n + 1} - ξ^{*} ∥^{2} - ϑ_{n + 1} ∥ u_{n} - ξ^{*} ∥^{2} + r_{n + 1} ∥ u_{n + 1} - u_{n} ∥^{2} + 4 λ L_{1} {∥ ρ_{n + 1} - v_{n} ∥}^{2} \\ - ∥ u_{n} - ξ^{*} ∥^{2} + ϑ_{n} ∥ u_{n - 1} - ξ^{*} ∥^{2} - r_{n} ∥ u_{n} - u_{n - 1} ∥^{2} - 4 λ L_{1} {∥ ρ_{n} - v_{n - 1} ∥}^{2} \\ = ∥ u_{n + 1} - ξ^{*} ∥^{2} - (1 + ϑ_{n + 1}) ∥ u_{n} - ξ^{*} ∥^{2} + ϑ_{n} ∥ u_{n - 1} - ξ^{*} ∥^{2} + 4 λ L_{1} {∥ ρ_{n + 1} - v_{n} ∥}^{2} \\ - 4 λ L_{1} ∥ ρ_{n} - v_{n - 1} ∥^{2} - r_{n} ∥ u_{n} - u_{n - 1} ∥^{2} + r_{n + 1} {∥ u_{n + 1} - u_{n} ∥}^{2} \\ \leq - q_{n} ∥ u_{n + 1} - u_{n} ∥^{2} + r_{n + 1} {∥ u_{n + 1} - u_{n} ∥}^{2} \\ = - (q_{n} - r_{n + 1}) {∥ u_{n + 1} - u_{n} ∥}^{2} . \end{matrix}

(23)

Next, we need to compute

\begin{matrix} q_{n} - r_{n + 1} \\ = σ_{n} (1 - ϑ_{n}) - 4 λ L_{1} ϑ_{n} (1 + ϑ_{n}) - ϑ_{n + 1} (1 + ϑ_{n + 1}) - σ_{n + 1} ϑ_{n + 1} (1 - ϑ_{n + 1}) \\ \geq σ_{n} (1 - ϑ_{n}) - 4 λ L_{1} ϑ_{n} (1 + ϑ_{n}) - ϑ_{n} (1 + ϑ_{n}) - σ_{n} ϑ_{n} (1 - ϑ_{n}) \\ \geq σ_{n} {(1 - ϑ)}^{2} - 4 λ L_{1} ϑ (1 + ϑ) - ϑ (1 + ϑ) \\ \geq \frac{1 - 2 λ L_{2} - 4 λ L_{1} (1 + ϑ)}{2} {(1 - ϑ)}^{2} - 4 λ L_{1} ϑ (1 + ϑ) - ϑ (1 + ϑ) \\ = (\frac{1}{2} - 2 ϑ - \frac{1}{2} ϑ^{2}) - λ (L_{2} {(1 - ϑ)}^{2} + 2 L_{1} (1 + ϑ + ϑ^{2} + ϑ^{3})) \\ \geq 0 . \end{matrix}

(24)

The expression (23) and (24) with some

δ \geq 0

, implies that

Γ_{n + 1} - Γ_{n} \leq - (q_{n} - r_{n + 1}) ∥ u_{n + 1} - u_{n} ∥^{2} \leq - δ {∥ u_{n + 1} - u_{n} ∥}^{2} \leq 0 .

(25)

The above relation (25) implies that the sequence

{Γ_{n}}

is non-increasing. From

Γ_{n + 1}

, we have

\begin{matrix} Γ_{n + 1} & = ∥ u_{n + 1} - ξ^{*} ∥^{2} - ϑ_{n + 1} ∥ u_{n} - ξ^{*} ∥^{2} + r_{n + 1} ∥ u_{n + 1} - u_{n} ∥^{2} + 4 λ L_{1} {∥ ρ_{n + 1} - v_{n} ∥}^{2} \\ \geq - ϑ_{n + 1} {∥ u_{n} - ξ^{*} ∥}^{2} . \end{matrix}

(26)

Additionally, from definition

Γ_{n}

, we have

\begin{matrix} ∥ u_{n} - ξ^{*} ∥^{2} & \leq Γ_{n} + ϑ_{n} {∥ u_{n - 1} - ξ^{*} ∥}^{2} \\ \leq Γ_{1} + ϑ {∥ u_{n - 1} - ξ^{*} ∥}^{2} \\ \leq \dots \leq Γ_{1} (ϑ^{n - 1} + \dots + 1) + ϑ^{n} {∥ u_{0} - ξ^{*} ∥}^{2} \\ \leq \frac{Γ_{1}}{1 - ϑ} + ϑ^{n} {∥ u_{0} - ξ^{*} ∥}^{2} . \end{matrix}

(27)

Combining the expression (26) and (27), we obtain

\begin{matrix} - Γ_{n + 1} & \leq ϑ_{n + 1} {∥ u_{n} - ξ^{*} ∥}^{2} \\ \leq ϑ ∥ u_{n} - ξ^{*} ∥^{2} \\ \leq ϑ \frac{Γ_{1}}{1 - ϑ} + ϑ^{n + 1} {∥ u_{0} - ξ^{*} ∥}^{2} . \end{matrix}

(28)

It continues to follow from (25) and (28), such that

\begin{matrix} δ \sum_{n = 1}^{k} {∥ u_{n + 1} - u_{n} ∥}^{2} & \leq Γ_{1} - Γ_{k + 1} \\ \leq Γ_{1} + ϑ \frac{Γ_{1}}{1 - ϑ} + ϑ^{n + 1} {∥ u_{0} - ξ^{*} ∥}^{2} \\ \leq \frac{Γ_{1}}{1 - ϑ} + {∥ u_{0} - ξ^{*} ∥}^{2}, \end{matrix}

(29)

letting

k \to \infty

in (29) implies that

\sum_{n = 1}^{\infty} ∥ u_{n + 1} - u_{n} ∥^{2} < + \infty implies ∥ u_{n + 1} - u_{n} ∥ \to 0 as n \to \infty .

(30)

From the relation (20) and (30), we obtain

∥ u_{n + 1} - ρ_{n} ∥ \to 0 as n \to \infty .

(31)

Next, the expression (28) implies that

- Λ_{n + 1} \leq ϑ \frac{Γ_{1}}{1 - ϑ} + ϑ^{n + 1} ∥ u_{0} - ξ^{*} ∥^{2} + r_{n + 1} {∥ u_{n + 1} - u_{n} ∥}^{2} .

(32)

From the relation (18) we have

\begin{matrix} (1 - 2 λ L_{2} - 4 λ L_{1} (1 + ϑ)) [∥ u_{n + 1} - v_{n} ∥^{2} + {∥ ρ_{n} - v_{n} ∥}^{2}] \\ \leq Λ_{n} - Λ_{n + 1} + ϑ (1 + ϑ) ∥ u_{n} - u_{n - 1} ∥^{2} + 4 λ L_{1} ϑ (1 + ϑ) {∥ u_{n + 1} - u_{n} ∥}^{2} . \end{matrix}

(33)

Set

k \in N

and using (33) for

n = 1, 2, \dots, k

, gives that

\begin{matrix} (1 - 2 L_{2} λ - 4 L_{1} λ (1 + ϑ)) \sum_{n = 1}^{k} [∥ u_{n + 1} - v_{n} ∥^{2} + {∥ ρ_{n} - v_{n} ∥}^{2}] \\ \leq Λ_{0} - Λ_{k + 1} + ϑ (1 + ϑ) \sum_{n = 1}^{k} ∥ u_{n} - u_{n - 1} ∥^{2} + 4 λ L_{1} ϑ (1 + ϑ) \sum_{n = 1}^{k} {∥ u_{n + 1} - u_{n} ∥}^{2} \\ \leq Λ_{0} + ϑ \frac{Γ_{1}}{1 - ϑ} + ϑ^{k + 1} ∥ u_{0} - ξ^{*} ∥^{2} + r_{k + 1} {∥ u_{k + 1} - u_{k} ∥}^{2} \\ + ϑ (1 + ϑ) \sum_{n = 1}^{k} ∥ u_{n} - u_{n - 1} ∥^{2} + 4 λ L_{1} ϑ (1 + ϑ) \sum_{n = 1}^{k} {∥ u_{n + 1} - u_{n} ∥}^{2}, \end{matrix}

(34)

letting

k \to \infty

in (34) implies that

\sum_{n = 1}^{\infty} ∥ u_{n + 1} - v_{n} ∥^{2} < + \infty and \sum_{n = 1}^{\infty} {∥ ρ_{n} - v_{n} ∥}^{2} < + \infty,

(35)

and

lim_{n \to \infty} ∥ u_{n + 1} - v_{n} ∥ = lim_{n \to \infty} ∥ ρ_{n} - v_{n} ∥ = 0 .

(36)

The following relation can easily be derived:

lim_{n \to \infty} ∥ u_{n} - v_{n} ∥ = lim_{n \to \infty} ∥ u_{n} - ρ_{n} ∥ = lim_{n \to \infty} ∥ v_{n - 1} - v_{n} ∥ = 0 .

