Open AccessArticle

On Strengthened Extragradient Methods Non-Convex Combination with Adaptive Step Sizes Rule for Equilibrium Problems

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

Applied Mathematics for Science and Engineering Research Unit (AMSERU), Program in Applied Statistics, Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT), Pathum Thani 12110, Thailand

Department of Mathematics, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand

⁴

Applied Mathematics for Science and Engineering Research Unit (AMSERU), Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT), Pathum Thani 12110, Thailand

Author to whom correspondence should be addressed.

Symmetry 2022, 14(5), 1045; https://doi.org/10.3390/sym14051045

Submission received: 12 April 2022 / Revised: 30 April 2022 / Accepted: 16 May 2022 / Published: 19 May 2022

(This article belongs to the Section Mathematics)

Download

Browse Figures

Versions Notes

Abstract

Symmetries play a vital role in the study of physical phenomena in diverse areas such as dynamic systems, optimization, physics, scientific computing, engineering, mathematical biology, chemistry, and medicine, to mention a few. These phenomena specialize mostly in solving equilibria-like problems in abstract spaces. Motivated by these facts, this research provides two innovative modifying extragradient strategies for solving pseudomonotone equilibria problems in real Hilbert space with the Lipschitz-like bifunction constraint. Such strategies make use of multiple step-size concepts that are modified after each iteration and are reliant on prior iterations. The excellence of these strategies comes from the fact that they were developed with no prior knowledge of Lipschitz-type parameters or any line search strategy. Mild assumptions are required to prove strong convergence theorems for proposed strategies. Various numerical tests have been reported to demonstrate the numerical behavior of the techniques and then contrast them with others.

Keywords:

Lipschitz-like conditions; equilibrium problem; strong convergence theorems; variational inequality problems; fixed-point problem

MSC:

47J25; 47H09; 47H06; 47J05

1. Introduction

Consider that

Σ

is a nonempty, convex, and closed subset of a real Hilbert space

Π .

The inner product and norm are indicated with

〈 ., . 〉

and

∥ . ∥,

respectively. Furthermore,

R

and

N

symbolize the set of real numbers and the set of natural numbers, respectively. Assume that

R : Π \times Π \to R

is indeed a bifunction with the equilibrium problem solution set

E P (R, Σ) .

Let

s^{*} = P_{E P (R, Σ)},

whereas

θ

represents a zero element in

Π .

In this case,

Σ

characterizes the subset of a Hilbert space

Π

and

R

as follows:

R : Π \times Π \to R

is a bifunction through

R (r_{1}, r_{1}) = 0,

for all

r_{1} \in Σ .

The equilibrium problem [1,2] for

R

Σ

is to:

Find s^{*} \in Σ such that R (s^{*}, r_{1}) \geq 0, \forall r_{1} \in Σ .

(1)

The above-mentioned framework is an appropriate mathematical framework that incorporates a variety of problems, including vector and scalar minimization problems, saddle point problems, variational inequality problems, complementarity problems, Nash equilibrium problems in non-cooperative games, and inverse optimization problems [1,3,4]. This issue is primarily connected to Ky Fan inequity on the grounds of his prior contributions to the field [2]. It is also important to consider an approximate solution if the problem does not have an exact solution or is difficult to calculate. Several methodologies have been proposed and tested to tackle various types of equilibrium problems (1). Many successful algorithmic techniques, as well as theoretical characteristics, have already been proposed to solve the (1) issue in both finite- and infinite-dimensional spaces.

The regularization technique is the most significant method for dealing with many ill-posed problems in various subfields of applied and pure mathematics. The regularization approach is distinguished by the use of monotone equilibrium problems to convert the original problem into a strongly monotone equilibrium subproblem. As a result, each computationally productive subproblem is strongly monotone and has a unique solution. The discovered subproblem, for example, may be more successfully resolved than the initial problem, and the regularization solutions may lead to some solution to the basic problem once the regularization variables look to have an adequate limit. The two most prevalent regularization methods are the proximal point and Tikhonov’s regularized approaches. These approaches were recently extended to equilibrium problems [5,6,7,8,9,10,11,12,13]. A few techniques to address non-monotone equilibrium problems can be found in [14,15,16,17,18,19,20,21,22,23,24,25,26].

The proximal method [27] is indeed an innovative approach for determining equilibrium problems that are founded on minimization problems. Along with Korpelevich’s contribution [28] technique to addressing the saddle point problem, this procedure has also been known as the two-step extragradient method in [29]. Tran et al. [29] constructed an iterative sequence of

{s_{k}}

in the following manner:

\{\begin{matrix} s_{1} \in Σ, \\ m_{k} = \underset{v \in Σ}{arg min} {λ R (s_{k}, v) + \frac{1}{2} ∥ s_{k} - v ∥^{2}}, \\ s_{k + 1} = \underset{v \in Σ}{arg min} {λ R (m_{k}, v) + \frac{1}{2} ∥ s_{k} - v ∥^{2}}, \end{matrix}

where

0 < λ < min \{\frac{1}{2 c_{1}}, \frac{1}{2 c_{2}}\} .

The iterative sequence created by the aforementioned approach exhibits weak convergence, and prior knowledge of Lipschitz-type variables is necessary in order to use it. Lipschitz-type parameters are frequently unknown or difficult to calculate. To address this issue, Hieu et al. [30] introduced the following adaptation of the approach in [31] for equilibrium: Let

{[t]}_{+} = max {t, 0}

and select

s_{1} \in Σ,

μ \in (0, 1)

with

λ_{0} > 0,

such that

\{\begin{matrix} m_{k} = \underset{v \in Σ}{arg min} {λ_{k} R (s_{k}, v) + \frac{1}{2} ∥ s_{k} - v ∥^{2}}, \\ s_{k + 1} = \underset{v \in Σ}{arg min} {λ_{k} R (m_{k}, v) + \frac{1}{2} ∥ s_{k} - v ∥^{2}}, \end{matrix}

along with

λ_{k + 1} = min \{λ_{k}, \frac{μ (∥ s_{k} - m_{k} ∥^{2} + ∥ s_{k + 1} - m_{k} ∥^{2})}{2 {[R (s_{k}, s_{k + 1}) - R (s_{k}, m_{k}) - R (m_{k}, s_{k + 1})]}_{+}}\} .