(37)

By the definition of

ρ_{n}

and using Cauchy inequality, we have

\begin{matrix} ∥ ρ_{n} - v_{n - 1} ∥^{2} & = ∥ u_{n} + ϑ_{n} (u_{n} - u_{n - 1}) - v_{n - 1} ∥^{2} \\ = ∥ (1 + ϑ_{n}) (u_{n} - v_{n - 1}) - ϑ_{n} (u_{n - 1} - v_{n - 1}) ∥^{2} \\ = (1 + ϑ_{n}) ∥ u_{n} - v_{n - 1} ∥^{2} - ϑ_{n} ∥ u_{n - 1} - v_{n - 1} ∥^{2} + ϑ_{n} (1 + ϑ_{n}) {∥ u_{n} - u_{n - 1} ∥}^{2} \\ \leq (1 + ϑ) ∥ u_{n} - v_{n - 1} ∥^{2} + ϑ (1 + ϑ) {∥ u_{n} - u_{n - 1} ∥}^{2} . \end{matrix}

(38)

Now, summing up the expression (38) for

n = 1, 2 \dots, k

, we obtain

\sum_{n = 1}^{k} ∥ ρ_{n} - v_{n - 1} ∥^{2} \leq (1 + ϑ) \sum_{n = 1}^{k} ∥ u_{n} - v_{n - 1} ∥^{2} + ϑ (1 + ϑ) \sum_{n = 1}^{k} {∥ u_{n} - u_{n - 1} ∥}^{2}

(39)

The above expression with (30) and (35) implies that

\sum ∥ ρ_{n} - v_{n - 1} ∥^{2} < + \infty .

(40)

It follows from the relation (16), we obtain

\begin{matrix} ∥ u_{n + 1} - ξ^{*} ∥^{2} & \leq (1 + ϑ) ∥ u_{n} - ξ^{*} ∥^{2} - ϑ ∥ u_{n - 1} - ξ^{*} ∥^{2} + ϑ (1 + ϑ) {∥ u_{n} - u_{n - 1} ∥}^{2} \\ + 4 L_{1} λ {∥ ρ_{n} - v_{n - 1} ∥}^{2}, \end{matrix}

(41)

above expression with (30), (40), (37) and Lemma 4 implies that limit of

∥ u_{n} - ξ^{*} ∥,

∥ ρ_{n} - ξ^{*} ∥

and

∥ v_{n} - ξ^{*} ∥

exists for every

ξ^{*} \in S O L_{E P (f, K)}

, means that the sequences

{u_{n}}

{ρ_{n}}

and

{v_{n}}

are bounded. Next, we need to show that each weak sequential limit point of the sequence

{u_{n}}

belongs to

S O L_{E P (f, K)}

. Let z be arbitrary weak cluster point of the sequence

{u_{n}}

, and then there exists a weak convergent subsequence

{u_{n_{k}}}

{u_{n}}

converges to

z

, this also implies that

{v_{n_{k}}}

also converge weakly to

z .

Now our aim to prove that

z \in S O L_{E P (f, K)} .

By Lemma 6, the bifunction Lipschitz-type condition and Lemma 8, we have

\begin{matrix} λ f (v_{n_{k}}, y) & \geq λ f (v_{n_{k}}, u_{n_{k} + 1}) + 〈 ρ_{n_{k}} - u_{n_{k} + 1}, y - u_{n_{k} + 1} 〉 \\ \geq λ f (v_{n_{k} - 1}, u_{n_{k} + 1}) - λ f (v_{n_{k} - 1}, v_{n_{k}}) - λ L_{1} {∥ v_{n_{k} - 1} - v_{n_{k}} ∥}^{2} \\ - λ L_{2} {∥ v_{n_{k}} - u_{n_{k} + 1} ∥}^{2} + 〈 ρ_{n_{k}} - u_{n_{k} + 1}, y - u_{n_{k} + 1} 〉 \\ \geq 〈 ρ_{n_{k}} - v_{n_{k}}, u_{n_{k} + 1} - v_{n_{k}} 〉 - λ L_{1} {∥ v_{n_{k} - 1} - v_{n_{k}} ∥}^{2} \\ - λ L_{2} {∥ v_{n_{k}} - u_{n_{k} + 1} ∥}^{2} + 〈 ρ_{n_{k}} - u_{n_{k} + 1}, y - u_{n_{k} + 1} 〉 \end{matrix}

(42)

where y be an any element in

H_{n} .

As a result with (31), (36), (37), and due to the boundedness of the sequence

{u_{n}}

the above inequality tends to zero. By given

λ > 0

, the assumption (A3) and

v_{n_{k}} ⇀ z

, we obtain

0 \leq \underset{k \to \infty}{lim sup} f (v_{n_{k}}, y) \leq f (z, y), \forall y \in H_{n} .

Due to

z \in K \subset H_{n}

, we obtain

f (z, y) \geq 0, \forall y \in K .

This implies that z belongs to

S O L_{E P (f, K)} .

Thus Lemma 5, ensures that

{ρ_{n}}

{u_{n}}

and

{v_{n}}

weakly converges to

ξ^{*}

n \to \infty .

Remark 2.

For

ϑ_{n} = ϑ = 0

in Algorithm 1 gives the results as in [35,36].

4. Inertial Popov’s Two-Step Subgradient Extragradient Algorithm for Strongly Pseudomonotone EP

The second algorithm is also an inertial algorithm that is able to solve the strongly pseudomonotone equilibrium problem. However, the advantage of this algorithm is that there is no need for prior information regarding the strongly pseudomonotone constant

γ

and Lipschitz constants

L_{1}, L_{2}

. Let

{λ_{n}} \subset (0, + \infty)

be a non-increasing sequence, so that the following conditions are satisfied:

(T 1) : lim_{n \to \infty} λ_{n} = 0 and (T 2) : \sum_{n = 1}^{\infty} λ_{n} = + \infty .

(43)

Assumption 2.

Let a bifunction

f : E \times E \to R

satisfies the following conditions:

(B1): $f (\tilde{u}, \tilde{u}) = 0, \forall \tilde{u} \in K$ and f is strongly pseudomontone on K;
(B2): f meet the Lipschitz-type condition on $E$ with two positive constants $L_{1}$ and $L_{2}$ ;
(B3): $f (\tilde{u}, .)$ is sub-differentiable and convex on $E$ for all $\tilde{u} \in E .$

Lemma 11.

Assume that

f : E \times E \to R

satisfies the conditions (B1)–(B3). Let the solution set

S O L_{E P (f, K)}

is nonempty. For each

ξ^{*} \in S O L_{E P (f, K)}

, we have

\begin{matrix} ∥ u_{n + 1} - ξ^{*} ∥^{2} & \leq ∥ ρ_{n} - ξ^{*} ∥^{2} - (1 - 4 λ_{n} L_{1}) ∥ ρ_{n} - v_{n} ∥^{2} - (1 - 2 λ_{n} L_{2}) {∥ u_{n + 1} - v_{n} ∥}^{2} \\ + 4 λ_{n} L_{1} ∥ ρ_{n} - v_{n - 1} ∥^{2} - 2 γ λ_{n} {∥ v_{n} - ξ^{*} ∥}^{2} . \end{matrix}

Now, we are in a position to provide our second convergence result of this work.

Theorem 2.

Assume that

f : E \times E \to R

satisfies the conditions (B1)–(B3). Let

{u_{n}}

{v_{n}}

and

{ρ_{n}}

are sequences in

E

generated by Algorithm 2 and

ϑ_{n}

is non-decreasing sequence with

0 \leq ϑ_{n} \leq ϑ < \sqrt{5} - 2

. Subsequently,

{u_{n}}

{v_{n}}

and

{ρ_{n}}

strongly converge to an element

ξ^{*}

S O L_{E P (f, K)}

Algorithm 2 (Two-step Subgradient Extragradient Algorithm for Strongly Pseudomonotone EP)

Initialization: Choose $u_{- 1}, u_{0}, v_{0} \in E$ , $0 \leq ϑ_{n} \leq ϑ < \sqrt{5} - 2$ and a sequence ${λ_{n}}$ satisfying (43). Set

$u_{1} = \underset{}{arg min} {λ_{0} f (v_{0}, y) + \frac{1}{2} ∥ ρ_{0} - y ∥^{2} : y \in K},$

$v_{1} = \arg \min {λ_{1} f (v_{0}, y) + \frac{1}{2} ∥ ρ_{1} - y ∥^{2} : y \in K},$

where $ρ_{0} = u_{0} + ϑ_{0} (u_{0} - u_{- 1})$ and $ρ_{1} = u_{1} + ϑ_{1} (u_{1} - u_{0})$ .
Iterative steps: Assume that $u_{n - 1}$ , $u_{n}$ , $v_{n - 1}$ and $v_{n}$ are known for $n \geq 1$ and

$H_{n} = {z \in E : 〈 ρ_{n} - λ_{n} ω_{n - 1} - v_{n}, z - v_{n} 〉 \leq 0},$

where $ω_{n - 1} \in \partial f (v_{n - 1}, v_{n})$ .
Step 1: Compute

$u_{n + 1} = \underset{}{arg min} {λ_{n} f (v_{n}, y) + \frac{1}{2} ∥ ρ_{n} - y ∥^{2} : y \in H_{n}},$

where $ρ_{n} = u_{n} + ϑ_{n} (u_{n} - u_{n - 1})$ .
Step 2: Compute

$v_{n + 1} = \underset{}{arg min} {λ_{n + 1} f (v_{n}, y) + \frac{1}{2} ∥ ρ_{n + 1} - y ∥^{2} : y \in K},$

where $ρ_{n + 1} = u_{n + 1} + ϑ_{n + 1} (u_{n + 1} - u_{n})$ .
Step 3: If $u_{n + 1} = ρ_{n}$ and $v_{n} = v_{n - 1}$ , then STOP. Otherwise set $n : = n + 1$ and go to Step 1.