To solve a pseudomonotone equilibrium problem, the authors have suggested a non-convex combination iterative technique in [32]. The availability of a strong convergence iterative sequence without the need for hybrid projection or viscosity techniques is the main contribution. The details of the algorithm are as follows: Choose

0 < λ_{k} < min \{\frac{1}{2 c_{1}}, \frac{1}{2 c_{2}}\},

δ_{k} \subset [δ, 1)

with

δ > 0

and

ϕ_{k}

such that

lim_{k \to + \infty} ϕ_{k} = 0 and \sum_{k = 1}^{+ \infty} ϕ_{k} = + \infty .

\{\begin{matrix} m_{k} = \underset{v \in Σ}{arg min} {λ_{k} R (s_{k}, v) + \frac{1}{2} ∥ s_{k} - v ∥^{2}}, \\ r_{k} = \underset{v \in Σ}{arg min} {λ_{k} R (m_{k}, v) + \frac{1}{2} ∥ s_{k} - v ∥^{2}}, \end{matrix}

and

s_{k + 1} = P_{Σ} [ϕ_{k} s_{k} + (1 - ϕ_{k}) r_{k} - ϕ_{k} δ_{k} s_{k}] .

The main objective of this study is to focus on using well-known projection algorithms that are, in general, easier to apply due to their efficient and easy mathematical computation. We design and adapt an explicit subgradient extragradient method to solve the problem of pseudomonotone equilibrium and other specific classes of variational inequality problems and fixed-point problems, inspired by the works of [30,33]. Our techniques are a variation on the approaches described in [32]. Strong convergence results matching the sequence of the two methods are achieved under specific, moderate circumstances. Some applications of variational inequality and fixed-point problems are given. Consequently, experimental investigations have shown that the proposed strategy is more successful than the current one [32].

The rest of the article is organized as follows: Section 2 includes basic definitions and lemmas. Section 3 proposes new methods and their convergence analysis theorems. Section 4 contains several applications of our findings to variational inequality and fixed-point problems. Section 5 contains numerical tests to demonstrate the computational effectiveness of our proposed methods.

2. Preliminaries

Suppose that a convex function

ℑ : Σ \to R

and subdifferential of ℑ at

r_{1} \in Σ

is expressed as follows:

\partial ℑ (r_{1}) = {r_{3} \in Π : ℑ (r_{2}) - ℑ (r_{1}) \geq 〈 r_{3}, r_{2} - r_{1} 〉, \forall r_{2} \in Σ} .

A normal cone of Σ at

r_{1} \in Σ

is expressed as follows:

N_{Σ} (r_{1}) = {r_{3} \in Π : 〈 r_{3}, r_{2} - r_{1} 〉 \leq 0, \forall r_{2} \in Σ} .

Lemma 1.

([34]) Suppose that a convex function

ℑ : Σ \to R

is subdifferentiable and lower semicontinuous upon

Σ .

Then

r_{1} \in Σ

is a minimizer of a function ℑ if and only if

0 \in \partial ℑ (r_{1}) + N_{Σ} (r_{1}),

where

\partial ℑ (r_{1})

and

N_{Σ} (r_{1})

denotes the subdifferential of ℑ at

r_{1} \in Σ

and the normal cone of Σ at

r_{1},

respectively.

Definition 1.

([35]) A metric projection

P_{Σ} (r_{1})

for

r_{1} \in Π

onto a convex and closed subset Σ of Π is stated as follows:

P_{Σ} (r_{1}) = \underset{}{arg min} {∥ r_{2} - r_{1} ∥ : r_{2} \in Σ} .

Lemma 2.

([36]) Consider that a metric projection

P_{Σ} : Π \to Σ .

Then

(i): For some $r_{2} \in Σ$ and $r_{1} \in Π$ in order that

$∥ r_{1} - P_{Σ} (r_{1}) ∥ \leq ∥ r_{1} - r_{2} ∥^{2} .$
(ii): $r_{3} = P_{Σ} (r_{1})$ if and only if

$〈 r_{1} - r_{3}, r_{2} - r_{3} 〉 \leq 0, \forall r_{2} \in Σ .$

Lemma 3.

([37]) For some

r_{1}, r_{2} \in Π

and

χ \in R .

Then

(i)

∥ χ r_{1} + (1 - χ) r_{2} ∥^{2} = χ ∥ r_{1} ∥^{2} + (1 - χ) ∥ r_{2} ∥^{2} - χ (1 - χ) {∥ r_{1} - r_{2} ∥}^{2};

(ii)

∥ r_{1} + r_{2} ∥^{2} \leq {∥ r_{1} ∥}^{2} + 2 〈 r_{2}, r_{1} + r_{2} 〉 .

Lemma 4.

([38]) Consider a sequence of non-negative real numbers

{χ_{k}}

such that

χ_{k + 1} \leq (1 - τ_{k}) χ_{k} + τ_{k} δ_{k}, \forall k \in N,

while

{τ_{k}} \subset (0, 1)

and

{δ_{k}} \subset R

conforming to the following parameters:

lim_{k \to + \infty} τ_{k} = 0, \sum_{k = 1}^{+ \infty} τ_{k} = + \infty, a n d \underset{k \to + \infty}{lim sup} δ_{k} \leq 0 .

Thus,

{lim}_{k \to + \infty} χ_{k} = 0 .

Lemma 5.

([39]) Assume that

{χ_{k}}

is a sequence of real numbers namely that there exists a subsequence

{k_{i}}

{k}

such that

χ_{k_{i}} < χ_{k_{i + 1}}, f o r a l l i \in N .

Then, there would be a nondecreasing sequence

{e_{k}} \subset N

, namely that

e_{k} \to + \infty

k \to + \infty,

and the following criteria are fulfilled by all (sufficiently big) integers

k \in N :

χ_{e_{k}} \leq χ_{m_{k + 1}} and χ_{k} \leq χ_{m_{k + 1}} .