Proof.

The proof is the identical as the proof of Theorem 1, but there are still few changes. We provide the proof for the readable purpose. By Lemma 11 and adding

4 L_{1} λ_{n} {∥ ρ_{n + 1} - v_{n} ∥}^{2}

in both sides, we have

\begin{matrix} ∥ u_{n + 1} - ξ^{*} ∥^{2} + 4 L_{1} λ_{n} {∥ ρ_{n + 1} - v_{n} ∥}^{2} \\ \leq ∥ ρ_{n} - ξ^{*} ∥^{2} - (1 - 4 L_{1} λ_{n}) ∥ ρ_{n} - v_{n} ∥^{2} - (1 - 2 L_{2} λ_{n}) {∥ u_{n + 1} - v_{n} ∥}^{2} \\ + 4 L_{1} λ_{n} ∥ ρ_{n} - v_{n - 1} ∥^{2} - 2 γ λ_{n} ∥ v_{n} - ξ^{*} ∥^{2} + 4 L_{1} λ_{n} {∥ ρ_{n + 1} - v_{n} ∥}^{2} . \end{matrix}

(44)

By using the definition of

ρ_{n}

in Algorithm 2, we have

\begin{matrix} ∥ ρ_{n} - ξ^{*} ∥^{2} & = (1 + ϑ_{n}) ∥ u_{n} - ξ^{*} ∥^{2} - ϑ_{n} ∥ u_{n - 1} - ξ^{*} ∥^{2} + ϑ_{n} (1 + ϑ_{n}) {∥ u_{n} - u_{n - 1} ∥}^{2} . \end{matrix}

(45)

By using the definition

ρ_{n + 1}

in Algorithm 2, we also have

\begin{matrix} ∥ ρ_{n + 1} - v_{n} ∥^{2} & \leq (1 + ϑ_{n}) ∥ u_{n + 1} - v_{n} ∥^{2} + ϑ_{n} (1 + ϑ_{n}) {∥ u_{n + 1} - u_{n} ∥}^{2} . \end{matrix}

(46)

Combining the expression (44)–(46), we obtain

\begin{matrix} ∥ u_{n + 1} - ξ^{*} ∥^{2} + 4 L_{1} λ_{n + 1} {∥ ρ_{n + 1} - v_{n} ∥}^{2} \\ \leq (1 + ϑ_{n}) ∥ u_{n} - ξ^{*} ∥^{2} - ϑ_{n} ∥ u_{n - 1} - ξ^{*} ∥^{2} + ϑ_{n} (1 + ϑ_{n}) {∥ u_{n} - u_{n - 1} ∥}^{2} \\ + 4 L_{1} λ_{n} ∥ ρ_{n} - v_{n - 1} ∥^{2} - (1 - 4 L_{1} λ_{n}) ∥ ρ_{n} - v_{n} ∥^{2} - (1 - 2 L_{2} λ_{n}) {∥ u_{n + 1} - v_{n} ∥}^{2} \\ + 4 L_{1} λ_{n} (1 + ϑ_{n}) ∥ u_{n + 1} - v_{n} ∥^{2} + 4 L_{1} λ_{n} ϑ_{n} (1 + ϑ_{n}) ∥ u_{n + 1} - u_{n} ∥^{2} - 2 γ λ_{n} {∥ v_{n} - ξ^{*} ∥}^{2} \end{matrix}

(47)

\begin{matrix} \leq (1 + ϑ_{n}) ∥ u_{n} - ξ^{*} ∥^{2} - ϑ_{n} ∥ u_{n - 1} - ξ^{*} ∥^{2} + 4 L_{1} λ_{n} {∥ ρ_{n} - v_{n - 1} ∥}^{2} \\ + ϑ_{n} (1 + ϑ_{n}) ∥ u_{n} - u_{n - 1} ∥^{2} + 4 L_{1} λ_{n} (1 + ϑ_{n}) ∥ u_{n + 1} - u_{n} ∥^{2} - 2 γ λ_{n} {∥ v_{n} - ξ^{*} ∥}^{2} \\ - \frac{(1 - 2 L_{2} λ_{n} - 4 L_{1} λ_{n} (1 + ϑ_{n}))}{2} [2 (∥ u_{n + 1} - v_{n} ∥^{2} + ∥ ρ_{n} - v_{n} ∥^{2})] . \end{matrix}

(48)

Next, we let

ϱ_{n} = \frac{1 - 2 L_{2} λ_{n} - 4 L_{1} λ_{n} (1 + ϑ_{n})}{2}

and

Φ_{n} = ∥ u_{n} - ξ^{*} ∥^{2} - ϑ_{n} ∥ u_{n - 1} - ξ^{*} ∥^{2} + 4 L_{1} λ_{n} {∥ ρ_{n} - v_{n - 1} ∥}^{2} .

Due to the above substituting the expression (48) turns into the following:

\begin{matrix} Φ_{n + 1} & \leq Φ_{n} + ϑ_{n} (1 + ϑ_{n}) ∥ u_{n} - u_{n - 1} ∥^{2} + 4 L_{1} λ_{n} ϑ_{n} (1 + ϑ_{n}) {∥ u_{n + 1} - u_{n} ∥}^{2} \\ - ϱ_{n} ∥ u_{n + 1} - ρ_{n} ∥^{2} - 2 γ λ_{n} {∥ v_{n} - ξ^{*} ∥}^{2}, \end{matrix}

(49)

By the definition

ρ_{n + 1}

, we have

\begin{matrix} ∥ u_{n + 1} - ρ_{n} ∥^{2} & \geq (1 - ϑ_{n}) ∥ u_{n + 1} - u_{n} ∥^{2} + (ϑ_{n}^{2} - ϑ_{n}) {∥ u_{n} - u_{n - 1} ∥}^{2} . \end{matrix}

(50)

Combining the expression (49) and (50), we obtain

\begin{matrix} Φ_{n + 1} & \leq Φ_{n} + ϑ_{n} (1 + ϑ_{n}) ∥ u_{n} - u_{n - 1} ∥^{2} + 4 L_{1} λ_{n} ϑ_{n} (1 + ϑ_{n}) {∥ u_{n + 1} - u_{n} ∥}^{2} \\ - 2 γ λ_{n} ∥ v_{n} - ξ^{*} ∥^{2} - ϱ_{n} (1 - ϑ_{n}) ∥ u_{n + 1} - u_{n} ∥^{2} - ϱ_{n} (ϑ_{n}^{2} - ϑ_{n}) {∥ u_{n} - u_{n - 1} ∥}^{2} \\ = Φ_{n} + R_{n} ∥ u_{n} - u_{n - 1} ∥^{2} - Q_{n} ∥ u_{n + 1} - u_{n} ∥^{2} - 2 γ λ_{n} {∥ v_{n} - ξ^{*} ∥}^{2}, \end{matrix}

(51)

where

R_{n} : = ϑ_{n} (1 + ϑ_{n}) + ϱ_{n} ϑ_{n} (1 - ϑ_{n})

and

Q_{n} : = ϱ_{n} (1 - ϑ_{n}) - 4 L_{1} λ_{n} ϑ_{n} (1 + ϑ_{n}) .

In addition, we also take

Ψ_{n} = Φ_{n} + R_{n} {∥ u_{n} - u_{n - 1} ∥}^{2} .

It follows from (51) that

Ψ_{n + 1} - Ψ_{n} \leq - (Q_{n} - R_{n + 1}) ∥ u_{n + 1} - u_{n} ∥^{2} - 2 γ λ_{n} {∥ v_{n} - ξ^{*} ∥}^{2} .

(52)

Since

λ_{n} \to 0

, then there exists a finite number

n_{0} \in N

such that

0 < λ_{n} < \frac{\frac{1}{2} - 2 ϑ - \frac{1}{2} ϑ^{2}}{L_{2} {(1 - ϑ)}^{2} + 2 L_{1} (1 + ϑ + ϑ^{2} + ϑ^{3})}, n \geq n_{0} .

Similarly, it follows from (24) and expression (52) implies that

Ψ_{n + 1} - Ψ_{n} \leq - δ {∥ u_{n + 1} - u_{n} ∥}^{2} \leq 0, n \geq n_{0} .

(53)

The above implies that the sequence

{Ψ_{n}}

is non-increasing for

n \geq n_{0} .

From the value of

Ψ_{n}

, we have

\begin{matrix} ∥ u_{n} - ξ^{*} ∥^{2} & \leq Ψ_{n} + ϑ_{n} {∥ u_{n - 1} - ξ^{*} ∥}^{2} \\ \leq Ψ_{n_{0}} + ϑ {∥ u_{n - 1} - ξ^{*} ∥}^{2} \\ \leq \dots \leq Ψ_{n_{0}} (ϑ^{n - n_{0}} + \dots + 1) + ϑ^{n - n_{0}} {∥ u_{n_{0}} - ξ^{*} ∥}^{2} \\ \leq \frac{Ψ_{n_{0}}}{1 - ϑ} + ϑ^{n - n_{0}} {∥ u_{n_{0}} - ξ^{*} ∥}^{2} . \end{matrix}

(54)

From the definition of

Ψ_{n + 1}

with the expression (54), we obtain

\begin{matrix} - Ψ_{n + 1} & \leq ϑ_{n + 1} {∥ u_{n} - ξ^{*} ∥}^{2} \\ \leq ϑ ∥ u_{n} - ξ^{*} ∥^{2} \\ \leq ϑ \frac{Ψ_{n_{0}}}{1 - ϑ} + ϑ^{n - n_{0} + 1} {∥ u_{n_{0}} - ξ^{*} ∥}^{2} \\ \leq ϑ \frac{Ψ_{n_{0}}}{1 - ϑ} + {∥ u_{n_{0}} - ξ^{*} ∥}^{2} . \end{matrix}

(55)

It is follows from (53) and (55) that

\begin{matrix} δ \sum_{n = n_{0}}^{k} {∥ u_{n + 1} - u_{n} ∥}^{2} & \leq Ψ_{n_{0}} - Ψ_{k + 1} \\ \leq Ψ_{n_{0}} + ϑ \frac{Ψ_{n_{0}}}{1 - ϑ} + {∥ u_{n_{0}} - ξ^{*} ∥}^{2} \\ \leq \frac{Ψ_{n_{0}}}{1 - ϑ} + {∥ u_{n_{0}} - ξ^{*} ∥}^{2}, \end{matrix}

(56)

letting

k \to \infty

in the expression (56), we obtain

\sum_{n = 1}^{\infty} ∥ u_{n + 1} - u_{n} ∥^{2} < + \infty implies that ∥ u_{n + 1} - u_{n} ∥ \to 0 as n \to \infty .

(57)

From the expression (20) and (57), we obtain

∥ u_{n + 1} - ρ_{n} ∥ \to 0 as n \to \infty .

(58)

The expression (55) implies that

- Φ_{n + 1} \leq ϑ \frac{Ψ_{n_{0}}}{1 - ϑ} + ∥ u_{n_{0}} - ξ^{*} ∥^{2} + R_{n + 1} {∥ u_{n + 1} - u_{n} ∥}^{2} .

(59)

It follows from (48) for all

n \geq n_{0}

, such that

\begin{matrix} (1 - 2 L_{2} λ_{n} - 4 L_{1} λ_{n} (1 + ϑ)) [∥ u_{n + 1} - v_{n} ∥^{2} + {∥ ρ_{n} - v_{n} ∥}^{2}] \\ \leq Φ_{n} - Φ_{n + 1} + ϑ (1 + ϑ) ∥ u_{n} - u_{n - 1} ∥^{2} + 4 L_{1} λ_{n} ϑ (1 + ϑ) {∥ u_{n + 1} - u_{n} ∥}^{2} . \end{matrix}

(60)

Consider the expression (60) for

n_{0}, n_{0} + 1, \dots, k .

Summing them up, we obtain

\begin{matrix} (1 - 2 L_{2} λ_{n} - 4 L_{1} λ_{n} (1 + ϑ)) \sum_{n = n_{0}}^{k} [∥ u_{n + 1} - v_{n} ∥^{2} + {∥ ρ_{n} - v_{n} ∥}^{2}] \\ \leq Φ_{n_{0}} - Φ_{k + 1} + ϑ (1 + ϑ) \sum_{n = n_{0}}^{k} ∥ u_{n} - u_{n - 1} ∥^{2} + \frac{4 L_{1}}{2 L_{2} + 4 L_{1}} ϑ (1 + ϑ) \sum_{n = n_{0}}^{k} {∥ u_{n + 1} - u_{n} ∥}^{2} \\ \leq Φ_{n_{0}} + ϑ \frac{Φ_{n_{0}}}{1 - ϑ} + ∥ u_{n_{0}} - ξ^{*} ∥^{2} + R_{k + 1} {∥ u_{k + 1} - u_{k} ∥}^{2} \\ + ϑ (1 + ϑ) \sum_{n = n_{0}}^{k} ∥ u_{n} - u_{n - 1} ∥^{2} + \frac{4 L_{1}}{2 L_{2} + 4 L_{1}} ϑ (1 + ϑ) \sum_{n = n_{0}}^{k} {∥ u_{n + 1} - u_{n} ∥}^{2} \\ = \frac{Φ_{n_{0}}}{1 - ϑ} + ∥ u_{n_{0}} - ξ^{*} ∥^{2} + R_{k + 1} {∥ u_{k + 1} - u_{k} ∥}^{2} \\ + ϑ (1 + ϑ) \sum_{n = n_{0}}^{k} ∥ u_{n} - u_{n - 1} ∥^{2} + \frac{4 L_{1}}{2 L_{2} + 4 L_{1}} ϑ (1 + ϑ) \sum_{n = n_{0}}^{k} {∥ u_{n + 1} - u_{n} ∥}^{2}, \end{matrix}

(61)

By letting

k \to \infty

in the expression (61) implies that

\sum_{n} ∥ u_{n + 1} - v_{n} ∥^{2} < + \infty and \sum_{n} {∥ ρ_{n} - v_{n} ∥}^{2} < + \infty,

(62)

and

lim_{n \to \infty} ∥ u_{n + 1} - v_{n} ∥ = lim_{n \to \infty} ∥ ρ_{n} - v_{n} ∥ = 0 .

(63)

We can easily derive the following relationship:

lim_{n \to \infty} ∥ u_{n} - v_{n} ∥ = lim_{n \to \infty} ∥ u_{n} - ρ_{n} ∥ = lim_{n \to \infty} ∥ v_{n - 1} - v_{n} ∥ = 0 .

(64)

By using the value

ρ_{n}

, we obtain

\begin{matrix} ∥ ρ_{n} - v_{n - 1} ∥^{2} & \leq (1 + ϑ) ∥ u_{n} - v_{n - 1} ∥^{2} + ϑ (1 + ϑ) {∥ u_{n} - u_{n - 1} ∥}^{2} . \end{matrix}

(65)

Now, summing up equation (65) for

n = n_{0}, n_{0} + 1 \dots, k

, we obtain

\sum_{n = n_{0}}^{k} ∥ ρ_{n} - v_{n - 1} ∥^{2} \leq (1 + ϑ) \sum_{n = n_{0}}^{k} ∥ u_{n} - v_{n - 1} ∥^{2} + ϑ (1 + ϑ) \sum_{n = n_{0}}^{k} {∥ u_{n} - u_{n - 1} ∥}^{2}

(66)

The above expression with (57) and (62) implies that

\sum_{n = 1}^{\infty} {∥ ρ_{n} - v_{n - 1} ∥}^{2} < + \infty .

(67)

Furthermore, the expression (47) gives that

\begin{matrix} ∥ u_{n + 1} - ξ^{*} ∥^{2} \\ \leq (1 + ϑ) ∥ u_{n} - ξ^{*} ∥^{2} - ϑ ∥ u_{n - 1} - ξ^{*} ∥^{2} + ϑ (1 + ϑ) ∥ u_{n} - u_{n - 1} ∥^{2} + 4 L_{1} λ_{n} {∥ ρ_{n} - v_{n - 1} ∥}^{2} . \end{matrix}

(68)

The above expression through (57), (67), and Lemma 4 implies that

lim_{n \to \infty} ∥ u_{n} - ξ^{*} ∥ = l .

(69)

The expression (64) with (69), we obtain

lim_{n \to \infty} ∥ ρ_{n} - ξ^{*} ∥ = lim_{n \to \infty} ∥ v_{n} - ξ^{*} ∥ = l .

(70)

Now, we are showing that the sequence

{u_{n}}

converges strongly to

ξ^{*} .

Due to the condition on

λ_{n}

for all

n \geq n_{0}

, we can easily observe the following inequality:

0 < λ_{n} < \frac{1}{2 L_{2} + 4 L_{1}}, \forall n \geq n_{0} .

It follows from Lemma 11, such that

2 γ λ_{n} ∥ v_{n} - ξ^{*} ∥^{2} \leq ∥ ρ_{n} - ξ^{*} ∥^{2} - ∥ u_{n + 1} - ξ^{*} ∥^{2} + 4 L_{1} λ_{n} {∥ ρ_{n} - v_{n - 1} ∥}^{2}, \forall n \geq n_{0} .