In fact,

e_{k} = max {j \leq k : χ_{j} \leq χ_{j + 1}} .

Now, we consider the following bifunction monotonicity notions (for more information, see [1,40]). A bifunction

R : Π \times Π \to R

Σ

for

ξ > 0

such that

(1) strongly monotone if

R (r_{1}, r_{2}) + R (r_{2}, r_{1}) \leq - ξ {∥ r_{1} - r_{2} ∥}^{2}, \forall r_{1}, r_{2} \in Σ;

(2) monotone if

R (r_{1}, r_{2}) + R (r_{2}, r_{1}) \leq 0, \forall r_{1}, r_{2} \in Σ;

(3) strongly pseudomonotone if

R (r_{1}, r_{2}) \geq 0 ⟹ R (r_{2}, r_{1}) \leq - ξ {∥ r_{1} - r_{2} ∥}^{2}, \forall r_{1}, r_{2} \in Σ;

(4) pseudomonotone if

R (r_{1}, r_{2}) \geq 0 ⟹ R (r_{2}, r_{1}) \leq 0, \forall r_{1}, r_{2} \in Σ .

Suppose that

R : Π \times Π \to R

meets the Lipschitz-type condition [41] over

Σ

c_{1}, c_{2} > 0,

such that

R (r_{1}, r_{3}) \leq R (r_{1}, r_{2}) + R (r_{2}, r_{3}) + c_{1} ∥ r_{1} - r_{2} ∥^{2} + c_{2} {∥ r_{2} - r_{3} ∥}^{2}, \forall r_{1}, r_{2}, r_{3} \in Σ .

We shall presume that the requirements listed below have been satisfied. A bifunction

R

meets the following criteria:

( $R$ 1): $R (r_{2}, r_{2}) = 0$ for all $r_{2} \in Σ$ and $R$ is pseudomonotone on feasible set $Σ;$
( $R$ 2): $R$ meet the Lipschitz-type condition on $Π$ with constants $c_{1}$ and $c_{2};$
( $R$ 3): $R (r_{1}, r_{2})$ is jointly weakly continuous on $Π \times Π$ ;
( $R$ 4): $R (r_{1}, .)$ need to be convex and subdifferentiable over $Π$ for each $r_{1} \in Π .$

3. Main Results

We add a method and have strong convergence results for that method. The following is a detailed algorithm:

The following lemma can be used to demonstrate that the step-size sequence

λ_{k}

generated by the previous formula decreases monotonically and is bounded, as required for iterative sequence convergence.

Lemma 6.

A sequence

{λ_{k}}

is decreasing monotonically with lower bound

min \{\frac{μ}{2 max {c_{1}, c_{2}}}, λ_{0}\}

and converge to

λ > 0 .

Proof.

It is straightforward that

{λ_{k}}

decreases monotonically. Let

R (s_{k}, r_{k}) - R (s_{k}, m_{k}) - R (m_{k}, r_{k}) > 0

, such that

\begin{matrix} \frac{μ (∥ s_{k} - m_{k} ∥^{2} + ∥ r_{k} - m_{k} ∥^{2})}{2 [R (s_{k}, r_{k}) - R (s_{k}, m_{k}) - R (m_{k}, r_{k})]} \\ \geq \frac{μ (∥ s_{k} - m_{k} ∥^{2} + ∥ r_{k} - m_{k} ∥^{2})}{2 [c_{1} ∥ s_{k} - m_{k} ∥^{2} + c_{2} ∥ r_{k} - m_{k} ∥^{2}]} \geq \frac{μ}{2 max {c_{1}, c_{2}}} . \end{matrix}

Thus, sequence

{λ_{k}}

has the lower bound

min \{\frac{μ}{2 max {c_{1}, c_{2}}}, λ_{0}\} .

Thus, there exists a real number

λ > 0,

to ensure that

{lim}_{k \to + \infty} λ_{k} = λ .

□

The following lemma can be used to verify the boundedness of an iterative sequence.

Lemma 7.

Let

R : Π \times Π \to R

be a bifunction that satisfies the conditions(

R

1)–(

R

4). For any

s^{*} \in E P (R, Σ) \neq \emptyset,

we have

∥ r_{k} - s^{*} ∥^{2} \leq ∥ s_{k} - s^{*} ∥^{2} - (1 - \frac{μ λ_{k}}{λ_{k + 1}}) ∥ s_{k} - m_{k} ∥^{2} - (1 - \frac{μ λ_{k}}{λ_{k + 1}}) {∥ r_{k} - m_{k} ∥}^{2} .

Proof.

By the value

r_{k}

and Lemma 1, we obtain

λ_{k} R (m_{k}, y) - λ_{k} R (m_{k}, r_{k}) \geq 〈 s_{k} - r_{k}, y - r_{k} 〉, \forall y \in Π_{k} .

(2)

From definition of

Π_{k},

we have

λ_{k} R (s_{k}, r_{k}) - λ_{k} R (s_{k}, m_{k}) \geq 〈 s_{k} - m_{k}, r_{k} - m_{k} 〉 .

(3)

Using the value of

λ_{k + 1},

we can write

R (s_{k}, r_{k}) - R (s_{k}, m_{k}) - R (m_{k}, r_{k}) \leq \frac{μ (∥ s_{k} - m_{k} ∥^{2} + ∥ r_{k} - m_{k} ∥^{2})}{2 λ_{k + 1}} .

(4)

Expressions (2)–(4) imply that (see Lemma 3.3 in [42]):

∥ r_{k} - s^{*} ∥^{2} \leq ∥ s_{k} - s^{*} ∥^{2} - (1 - \frac{μ λ_{k}}{λ_{k + 1}}) ∥ s_{k} - m_{k} ∥^{2} - (1 - \frac{μ λ_{k}}{λ_{k + 1}}) {∥ r_{k} - m_{k} ∥}^{2} .

□

The strong convergence analysis for Algorithm 1 is presented in the following theorem. The details of the convergence theorems are given below.