(71)

From the expression (45) and (71), we obtain

\begin{matrix} 2 γ λ_{n} {∥ v_{n} - ξ^{*} ∥}^{2} & \leq - ∥ u_{n + 1} - ξ^{*} ∥^{2} + (1 + ϑ_{n}) ∥ u_{n} - ξ^{*} ∥^{2} - ϑ_{n} {∥ u_{n - 1} - ξ^{*} ∥}^{2} \\ + ϑ_{n} (1 + ϑ_{n}) ∥ u_{n} - u_{n - 1} ∥^{2} + 4 L_{1} λ_{n} {∥ ρ_{n} - v_{n - 1} ∥}^{2} \\ \leq (∥ u_{n} - ξ^{*} ∥^{2} - ∥ u_{n + 1} - ξ^{*} ∥^{2}) + 2 ϑ ∥ u_{n} - u_{n - 1} ∥^{2} \\ + (ϑ_{n} ∥ u_{n} - ξ^{*} ∥^{2} - ϑ_{n - 1} ∥ u_{n - 1} - ξ^{*} ∥^{2}) + 4 L_{1} λ_{n} {∥ ρ_{n} - v_{n - 1} ∥}^{2} . \end{matrix}

(72)

It follows from expression (72) that

\begin{matrix} \sum_{n = n_{0}}^{k} 2 γ λ_{n} {∥ v_{n} - ξ^{*} ∥}^{2} \\ \leq (∥ u_{n_{0}} - ξ^{*} ∥^{2} - ∥ u_{k + 1} - ξ^{*} ∥^{2}) + 2 ϑ \sum_{n = n_{0}}^{k} {∥ u_{n} - u_{n - 1} ∥}^{2} \\ + (ϑ_{k} ∥ u_{k} - ξ^{*} ∥^{2} - ϑ_{n_{0} - 1} ∥ u_{n_{0} - 1} - ξ^{*} ∥^{2}) + \frac{4 L_{1}}{2 L_{2} + 4 L_{1}} \sum_{n = n_{0}}^{k} {∥ ρ_{n} - v_{n - 1} ∥}^{2} \\ \leq ∥ u_{n_{0}} - ξ^{*} ∥^{2} + ϑ ∥ u_{k} - ξ^{*} ∥^{2} + 2 ϑ \sum_{n = n_{0}}^{k} ∥ u_{n} - u_{n - 1} ∥^{2} + \frac{4 L_{1}}{2 L_{2} + 4 L_{1}} \sum_{n = n_{0}}^{k} {∥ ρ_{n} - v_{n - 1} ∥}^{2} \\ \leq M, \end{matrix}

(73)

for

M \geq 0 .

It implies that

\sum_{n = 1}^{\infty} 2 γ λ_{n} {∥ v_{n} - ξ^{*} ∥}^{2} < + \infty .

(74)

By the Lemma 2 and (74) implies that

lim inf ∥ v_{n} - ξ^{*} ∥ = 0 .

(75)

Finally, expression (69) and (75) provide that

{lim}_{n \to \infty} ∥ u_{n} - ξ^{*} ∥ = 0 .

This completes the proof. □

5. Application to Variational Inequality Problems

For considering Algorithm 1 and Theorem 1, we can able to write the next result for solving variational inequality problems that involve pseudomonotone and Lipschitz continuous operator.

Corollary 1.

Assume that

H : K \to E

be a Lipschitz continuous with the constant L and pseudomonotone operator. Let

{u_{n}}

{v_{n}}

and

{ρ_{n}}

be sequences generated, as follows:

(i): Choose $u_{- 1}, u_{0}, v_{0} \in E$ , $0 \leq ϑ_{n} \leq ϑ < \sqrt{5} - 2$ and $λ (ϑ, L_{1}, L_{2}) > 0 .$ Compute

$\{\begin{matrix} u_{1} = P_{K} (ρ_{0} - λ H v_{0}), where ρ_{0} = u_{0} + ϑ_{0} (u_{0} - u_{- 1}), \\ v_{1} = P_{K} (ρ_{1} - λ H v_{0}), where ρ_{1} = u_{1} + ϑ_{1} (u_{1} - u_{0}) . \end{matrix}$
(ii): Given $u_{n - 1}, u_{n}$ , $v_{n - 1},$ and $v_{n}$ for each $n \geq 1,$ and construct the half-space first as

$H_{n} = {z \in E : 〈 ρ_{n} - λ H v_{n - 1} - v_{n}, z - v_{n} 〉 \leq 0} .$
(iii): Evaluate

$\{\begin{matrix} u_{n + 1} = P_{H_{n}} (ρ_{n} - λ H v_{n}), where ρ_{n} = u_{n} + ϑ_{n} (u_{n} - u_{n - 1}), \\ v_{n + 1} = P_{K} (ρ_{n + 1} - λ H v_{n}), where ρ_{n + 1} = u_{n + 1} + ϑ_{n + 1} (u_{n + 1} - u_{n}), \end{matrix}$

where

λ > 0

, such that

0 < λ < \frac{\frac{1}{2} - 2 ϑ - \frac{1}{2} ϑ^{2}}{L_{2} {(1 - ϑ)}^{2} + 2 L_{1} (1 + ϑ + ϑ^{2} + ϑ^{3})} and 0 \leq ϑ_{n} \leq ϑ < \sqrt{5} - 2,

with

L_{1} = L_{2} = \frac{L}{2}

. Subsequently, sequence

{u_{n}}

{ρ_{n}}

and

{v_{n}}

converge weakly to

ξ^{*} \in S O L_{V I (H, K)}

From the consideration on Algorithm 2 and Theorem 2, we state the following result for the class of variational inequality problems involving strongly pseudomonotone and Lipschitz continuous operator.

Corollary 2.

Assume that

H : K \to E

is a Lipschitz continuous and strongly pseudomonotone operator with the constant

L

. Let

{u_{n}}

{v_{n}}

and

{ρ_{n}}

are the sequences generated as follows:

(i): Choose $u_{- 1}, u_{0}, v_{0} \in E$ , $0 \leq ϑ_{n} \leq ϑ < \sqrt{5} - 2$ and a sequence ${λ_{n}}$ satisfying (43). Compute

$\{\begin{matrix} u_{1} = P_{K} (ρ_{0} - λ_{0} H v_{0}), where ρ_{0} = u_{0} + ϑ_{0} (u_{0} - u_{- 1}), \\ v_{1} = P_{K} (ρ_{1} - λ_{1} H v_{0}), where ρ_{1} = u_{1} + ϑ_{1} (u_{1} - u_{0}) . \end{matrix}$
(ii): Given $u_{n - 1}, u_{n}$ , $v_{n - 1},$ and $v_{n}$ create a half space for each $n \geq 1,$ such that

$H_{n} = {z \in E : 〈 ρ_{n} - λ_{n} H v_{n - 1} - v_{n}, z - v_{n} 〉 \leq 0} .$
(iii): Compute

$\{\begin{matrix} u_{n + 1} = P_{H_{n}} (ρ_{n} - λ_{n} H v_{n}), where ρ_{n} = u_{n} + ϑ_{n} (u_{n} - u_{n - 1}), \\ v_{n + 1} = P_{K} (ρ_{n + 1} - λ_{n + 1} H v_{n}), where ρ_{n + 1} = u_{n + 1} + ϑ_{n + 1} (u_{n + 1} - u_{n}), \end{matrix}$

where

0 \leq ϑ_{n} \leq ϑ < \sqrt{5} - 2

, with

L_{1} = L_{2} = \frac{L}{2}

. The sequence

{u_{n}}

{ρ_{n}}

and

{v_{n}}

converge strongly to

ξ^{*} \in S O L_{V I (H, K)}

6. Computational Experiment

Some numerical results will be presented in this section to show the performance of our proposed methods. The MATLAB codes run in MATLAB version 9.5 (R2018b) on a PC (with Intel(R) Core(TM)i3-4010U CPU @ 1.70GHz 1.70GHz, RAM 4.00 GB).

6.1. Nash-Cournot Equilibrium Model of Electricity Markets

The Nash–Cournot equilibrium model of electricity markets in [20] is considered in this example. Assume that there are three companies

(i = 1, 2, 3)

generating electricity. These three companies has generating units denoted as

U_{1} = {1}

U_{2} = {2, 3}

and

U_{3} = {4, 5, 6}

, respectively. Let

u_{j}

denote the generating power of the each unit for

i = {1, 2, 3, 4, 5, 6} .

Next, we take the electricity price P as

P = 378.4 - 2 \sum_{j = 1}^{6} u_{j} .

The cost of generating the j unit is written as:

c_{j} (u_{j}) : = max {\overset{\circ}{c_{j}} (u_{j}), \overset{•}{c_{j}} (u_{j})},

where

\overset{\circ}{c_{j}} (u_{j}) : = \frac{\overset{\circ}{α_{j}}}{2} u_{j}^{2} + \overset{\circ}{β_{j}} u_{j} + \overset{\circ}{γ_{j}}

and

\overset{•}{c_{j}} (u_{j}) : = \overset{•}{α_{j}} u_{j} + \frac{\overset{•}{β_{j}}}{\overset{•}{β_{j}} + 1} {\overset{•}{γ_{j}}}^{\frac{- 1}{\overset{•}{β_{j}}}} {(u_{j})}^{\frac{(\overset{•}{β_{j}} + 1)}{\overset{•}{β_{j}}}} .

Table 1 provides the values of the unknown parameters. Consider that the profit of the firm i is

F_{i} (u) : = P \sum_{j \in I_{i}} u_{j} - \sum_{j \in I_{i}} c_{j} (u_{j}) = (378.4 - 2 \sum_{l = 1}^{6} u_{l}) \sum_{j \in I_{i}} u_{j} - \sum_{j \in I_{i}} c_{j} (u_{j}),

with

u = {(u_{1}, \dots, u_{6})}^{T}

corresponding to the constraint set

u \in C : = {u \in R^{6} : u_{j}^{min} \leq u_{j} \leq u_{j}^{max}}

, with

u_{j}^{min}

and

u_{j}^{max}

values given in Table 2. Consider the equilibrium function f by

f (u, v) : = \sum_{i = 1}^{3} (ϕ_{i} (u, u) - ϕ_{i} (u, v)),

where

ϕ_{i} (u, v) : = [378.4 - 2 (\sum_{j \notin I_{i}} u_{j} + \sum_{j \in I_{i}} v_{j})] \sum_{j \in I_{i}} v_{j} - \sum_{j \in I_{i}} c_{j} (v_{j}) .

The Nash–Cournot equilibrium models of electricity markets can be seen as an equilibrium problem in the following way (see [44] for more details):

Find ξ^{*} \in K such that f (ξ^{*}, y) \geq 0, \forall y \in K .

During the numerical example in Section 6.1, we take the values

u_{- 1} = {(10, 10, 20, 17, 8, 14)}^{T}

u_{0} = {(10, 20, 30, 10, 0, 1)}^{T}

v_{0} = {(48, 48, 30, 27, 18, 24)}^{T}

6.1.1. Algorithm 1 Behaviour for Different Values of $ϑ_{n}$ :

Figure 1 and Table 3 characterize the behaviour of error term

D_{n} = ∥ u_{n + 1} - u_{n} ∥ \leq T O L

regarding Algorithm 1 (Algo1) with respect to different values of

ϑ_{n}

in terms of the number of iterations and elapsed time, respectively.

6.1.2. Algorithm 1 Comparison with Existing Algorithms

Figure 2 and Table 4 explain the numerical comparison between Algorithm 1 (EgA) in [19], Algorithm 1 (PEgA) in [21], Algorithm 3.1 (PSgEgA) in [35,36] and Algorithm 1(Algo1).

Algorithm 1 (EgA) in [19]: Choose

u_{0} \in E

and

0 < λ < min {\frac{1}{2 L_{1}}, \frac{1}{2 L_{2}}}

\{\begin{matrix} v_{n} = \underset{}{arg min} {λ f (u_{n}, y) + \frac{1}{2} ∥ u_{n} - y ∥^{2} : y \in K}, \\ u_{n + 1} = \underset{}{arg min} {λ f (v_{n}, y) + \frac{1}{2} ∥ u_{n} - y ∥^{2} : y \in K} . \end{matrix}

(76)

Algorithm 1 (PEgA) in [21]: Choose

u_{0}, v_{0} \in E

and

0 < λ < min \frac{1}{2 L_{2} + 4 L_{1}}

\{\begin{matrix} u_{n + 1} = \underset{}{arg min} {λ f (v_{n}, y) + \frac{1}{2} ∥ u_{n} - y ∥^{2} : y \in K}, \\ v_{n + 1} = \underset{}{arg min} {λ f (v_{n}, y) + \frac{1}{2} ∥ u_{n + 1} - y ∥^{2} : y \in K} . \end{matrix}

(77)

Algorithm 3.1 (PSgEgA) in [35,36]: Choose

u_{0}, v_{0} \in E

and

0 < λ < min \frac{1}{2 L_{2} + 4 L_{1}}

(i): $\{\begin{matrix} u_{1} = \underset{}{arg min} {λ f (v_{0}, y) + \frac{1}{2} ∥ u_{0} - y ∥^{2} : y \in K}, \\ v_{1} = \underset{}{arg min} {λ f (v_{0}, y) + \frac{1}{2} ∥ u_{1} - y ∥^{2} : y \in K} . \end{matrix}$
(ii): Given $u_{n - 1}, u_{n}$ , $v_{n - 1}$ , $v_{n}$ for $n \geq 1$ and construct a half space as

$H_{n} = {z \in E : 〈 u_{n} - λ ω_{n - 1} - v_{n}, z - v_{n} 〉 \leq 0}, where ω_{n - 1} \in \partial f (v_{n - 1}, v_{n}) .$
(iii): $\{\begin{matrix} u_{n + 1} = \underset{}{arg min} {λ f (v_{n}, y) + \frac{1}{2} ∥ u_{n} - y ∥^{2} : y \in H_{n}}, \\ v_{n + 1} = \underset{}{arg min} {λ f (v_{n}, y) + \frac{1}{2} ∥ u_{n + 1} - y ∥^{2} : y \in K} . \end{matrix}$

(78)

6.1.3. Algorithm 2 Behaviour by Using Different Step-Size Sequences $λ_{n}$

Figure 3 and Table 5 describe the numerical results for error term

D_{n} = ∥ u_{n + 1} - u_{n} ∥ \leq T O L

for Algorithm 2 (Algo2).

6.2. Example 2

Assume that

f : R \times R \to R

is defined by

f (u, v) = {tan}^{- 1} (u) (v - u), \forall u, v \in R,

where

K = [0, 1] .

We can easily see that

f (u, v)

satisfy all of the conditions (A1)–(A4) with Lipschitz-type constants are

L_{1} = L_{2} = \frac{1}{2}

(for more details, see [36]).

6.2.1. Algorithm 1 Performance for Different Values of Extrapolation Factor $ϑ_{n}$ :

Figure 4 and Table 6 show the numerical results regarding the error term

D_{n} = ∥ u_{n} ∥

of Algorithm 1 using different values of

ϑ_{n}

in term of the no.of iterations. For these results, we use values

u_{- 1} = \frac{1}{2}

u_{0} = 1

v_{0} = 1

and y-axes depict

D_{n}

value, whereas x-axes are depicted as the number of iterations. The input and output values of the parameters are shown in Table 6, which are useful for choosing the best extrapolation factor value.

6.2.2. Algorithm 1 Comparison with Existing Algorithm

Figure 5 and Table 7 illustrate the comparison of our proposed Algorithm 1 (Algo1) with the existing Algorithm 3.1 (PSgEgA) that appears in the paper of Liu [36]. For these results, the stopping criterion is (

D_{n} = ∥ u_{n} ∥

) and y-axes depict

D_{n}

value, whereas the x-axes are depicted as the number of iterations. The input and output values for the parameters are written in Table 7.

6.3. Nash–Cournot Oligopolistic Equilibrium Model

Consider a Nash–Cournot oligopolistic equilibrium model [19] based on n companies that manufacture the same commodity. Each company produces

u_{i}

amount of commodity and u denotes a vector whose entries

u_{i} .

The price function for each company i is defined by

P_{i} (S) = ϕ_{i} - ψ_{i} S

, where

S = \sum_{i = 1}^{m} u_{i}

and

ϕ_{i} > 0

ψ_{i} > 0 .

Now, consider a profit function for each company i are

F_{i} (u) = P_{i} (S) u_{i} - t_{i} (u_{i})

, where

t_{i} (u_{i})

is the value tax and fee for producing

u_{i} .

Let

K_{i} = [u_{i}^{min}, u_{i}^{max}]

is the set of action of each company i and accumulated actions for whole model taken the form as

K : = K_{1} \times K_{2} \times \dots \times K_{n} .

In addition, each company wants to get peak revenue on the assertion that the output of the other companies is an input parameter. The strategy being used to deal with this sort of model mainly focuses on the well-known Nash equilibrium idea. A point

u^{*} \in K = K_{1} \times K_{2} \times \dots \times K_{n}

is equilibrium point of the model if

F_{i} (u^{*}) \geq F_{i} (u^{*} [u_{i}]), \forall u_{i} \in K_{i}, \forall i = 1, 2, \dots, n,

with vector

u^{*} [u_{i}]

denote a vector achievement from

u^{*}

by considering

u_{i}^{*}

with

u_{i} .

Let

f (u, v) : = φ (u, v) - φ (u, u)

with

φ (u, v) : = - \sum_{i = 1}^{n} F_{i} (u [v_{i}])

and the problem of determine the Nash equilibrium point is

Find u^{*} \in K : f (u^{*}, v) \geq 0, \forall v \in K .

Next, the bifunction f is written as

f (u, v) = 〈 P u + Q v + q, v - u 〉,

where

q \in R^{m}

and the matrices P, Q are

Q = (\begin{matrix} 1.6 & 1 & 0 & 0 & 0 \\ 1 & 1.6 & 0 & 0 & 0 \\ 0 & 0 & 1.5 & 0 & 0 \\ 0 & 0 & 1 & 1.5 & 0 \\ 0 & 0 & 0 & 0 & 2 \end{matrix}), P = (\begin{matrix} 3.1 & 2 & 0 & 0 & 0 \\ 3 & 3.6 & 0 & 0 & 0 \\ 0 & 0 & 3.5 & 2 & 0 \\ 0 & 0 & 2 & 3.3 & 0 \\ 0 & 0 & 0 & 0 & 3 \end{matrix})

with

q = {(1, 2, - 1, 2, - 1)}^{T} and K = {u \in R^{5} : - 2 \leq u_{i} \leq 5}

. During this example, we use the values of the parameters

u_{- 1} = {(1, 2, 1, 2, 0)}^{T}

u_{0} = {(1, 3, 1, 1, 2)}^{T}

and

v_{0} = {(1, 2, 1, 1, 2)}^{T} .