Algorithm 1 Self-Adaptive Explicit Extragradient Method with Non-Convex Combination

Step 0: Let $s_{1} \in Π,$ $λ_{0} > 0$ , $μ \in (0, 1),$ $δ_{k} \subset [δ, 1)$ through $δ > 0$ and $ϕ_{k} \subset (0, 1)$ such that

$lim_{k \to + \infty} ϕ_{k} = 0 and \sum_{k = 1}^{+ \infty} ϕ_{k} = + \infty .$
Step 1: Compute

$m_{k} = \underset{v \in Σ}{arg min} {λ_{k} R (s_{k}, v) + \frac{1}{2} ∥ s_{k} - v ∥^{2}} .$

In the case that $m_{k} = s_{k},$ stop and $s_{k} \in E P (R, Σ)$ . Otherwise, go to the next step.
Step 2: First, choose $ω_{k} \in \partial R (s_{k}, m_{k})$ satisfying $s_{k} - λ_{k} ω_{k} - m_{k} \in N_{K} (m_{k})$ and generate a half-space

$Π_{k} = {z \in Π : 〈 s_{k} - λ_{k} ω_{k} - m_{k}, z - m_{k} 〉 \leq 0} .$

Solve $r_{k} = \underset{v \in Π_{k}}{arg min} {λ_{k} R (m_{k}, v) + \frac{1}{2} ∥ s_{k} - v ∥^{2}} .$
Step 3: Compute

$s_{k + 1} = P_{Σ} [ϕ_{k} s_{k} + (1 - ϕ_{k}) r_{k} - ϕ_{k} δ_{k} s_{k}] .$
Step 4: Revise the step size as follows and continue:

$λ_{k + 1} = \{\begin{cases} min \{λ_{k}, \frac{μ (∥ s_{k} - m_{k} ∥^{2} + ∥ r_{k} - m_{k} ∥^{2})}{2 [R (s_{k}, r_{k}) - R (s_{k}, m_{k}) - R (m_{k}, r_{k})]}\} \\ if R (s_{k}, r_{k}) - R (s_{k}, m_{k}) - R (m_{k}, r_{k}) > 0 \\ λ_{k} otherwise . \end{cases}$
Set $k : = k + 1$ and move back to Step 1.

Theorem 1.

Let a sequence

{s_{k}}

be generated by Algorithm 1. Then, sequence

{s_{k}}

converges strongly to

s^{*} \in E P (R, Σ) .

Proof.

Given that

λ_{k} \to λ,

then

ϵ \in (0, 1 - μ),

is a number such that

lim_{k \to + \infty} (1 - \frac{μ λ_{k}}{λ_{k + 1}}) = 1 - μ > ϵ > 0 .

As a result, there exists a finite number

k_{1} \in N

such that

(1 - \frac{μ λ_{k}}{λ_{k + 1}}) > ϵ > 0, \forall k \geq k_{1} .

(5)

Using Lemma 7, we have

∥ r_{k} - s^{*} ∥^{2} \leq {∥ s_{k} - s^{*} ∥}^{2}, \forall k \geq k_{1} .

(6)

We derive using Lemma 3 (i) for any

k \geq k_{1},

such that

\begin{matrix} ∥ s_{k + 1} - s^{*} ∥^{2} & = ∥ P_{Σ} [ϕ_{k} (1 - δ_{k}) s_{k} + (1 - ϕ_{k}) r_{k}] - P_{Σ} (s^{*}) ∥^{2} \\ \leq ∥ ϕ_{k} (1 - δ_{k}) s_{k} + (1 - ϕ_{k}) r_{k} - s^{*} ∥^{2} \\ = ∥ ϕ_{k} [(1 - δ_{k}) s_{k} - s^{*}] + (1 - ϕ_{k}) (r_{k} - s^{*}) ∥^{2} \\ \leq ϕ_{k} ∥ (1 - δ_{k}) s_{k} - s^{*} ∥^{2} + (1 - ϕ_{k}) {∥ r_{k} - s^{*} ∥}^{2} \\ \leq ϕ_{k} [∥ (1 - δ_{k}) (s_{k} - s^{*}) + δ_{k} s^{*} ∥^{2}] + (1 - ϕ_{k}) {∥ s_{k} - s^{*} ∥}^{2} \\ - (1 - ϕ_{k}) [(1 - \frac{μ λ_{k}}{λ_{k + 1}}) ∥ s_{k} - m_{k} ∥^{2} + (1 - \frac{μ λ_{k}}{λ_{k + 1}}) {∥ r_{k} - m_{k} ∥}^{2}] \\ \leq ϕ_{k} [(1 - δ_{k}) ∥ s_{k} - s^{*} ∥^{2} + δ_{k} {∥ s^{*} ∥}^{2}] + (1 - ϕ_{k}) {∥ s_{k} - s^{*} ∥}^{2} \\ - (1 - ϕ_{k}) [ϵ ∥ s_{k} - m_{k} ∥^{2} + ϵ {∥ r_{k} - m_{k} ∥}^{2}] \\ = (1 - ϕ_{k} δ_{k}) ∥ s_{k} - s^{*} ∥^{2} + ϕ_{k} δ_{k} {∥ s^{*} ∥}^{2} \end{matrix}

\begin{matrix} - ϵ (1 - ϕ_{k}) [∥ s_{k} - m_{k} ∥^{2} + {∥ r_{k} - m_{k} ∥}^{2}] \\ \leq max {∥ s_{k} - s^{*} ∥^{2}, ∥ s^{*} ∥^{2}} \end{matrix}

(7)

\begin{matrix} \leq max {∥ s_{k_{1}} - s^{*} ∥^{2}, ∥ s^{*} ∥^{2}} . \end{matrix}

(8)

It is deduced that sequence

{s_{k}}

is a bounded sequence. Let

q_{k} = ϕ_{k} s_{k} + (1 - ϕ_{k}) r_{k},

for any

k \in N .