6.3.1. Algorithm 2. Behaviour for Different Step-Size Sequences $λ_{n}$ :

The class of step-size sequences

{λ_{n}}

used in the experiments are:

(I): $λ_{n} = \frac{1}{{(n + 2)}^{q}}, q \in {1.0; 0.8; 0.5; 0.3; 0.1};$
(II): $λ_{n} = \frac{1}{{(log (n + 3))}^{q}}, q \in {7; 5; 3; 2; 0.5} .$

Figure 6 and Figure 7 describe the numerical results for Algorithm 2 (Algo2) by using the above define classes of step-size sequences.

6.3.2. Algorithm 2. Comparison with Existing Algorithms

Figure 8 describes the numerical results of Algorithm 2 (Algo2) using the stepsize sequences

λ_{n} = \frac{1}{n + 1}

Discussion About Numerical Experiments: We have the following observations regarding the above-mentioned experiments:

(1)

Figure 1 and Figure 4 and Table 3 and Table 6 reported results for Algorithm 1 while using different values for

ϑ_{n} .

From these results, we can see that the value of

θ_{n}

nearer the upper bound value

\sqrt{5} - 2

is more appropriate and enhances the effectiveness of the suggested algorithms.

(2)

It can also be acknowledged that the efficiency of the algorithm depends on the complexity of the problem and tolerance of the error term. More time and a significant number of iterations are required in the case of large-scale problems. In this situation, we can see that the certain value of the step-size enhances the performance of the algorithm and boosts the convergence rate.

(3)

From Figure 5 and Table 7, it can also be noted that the choice of the initial points and the complexity of the bifunction affect the performance of algorithms in terms of the number of iterations and time of execution in seconds.

(4)

We have the following observation from Figure 3 and Figure 6, Figure 7 and Figure 8 with Table 5.

(i): No previous information of Lipschitz-constant $L_{1}, L_{2}$ is required for running algorithms on Matlab.
(ii): In fact, the convergence rate of algorithms depends entirely on the convergence rate of step-size sequences $λ_{n} .$
(iii): The convergence rate of the iterative sequence often depends on the complexity of the problem as well as on the size of the problem.
(iv): Due to the variable step-size sequence, a specific step-size value that is not appropriate for the current iteration of the method often causes inconsistency and a hump in the behavior of the iterative sequence.

7. Conclusions

Two different approaches are proposed in this paper to deal with two families of equilibrium problems. The first algorithm is an inertial two-step step proximal-like method that generates a weak converging iterative sequence and it can solve pseudomonoton equilibrium problems. In addition, we use the diminishing and non-summable step-size sequence for the second algorithm to achieve the strong convergence. The key advantage of the second algorithm is that iterative sequences have been developed with no prior knowledge of a strong pseudomonotonicity and Lipschitz-type constants of a bifunction. Numerical findings were mentioned to show the numerical efficiency of algorithms as compared to other algorithms. Such numerical studies imply that the inertial effects normally enhance the effectiveness of the iterative sequence in this context.

Author Contributions

Conceptualization, H.u.R. and P.K.; methodology, M.S., N.A.A. and W.K.; writing—original draft preparation, H.u.R., P.K. and W.K.; writing—review and editing, H.u.R., P.K., M.S. and N.A.A.; software, H.u.R., M.S. and N.A.A.; supervision, P.K., M.S. and W.K.; project administration and funding acquisition, P.K. and W.K. All authors have read and agreed to the published version of the manuscript.

Funding

This research work was financially supported by King Mongkut’s University of Technology Thonburi through the ‘KMUTT 55th Anniversary Commemorative Fund’. Moreover, this project was supported by Theoretical and Computational Science (TaCS) Center under Computational and Applied Science for Smart research Innovation research Cluster (CLASSIC), Faculty of Science, KMUTT. In particular, Habib ur Rehman was financed by the Petchra Pra Jom Doctoral Scholarship Academic for Ph.D. Program at KMUTT [grant number 39/2560]. Furthermore, Wiyada Kumam was financially supported by the Rajamangala University of Technology Thanyaburi (RMUTTT) (Grant No. NSF62D0604).

Acknowledgments

The first author would like to thank the “Petchra Pra Jom Klao Ph.D. Research Scholarship from King Mongkut’s University of Technology Thonburi”. We are very grateful to editor and the anonymous referees for their valuable and useful comments, which helps in improving the quality of this work.

Conflicts of Interest

The authors declare that they have conflict of interest.

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Figure 1. Experiment in Section 6.1.1: Algorithm 1 behaviour for different values of

ϑ_{n}

Figure 1. Experiment in Section 6.1.1: Algorithm 1 behaviour for different values of

ϑ_{n}

Figure 2. Comparison of Algorithm 1 with Algorithm 1 in [19], Algorithm 1 in [21], and Algorithm 3.1 in [35,36].

Figure 3. Algorithm 2 behaviour with respect to different step-size sequences

λ_{n}

Figure 3. Algorithm 2 behaviour with respect to different step-size sequences

λ_{n}

Figure 4. Experiment in Section 6.2.1: Algorithm 1 behaviour regarding different values of

ϑ_{n}

Figure 4. Experiment in Section 6.2.1: Algorithm 1 behaviour regarding different values of

ϑ_{n}

Figure 5. Experiment in Section 6.2.2: Comparison of Algorithm 1 with Algorithm 3.1 in [35,36].

Figure 6. Experiment in Section 6.3.1: Algorithm 2 behaviour with respect to step-size sequences

λ_{n} = \frac{1}{{(n + 2)}^{q}}

Figure 6. Experiment in Section 6.3.1: Algorithm 2 behaviour with respect to step-size sequences

λ_{n} = \frac{1}{{(n + 2)}^{q}}

Figure 7. Experiment in Section 6.3.1: Algorithm 2 behaviour with respect to step-size sequences

λ_{n} = \frac{1}{{(log (n + 3))}^{q}}

Figure 7. Experiment in Section 6.3.1: Algorithm 2 behaviour with respect to step-size sequences

λ_{n} = \frac{1}{{(log (n + 3))}^{q}}

Figure 8. Experiment in Section 6.3.2: Comparison of Algorithm 2 with Algorithm 1 (EgM) in [23] and Algorithm 3.1 (PEgM) in [45].

Table 1. The values of parameters are used in the cost function.

j	$\overset{\circ}{α_{j}}$	$\overset{\circ}{β_{j}}$	$\overset{•}{α_{j}}$	$\overset{•}{β_{j}}$	$\overset{•}{γ_{j}}$
1	0.0400	2.00	2.0000	1.0000	25.0000
2	0.0350	1.75	1.7500	1.0000	28.5714
3	0.1250	1.00	1.0000	1.0000	8.0000
4	0.0116	3.25	3.2500	1.0000	86.2069
5	0.0500	3.00	3.0000	1.0000	20.0000
6	0.0500	3.00	3.0000	1.0000	20.0000

Table 2. The parameter values use for constraint set.

j	1	2	3	4	5	6
$u_{j}^{m i n}$	0	0	0	0	0	0
$u_{j}^{m a x}$	80	80	50	55	30	40

Table 3. Experiment in Section 6.1.1: Algorithm 1 performance for varying parameters extrapolation factor

ϑ_{n}

Table 3. Experiment in Section 6.1.1: Algorithm 1 performance for varying parameters extrapolation factor

ϑ_{n}

Algo.name	$ϑ_{n}$	$λ$	$ξ^{*}$	Iter.	Time	TOL
`Algo1`	0.22	0.02	${(46.6525, 32.1462, 15.0018, 25.0170, 10.8987, 10.8982)}^{T}$	4824	138.915365	$10^{- 4}$
`Algo1`	0.18	0.02	${(46.6525, 32.1460, 15.0020, 25.0104, 10.9019, 10.9016)}^{T}$	4949	166.620335	$10^{- 4}$
`Algo1`	0.14	0.02	${(46.6525, 32.1460, 15.0020, 25.0035, 10.9050, 10.9053)}^{T}$	5193	127.834772	$10^{- 4}$
`Algo1`	0.10	0.02	${(46.6726, 32.1460, 15.0020, 24.9969, 10.9080, 10.9089)}^{T}$	5432	136.310422	$10^{- 4}$
`Algo1`	0.05	0.02	${(46.6526, 32.1460, 15.0020, 24.9885, 10.9118, 10.9134)}^{T}$	5721	142.108161	$10^{- 4}$
`Algo1`	0.01	0.02	${(46.6526, 32.1460, 15.0020, 24.9818, 10.9149, 10.9170)}^{T}$	5945	144.356535	$10^{- 4}$
`Algo1`	0.001	0.02	${(46.6726, 32.1460, 15.0021, 24.9787, 10.9163, 10.9187)}^{T}$	6043	157.711757	$10^{- 4}$

Table 4. Experiment in Section 6.1.2: Algorithm 1 comparison with existing algorithms using two different values of