By Lemma 3 (i), we have

\begin{matrix} ∥ q_{k} - s^{*} ∥^{2} & = ∥ ϕ_{k} s_{k} + (1 - ϕ_{k}) r_{k} - s^{*} ∥^{2} \leq {∥ s_{k} - s^{*} ∥}^{2}, \forall k \geq k_{1} . \end{matrix}

(9)

Notice that there is

\begin{matrix} s_{k + 1} & = P_{Σ} (q_{k} - ϕ_{k} δ_{k} s_{k}) = P_{Σ} [(1 - ϕ_{k} δ_{k}) q_{k} + ϕ_{k} δ_{k} (1 - ϕ_{k}) (r_{k} - s_{k})] . \end{matrix}

(10)

By Lemma 3 (ii) and (9), (10) implies that (see Equation (3.6) [32])

\begin{matrix} ∥ s_{k + 1} - s^{*} ∥^{2} \\ = ∥ P_{Σ} [(1 - ϕ_{k} δ_{k}) q_{k} + ϕ_{k} δ_{k} (1 - ϕ_{k}) (r_{k} - s_{k})] - P_{Σ} (s^{*}) ∥^{2} \\ \leq (1 - ϕ_{k} δ_{k}) {∥ s_{k} - s^{*} ∥}^{2} + 2 ϕ_{k} δ_{k} (1 - ϕ_{k}) \\ 〈r_{k} - s_{k}, (1 - ϕ_{k} δ_{k}) q_{k} + ϕ_{k} δ_{k} (1 - ϕ_{k}) (r_{k} - s_{k}) - s^{*}〉 \\ + 2 ϕ_{k} δ_{k} (1 - ϕ_{k}) 〈- s^{*}, r_{k} - s_{k}〉 + 2 ϕ_{k} δ_{k} 〈- s^{*}, s_{k} - s^{*}〉 + 2 ϕ_{k}^{2} δ_{k}^{2} 〈s^{*}, s_{k}〉 . \end{matrix}

(11)

The remains of the proof can be split into two parts:

Case 1: Let

k_{2} \in N

(k_{2} \geq k_{1})

such that

∥ s_{k + 1} - s^{*} ∥ \leq ∥ s_{k} - s^{*} ∥, \forall k \geq k_{2} .

Thus,

{lim}_{k \to + \infty} ∥ s_{k} - s^{*} ∥

, exists and let

{lim}_{k \to + \infty} ∥ s_{k} - s^{*} ∥ = l .

By relationship (7), we have

\begin{matrix} ϵ [∥ s_{k} - m_{k} ∥^{2} + {∥ r_{k} - m_{k} ∥}^{2}] & \leq ∥ s_{k} - s^{*} ∥^{2} - ∥ s_{k + 1} - s^{*} ∥^{2} + ϕ_{k} δ_{k} {∥ s^{*} ∥}^{2} \\ + ϵ ϕ_{k} [∥ s_{k} - m_{k} ∥^{2} + {∥ r_{k} - m_{k} ∥}^{2}], \forall k \geq k_{2} . \end{matrix}

(12)

The existence of

{lim}_{k \to + \infty} ∥ s_{k} - s^{*} ∥ = l,

provides that

lim_{k \to + \infty} ∥ s_{k} - m_{k} ∥ = lim_{k \to + \infty} ∥ r_{k} - m_{k} ∥ = 0,

(13)

and accordingly

lim_{k \to + \infty} ∥ s_{k} - r_{k} ∥ \leq lim_{k \to + \infty} ∥ s_{k} - m_{k} ∥ + lim_{k \to + \infty} ∥ m_{k} - r_{k} ∥ = 0 .

(14)

Thus, the sequence

{s_{k}}

is a bounded sequence. Hence, we may select a subsequence

{s_{k_{j}}}

{s_{k}}

such that

{s_{k_{j}}}

converges weakly to a certain

s \in Σ

such that

\underset{k \to + \infty}{lim sup} 〈 - s^{*}, s_{k} - s^{*} 〉 = \underset{j \to + \infty}{lim sup} 〈 - s^{*}, s_{k_{j}} - s^{*} 〉 = 〈 - s^{*}, s - s^{*} 〉 .

(15)

From (13) the subsequence

{m_{k_{j}}}

also converges weakly to s as

j \to + \infty .

Due to the expression (3), we obtain

λ_{k_{j}} R (s_{k_{j}}, y) - λ_{k_{j}} R (s_{k_{j}}, m_{k_{j}}) \geq 〈 s_{k_{j}} - m_{k_{j}}, y - m_{k_{j}} 〉, \forall y \in Σ .

(16)

Allowing

j \to + \infty

entails that

R (s, y) \geq 0, \forall y \in Σ .

(17)

As a result,

s \in E P (R, Σ) .

Eventually, using (15) and Lemma 2 (ii), we derive

\begin{matrix} \underset{k \to + \infty}{lim sup} 〈 - s^{*}, s_{k} - s^{*} 〉 & = \underset{j \to + \infty}{lim sup} 〈 - s^{*}, s_{k_{j}} - s^{*} 〉 \\ = 〈 - s^{*}, s - s^{*} 〉 . \\ = 〈 θ - P_{E P (R, Σ)}, s - P_{E P (R, Σ)} 〉 . \\ \leq 0 . \end{matrix}

(18)

We have the desired results from of the assertion on

ϕ_{k},

δ_{k}

, (11), (13), (14), (18) and Lemma 4.

Case 2: Assume that there exists a subsequence

{k_{i}}

{k}

such that

∥ s_{k_{i}} - s^{*} ∥ \leq ∥ s_{k_{i + 1}} - s^{*} ∥, \forall i \in N .

Consequently, according to Lemma 5, there is indeed a sequence

{n_{j}} \subset N

such that

n_{j} \to + \infty,

we have

∥ s_{n_{j}} - s^{*} ∥ \leq ∥ s_{m_{j + 1}} - s^{*} ∥ and ∥ s_{j} - s^{*} ∥ \leq ∥ s_{m_{j + 1}} - s^{*} ∥, for all j \in N .

(19)

By the expression (7), we have

\begin{matrix} ϵ [∥ s_{n_{j}} - m_{n_{j}} ∥^{2} + {∥ r_{n_{j}} - m_{n_{j}} ∥}^{2}] & \leq ∥ s_{n_{j}} - s^{*} ∥^{2} - ∥ s_{n_{j} + 1} - s^{*} ∥^{2} + ϕ_{n_{j}} δ_{n_{j}} {∥ s^{*} ∥}^{2} \\ + ϵ ϕ_{n_{j}} [∥ s_{n_{j}} - m_{n_{j}} ∥^{2} + {∥ r_{n_{j}} - m_{n_{j}} ∥}^{2}], \forall n_{j} \geq k_{1} . \end{matrix}

(20)

The above expressions imply that

lim_{j \to + \infty} ∥ s_{n_{j}} - m_{n_{j}} ∥ = lim_{j \to + \infty} ∥ r_{n_{j}} - m_{n_{j}} ∥ = 0,

(21)

thus

lim_{j \to + \infty} ∥ s_{n_{j}} - r_{n_{j}} ∥ \leq lim_{j \to + \infty} ∥ s_{n_{j}} - m_{n_{j}} ∥ + lim_{j \to + \infty} ∥ m_{n_{j}} - r_{n_{j}} ∥ = 0 .

(22)

By statements identical to those in expression (18), we have

\underset{j \to + \infty}{lim sup} 〈 - s^{*}, s_{n_{j}} - s^{*} 〉 \leq 0 .

(23)

From expression (11), we obtain

\begin{matrix} ∥ s_{n_{j} + 1} - s^{*} ∥^{2} \\ \leq (1 - ϕ_{n_{j}} δ_{n_{j}}) {∥ s_{n_{j}} - s^{*} ∥}^{2} + 2 ϕ_{n_{j}} δ_{n_{j}} (1 - ϕ_{n_{j}}) \\ 〈r_{n_{j}} - s_{n_{j}}, (1 - ϕ_{n_{j}} δ_{n_{j}}) q_{n_{j}} + ϕ_{n_{j}} δ_{n_{j}} (1 - ϕ_{n_{j}}) (r_{n_{j}} - s_{n_{j}}) - s^{*}〉 \\ + 2 ϕ_{n_{j}} δ_{n_{j}} (1 - ϕ_{n_{j}}) 〈- s^{*}, r_{n_{j}} - s_{n_{j}}〉 + 2 ϕ_{n_{j}} δ_{n_{j}} 〈- s^{*}, s_{n_{j}} - s^{*}〉 + 2 ϕ_{n_{j}}^{2} δ_{n_{j}}^{2} 〈s^{*}, s_{n_{j}}〉 . \end{matrix}

(24)

It is given that

∥ s_{n_{j}} - s^{*} ∥ \leq ∥ s_{m_{j + 1}} - s^{*} ∥,

implies that

\begin{matrix} ∥ s_{n_{j} + 1} - s^{*} ∥^{2} \\ \leq (1 - ϕ_{n_{j}} δ_{n_{j}}) {∥ s_{m_{j + 1}} - s^{*} ∥}^{2} + 2 ϕ_{n_{j}} δ_{n_{j}} (1 - ϕ_{n_{j}}) \\ 〈r_{n_{j}} - s_{n_{j}}, (1 - ϕ_{n_{j}} δ_{n_{j}}) q_{n_{j}} + ϕ_{n_{j}} δ_{n_{j}} (1 - ϕ_{n_{j}}) (r_{n_{j}} - s_{n_{j}}) - s^{*}〉 \\ + 2 ϕ_{n_{j}} δ_{n_{j}} (1 - ϕ_{n_{j}}) 〈- s^{*}, r_{n_{j}} - s_{n_{j}}〉 + 2 ϕ_{n_{j}} δ_{n_{j}} 〈- s^{*}, s_{n_{j}} - s^{*}〉 + 2 ϕ_{n_{j}}^{2} δ_{n_{j}}^{2} 〈s^{*}, s_{n_{j}}〉 . \end{matrix}

(25)

The expression (19) and (25) implies that

\begin{matrix} ∥ s_{j} - s^{*} ∥^{2} \leq {∥ s_{n_{j} + 1} - s^{*} ∥}^{2} \\ \leq 2 (1 - ϕ_{n_{j}}) 〈r_{n_{j}} - s_{n_{j}}, (1 - ϕ_{n_{j}} δ_{n_{j}}) q_{n_{j}} + ϕ_{n_{j}} δ_{n_{j}} (1 - ϕ_{n_{j}}) (r_{n_{j}} - s_{n_{j}}) - s^{*}〉 \\ + 2 (1 - ϕ_{n_{j}}) 〈- s^{*}, r_{n_{j}} - s_{n_{j}}〉 + 2 〈- s^{*}, s_{n_{j}} - s^{*}〉 + 2 ϕ_{n_{j}} δ_{n_{j}} 〈s^{*}, s_{n_{j}}〉, \forall k \geq k_{1} . \end{matrix}

(26)

Because

ϕ_{n_{j}} \to 0,

it derives via expressions (21), (22) such that

lim_{k \to + \infty} ∥ s_{j} - s^{*} ∥^{2} \leq lim_{k \to + \infty} {∥ s_{n_{j} + 1} - s^{*} ∥}^{2} \leq 0 .

(27)

Consequently,

s_{k} \to s^{*} .

This is the required result. □

Now, a modification of Algorithm 1 proves a strong convergence theorem for it. For the purpose of simplicity, we will adopt the notation

{[t]}_{+} = max {0, t}

and the conventional

\frac{0}{0} = + \infty

and

\frac{a}{0} = + \infty

(

a \neq 0

). The following is a more detailed algorithm:

Lemma 8.

Let

R : Π \times Π \to R

be a bifunction satisfies the conditions(

R

1)–(

R

4). For any

s^{*} \in E P (R, Σ) \neq \emptyset,

we have

∥ r_{k} - s^{*} ∥^{2} \leq ∥ P_{Σ} (s_{k}) - s^{*} ∥^{2} - (1 - \frac{μ λ_{k}}{λ_{k + 1}}) ∥ P_{Σ} (s_{k}) - m_{k} ∥^{2} - (1 - \frac{μ λ_{k}}{λ_{k + 1}}) {∥ r_{k} - m_{k} ∥}^{2} .