ϑ_{n}

Table 4. Experiment in Section 6.1.2: Algorithm 1 comparison with existing algorithms using two different values of

ϑ_{n}

Algo.name	$ϑ_{n}$	$λ$	$ξ^{*}$	Iter.	Time	TOL
`EgA`	–	0.02	${(46.6526, 32.1469, 15.0012, 24.9783, 10.9154, 10.9200)}^{T}$	7180	264.156236	$10^{- 4}$
`PEgA`	–	0.02	${(46.6526, 32.1460, 15.0021, 24.9784, 10.8164, 10.9188)}^{T}$	6055	210.681669	$10^{- 4}$
`PSgEgA`	–	0.02	${(46.6525, 32.1463, 15.0017, 25.0004, 10.9058, 10.9076)}^{T}$	5515	175.840493	$10^{- 4}$
`Algo1`	0.12	0.02	${(46.6725, 32.1463, 15.0017, 25.0181, 10.8976, 10.8982)}^{T}$	4894	134.245610	$10^{- 4}$
`Algo1`	0.20	0.02	${(46.6725, 32.1463, 15.0017, 25.0326, 10.8910, 10.8904)}^{T}$	4333	115.599023	$10^{- 4}$

Table 5. Experiment in Section 6.1.3: Algorithm 2 numerical values by using different step-size sequences

λ_{n}

Table 5. Experiment in Section 6.1.3: Algorithm 2 numerical values by using different step-size sequences

λ_{n}

Algo.name	$ϑ_{n}$	$λ$	$ξ^{*}$	Iter.	Time	TOL
`Algo2`	0.12	$\frac{1}{n + 1}$	${(46.6526, 32.1467, 15.0011, 25.1260, 10.8442, 10.8442)}^{T}$	1254	61.898186	$10^{- 4}$
`Algo2`	0.12	$\frac{1}{log (n + 1)}$	${(46.6523, 32.1467, 15.0011, 25.1409, 10.8368, 10.8368)}^{T}$	442	29.006584	$10^{- 4}$
`Algo2`	0.12	$\frac{1}{(n + 1) (log (n + 3))}$	${(46.6524, 32.1467, 15.0011, 25.1011, 10.8566, 10.8566)}^{T}$	2311	70.849546	$10^{- 4}$
`Algo2`	0.12	$\frac{log (n + 3)}{n + 1}$	${(46.6523, 32.1467, 15.0011, 25.1371, 10.8387, 10.8387)}^{T}$	662	44.766232	$10^{- 4}$
`Algo2`	0.12	$\frac{1}{log (log (n + 20))}$	${(46.6525, 32.1467, 15.0011, 25.1464, 10.8341, 10.8341)}^{T}$	434	31.504484	$10^{- 4}$

Table 6. Experiment in Section 6.2.1: Algorithm 1 performance for varying parameters extrapolation factor

ϑ_{n}

Table 6. Experiment in Section 6.2.1: Algorithm 1 performance for varying parameters extrapolation factor

ϑ_{n}

$ϑ_{n}$	$λ$	$ξ^{*}$	Iter.	Time	TOL
0.20	0.050	6.5877 $\times 10^{- 9}$	70	0.008866	$10^{- 8}$
0.15	0.050	6.6948 $\times 10^{- 9}$	75	0.010382	$10^{- 8}$
0.10	0.050	6.4466 $\times 10^{- 9}$	80	0.008518	$10^{- 8}$
0.05	0.050	7.5191 $\times 10^{- 9}$	84	0.008378	$10^{- 8}$
0.01	0.050	6.9392 $\times 10^{- 9}$	88	0.008989	$10^{- 8}$

Table 7. Experiment in Section 6.2.2: Algorithm 1 comparison with Algorithm 3.1 in [35,36].

Algorithm	$u_{- 1}$	$u_{0}$	$v_{0}$	$ϑ_{n}$	$λ$	$ξ^{*}$	Iter.	Time	TOL
`PSgEgA`	—	1	1	—	0.1	7.9278 $\times 10^{- 11}$	110	0.001014	$10^{- 10}$
`Algo1`	0.5	1	1	0.16	0.1	6.5112 $\times 10^{- 11}$	92	0.006082	$10^{- 10}$
`PSgEgA`	—	0.5	0.5	—	0.1	6.9204 $\times 10^{- 11}$	107	0.006580	$10^{- 10}$
`Algo1`	1	0.5	0.5	0.16	0.1	7.1870 $\times 10^{- 11}$	87	0.006919	$10^{- 10}$
`PSgEgA`	—	0.2	0.2	—	0.1	7.7873 $\times 10^{- 11}$	102	0.007282	$10^{- 10}$
`Algo1`	1	0.2	0.2	0.16	0.1	7.1827 $\times 10^{- 11}$	66	0.000688	$10^{- 10}$

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Rehman, H.u.; Kumam, P.; Shutaywi, M.; Alreshidi , N.A.; Kumam, W. Inertial Optimization Based Two-Step Methods for Solving Equilibrium Problems with Applications in Variational Inequality Problems and Growth Control Equilibrium Models. Energies 2020, 13, 3292. https://doi.org/10.3390/en13123292

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Rehman Hu, Kumam P, Shutaywi M, Alreshidi NA, Kumam W. Inertial Optimization Based Two-Step Methods for Solving Equilibrium Problems with Applications in Variational Inequality Problems and Growth Control Equilibrium Models. Energies. 2020; 13(12):3292. https://doi.org/10.3390/en13123292

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Rehman, Habib ur, Poom Kumam, Meshal Shutaywi, Nasser Aedh Alreshidi , and Wiyada Kumam. 2020. "Inertial Optimization Based Two-Step Methods for Solving Equilibrium Problems with Applications in Variational Inequality Problems and Growth Control Equilibrium Models" Energies 13, no. 12: 3292. https://doi.org/10.3390/en13123292

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Rehman, H. u., Kumam, P., Shutaywi, M., Alreshidi , N. A., & Kumam, W. (2020). Inertial Optimization Based Two-Step Methods for Solving Equilibrium Problems with Applications in Variational Inequality Problems and Growth Control Equilibrium Models. Energies, 13(12), 3292. https://doi.org/10.3390/en13123292

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Inertial Optimization Based Two-Step Methods for Solving Equilibrium Problems with Applications in Variational Inequality Problems and Growth Control Equilibrium Models

Abstract

1. Introduction

2. Background

3. Inertial Popov’s Two-Step Subgradient Extragradient Algorithm for Pseudomonotone EP

4. Inertial Popov’s Two-Step Subgradient Extragradient Algorithm for Strongly Pseudomonotone EP

5. Application to Variational Inequality Problems

6. Computational Experiment

6.1. Nash-Cournot Equilibrium Model of Electricity Markets

6.1.1. Algorithm 1 Behaviour for Different Values of $ϑ_{n}$ :

6.1.2. Algorithm 1 Comparison with Existing Algorithms

6.1.3. Algorithm 2 Behaviour by Using Different Step-Size Sequences $λ_{n}$

6.2. Example 2

6.2.1. Algorithm 1 Performance for Different Values of Extrapolation Factor $ϑ_{n}$ :

6.2.2. Algorithm 1 Comparison with Existing Algorithm

6.3. Nash–Cournot Oligopolistic Equilibrium Model

6.3.1. Algorithm 2. Behaviour for Different Step-Size Sequences $λ_{n}$ :

6.3.2. Algorithm 2. Comparison with Existing Algorithms

7. Conclusions

Author Contributions

Funding

Acknowledgments

Conflicts of Interest

References

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Inertial Optimization Based Two-Step Methods for Solving Equilibrium Problems with Applications in Variational Inequality Problems and Growth Control Equilibrium Models

Abstract

1. Introduction

2. Background

3. Inertial Popov’s Two-Step Subgradient Extragradient Algorithm for Pseudomonotone EP

4. Inertial Popov’s Two-Step Subgradient Extragradient Algorithm for Strongly Pseudomonotone EP

5. Application to Variational Inequality Problems

6. Computational Experiment

6.1. Nash-Cournot Equilibrium Model of Electricity Markets

6.1.1. Algorithm 1 Behaviour for Different Values of ϑ n :

6.1.2. Algorithm 1 Comparison with Existing Algorithms

6.1.3. Algorithm 2 Behaviour by Using Different Step-Size Sequences λ n

6.2. Example 2

6.2.1. Algorithm 1 Performance for Different Values of Extrapolation Factor ϑ n :

6.2.2. Algorithm 1 Comparison with Existing Algorithm

6.3. Nash–Cournot Oligopolistic Equilibrium Model

6.3.1. Algorithm 2. Behaviour for Different Step-Size Sequences λ n :

6.3.2. Algorithm 2. Comparison with Existing Algorithms

7. Conclusions

Author Contributions

Funding

Acknowledgments

Conflicts of Interest

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

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6.1.1. Algorithm 1 Behaviour for Different Values of $ϑ_{n}$ :

6.1.3. Algorithm 2 Behaviour by Using Different Step-Size Sequences $λ_{n}$

6.2.1. Algorithm 1 Performance for Different Values of Extrapolation Factor $ϑ_{n}$ :

6.3.1. Algorithm 2. Behaviour for Different Step-Size Sequences $λ_{n}$ :