The strong convergence analysis for Algorithm 2 is presented in the following theorem. The details of the convergence theorems are given below.

Algorithm 2 Modified Self-Adaptive Explicit Extragradient Method with Non-Convex Combination

Step 0: Let $s_{1} \in Π,$ $λ_{0} > 0$ , $μ \in (0, 1)$ , $δ_{k} \subset [δ, 1)$ with $δ > 0$ and $ϕ_{k}, φ_{k} \subset (0, 1)$ such that

$lim_{k \to + \infty} ϕ_{k} = 0, \sum_{k = 1}^{+ \infty} ϕ_{k} = + \infty and \underset{k \to + \infty}{lim inf} φ_{k} (1 - φ_{k}) > 0 .$
Step 1: Compute

$m_{k} = \underset{v \in Σ}{arg min} {λ_{k} R (P_{Σ} (s_{k}), v) + \frac{1}{2} ∥ P_{Σ} (s_{k}) - v ∥^{2}} .$

If $m_{k} = s_{k},$ then $s_{k}$ is the solution of problem (EP). Otherwise, go to next step.
Step 2: First, choose $ω_{k} \in \partial R (P_{Σ} (s_{k}), m_{k})$ satisfying $P_{Σ} (s_{k}) - λ_{k} ω_{k} - m_{k} \in N_{K} (m_{k})$ and generate a half-space

$Π_{k} = {z \in Π : 〈 P_{Σ} (s_{k}) - λ_{k} ω_{k} - m_{k}, z - m_{k} 〉 \leq 0} .$

Solve

$r_{k} = \underset{v \in Π_{k}}{arg min} {λ_{k} R (m_{k}, v) + \frac{1}{2} ∥ P_{Σ} (s_{k}) - v ∥^{2}} .$
Step 3: Compute

$s_{k + 1} = ϕ_{k} (1 - δ_{k}) s_{k} + (1 - ϕ_{k}) [φ_{k} r_{k} + (1 - φ_{k}) s_{k})] .$
Step 4: Modify step size as follows:

$λ_{k + 1} = min \{λ_{k}, \frac{μ (∥ P_{Σ} (s_{k}) - m_{k} ∥^{2} + ∥ r_{k} - m_{k} ∥^{2})}{2 {[R (P_{Σ} (s_{k}), r_{k}) - R (P_{Σ} (s_{k}), m_{k}) - R (m_{k}, r_{k})]}_{+}}\} .$

Set $k : = k + 1$ and go back to Step 1.

Theorem 2.

Let a sequence

{s_{k}}

be generated by Algorithm 2 and satisfy the conditions(

R

1)–(

R

4). Then, a sequence

{s_{k}}

is strongly convergent to an element

s^{*}

E P (R, Σ) .

Proof.

Using Lemma 8, we have

\begin{matrix} ∥ s_{k + 1} - s^{*} ∥^{2} \\ = ∥ ϕ_{k} (1 - δ_{k}) s_{k} + (1 - ϕ_{k}) [φ_{k} r_{k} + (1 - φ_{k}) s_{k}] - s^{*} ∥^{2} \\ = ∥ ϕ_{k} [(1 - δ_{k}) s_{k} - s^{*}] + (1 - ϕ_{k}) [φ_{k} (r_{k} - s^{*}) + (1 - φ_{k}) (s_{k} - s^{*})] ∥^{2} \\ \leq ϕ_{k} ∥ (1 - δ_{k}) s_{k} - s^{*} ∥^{2} + (1 - ϕ_{k}) {∥ φ_{k} (r_{k} - s^{*}) + (1 - φ_{k}) (s_{k} - s^{*}) ∥}^{2} \\ \leq ϕ_{k} {∥ (1 - δ_{k}) (s_{k} - s^{*}) + δ_{k} s^{*} ∥}^{2} \\ + (1 - ϕ_{k}) [φ_{k} ∥ r_{k} - s^{*} ∥^{2} + (1 - φ_{k}) ∥ s_{k} - s^{*} ∥^{2} - φ_{k} (1 - φ_{k}) {∥ r_{k} - s_{k} ∥}^{2}] \\ \leq ϕ_{k} [(1 - δ_{k}) ∥ s_{k} - s^{*} ∥^{2} + δ_{k} {∥ s^{*} ∥}^{2}] + (1 - ϕ_{k}) [{∥ s_{k} - s^{*} ∥}^{2} \\ - φ_{k} (1 - \frac{μ λ_{k}}{λ_{k + 1}}) ∥ P_{Σ} (s_{k}) - m_{k} ∥^{2} - φ_{k} (1 - \frac{μ λ_{k}}{λ_{k + 1}}) ∥ r_{k} - m_{k} ∥^{2} - φ_{k} (1 - φ_{k}) {∥ r_{k} - s_{k} ∥}^{2}] \\ \leq (1 - ϕ_{k} δ_{k}) ∥ s_{k} - s^{*} ∥^{2} + ϕ_{k} δ_{k} ∥ s^{*} ∥^{2} - (1 - ϕ_{k}) [φ_{k} (1 - \frac{μ λ_{k}}{λ_{k + 1}}) {∥ P_{Σ} (s_{k}) - m_{k} ∥}^{2} \\ + φ_{k} (1 - \frac{μ λ_{k}}{λ_{k + 1}}) ∥ r_{k} - m_{k} ∥^{2} + φ_{k} (1 - φ_{k}) {∥ r_{k} - s_{k} ∥}^{2}] . \end{matrix}

(28)

It is given that

λ_{k} \to λ,

there exists a fixed number

ϵ_{0} \in (0, 1 - μ),

which is indeed a specific number such that

lim_{k \to + \infty} (1 - \frac{μ λ_{k}}{λ_{k + 1}}) = 1 - μ > ϵ_{0} > 0 .

Thus, there exists a fixed number

m_{1} \in N

such that

(1 - \frac{μ λ_{k}}{λ_{k + 1}}) > ϵ_{0} > 0, \forall k \geq m_{1} .

(29)

Combining the expression (28) and (29), we obtain

\begin{matrix} ∥ s_{k + 1} - s^{*} ∥^{2} & \leq max {∥ s_{k} - s^{*} ∥^{2}, ∥ s^{*} ∥^{2}} \leq max {∥ s_{m_{1}} - s^{*} ∥^{2}, ∥ s^{*} ∥^{2}} . \end{matrix}

(30)

The value of

s_{k + 1}

with Lemma 3 provides (see Equation (3.17) [32])

\begin{matrix} ∥ s_{k + 1} - s^{*} ∥^{2} & \leq (1 - ϕ_{k} δ_{k}) {∥ s_{k} - s^{*} ∥}^{2} + 2 ϕ_{k} δ_{k} (1 - ϕ_{k}) φ_{k} 〈 r_{k} - s_{k}, s_{k + 1} - s^{*} 〉 \\ + 2 ϕ_{k} δ_{k} 〈 - s^{*}, s_{k + 1} - s^{*} 〉 . \end{matrix}

(31)

The rest of the discussion will be divided into two parts:

Case 1: Assume that there exists an integer

m_{2} \in N

(m_{2} \geq m_{1})

such that

∥ s_{k + 1} - s^{*} ∥ \leq ∥ s_{k} - s^{*} ∥, \forall k \geq m_{2} .

(32)

Thus, the

{lim}_{k \to + \infty} ∥ s_{k} - s^{*} ∥

exists. By expression (28), we have

\begin{matrix} ϵ_{0} φ_{k} [∥ P_{Σ} (s_{k}) - m_{k} ∥^{2} + {∥ r_{k} - m_{k} ∥}^{2}] + φ_{k} (1 - φ_{k}) {∥ r_{k} - s_{k} ∥}^{2} \\ \leq ∥ s_{k} - s^{*} ∥^{2} - ∥ s_{k + 1} - s^{*} ∥^{2} + ϕ_{k} δ_{k} {∥ s^{*} ∥}^{2} \\ + ϵ_{0} ϕ_{k} φ_{k} [∥ P_{Σ} (s_{k}) - m_{k} ∥^{2} + {∥ r_{k} - m_{k} ∥}^{2}] + ϕ_{k} φ_{k} (1 - φ_{k}) {∥ r_{k} - s_{k} ∥}^{2} . \end{matrix}

(33)

The above, together with the assumptions on

λ_{k}

ϕ_{k}

and

φ_{k},

yields that

\begin{matrix} lim_{k \to + \infty} ∥ P_{Σ} (s_{k}) - m_{k} ∥ = lim_{k \to + \infty} ∥ r_{k} - m_{k} ∥ = 0 = lim_{k \to + \infty} ∥ s_{k} - r_{k} ∥ = 0 . \end{matrix}

(34)

As a result,

{s_{k}}

is bounded, and we may choose a subsequence

{s_{k_{j}}}

{s_{k}}

such that

{s_{k_{j}}}

converges weakly to

s \in Σ

and

\underset{k \to + \infty}{lim sup} 〈- s^{*}, s_{k} - s^{*}〉 = \underset{j \to + \infty}{lim sup} 〈- s^{*}, s_{k_{j}} - s^{*}〉 = 〈- s^{*}, s - s^{*}〉 .

(35)

As with expression (3) with (34), we have

λ_{k} R (P_{Σ} (s_{k_{j}}), y) - λ_{k_{j}} R (P_{Σ} (s_{k_{j}}), m_{k}) \geq 〈P_{Σ} (s_{k_{j}}) - m_{k_{j}}, y - m_{k_{j}}〉, \forall y \in Σ .

(36)

Allowing

j \to + \infty,

indicates that

R (s, y) \geq 0, \forall y \in Σ .

It continues that

s \in E P (R, Σ) .

In the end, by expression (35) and Lemma 2, we may obtain

\begin{matrix} \underset{k \to + \infty}{lim sup} 〈 - s^{*}, s_{k} - s^{*} 〉 & = \underset{j \to + \infty}{lim sup} 〈 - s^{*}, s_{k_{j}} - s^{*} 〉 \\ = 〈 - s^{*}, s - s^{*} 〉 . \\ = 〈 θ - P_{E P (R, Σ)}, s - P_{E P (R, Σ)} 〉 . \\ \leq 0 . \end{matrix}

(37)

The needed result is obtained using Equation (31) and the Lemma 4.

Case 2: Assume that a subsequence

{k_{i}}

{k}

such that

∥ s_{k_{i}} - s^{*} ∥ \leq ∥ s_{k_{i + 1}} - s^{*} ∥, \forall i \in N .

Thus, by Lemma 5 there exists a nondecreasing sequence

{n_{j}} \subset N

such that

{n_{j}} \to + \infty,

which gives

∥ s_{n_{j}} - s^{*} ∥ \leq ∥ s_{m_{j + 1}} - s^{*} ∥ and ∥ s_{j} - s^{*} ∥ \leq ∥ s_{m_{j + 1}} - s^{*} ∥, for all j \in N .

(38)

Using expression (31), we have

\begin{matrix} ∥ s_{n_{j} + 1} - s^{*} ∥^{2} \\ \leq (1 - ϕ_{n_{j}} δ_{n_{j}}) {∥ s_{n_{j}} - s^{*} ∥}^{2} + 2 ϕ_{n_{j}} δ_{n_{j}} (1 - ϕ_{n_{j}}) φ_{n_{j}} 〈 r_{n_{j}} - s_{n_{j}}, s_{n_{j} + 1} - s^{*} 〉 \\ + 2 ϕ_{n_{j}} δ_{n_{j}} 〈 - s^{*}, s_{n_{j} + 1} - s^{*} 〉 \end{matrix}

(39)

The remaining proof is analogous to Case 2 in Theorem 1. □

4. Applications

In this section, we derive our main results, which are used to solve fixed-point and variational inequality problems. An operator

T : Σ \subset Π \to Σ

is said to be

(i) κ-strict pseudocontraction [43] on

Σ

∥ T r_{1} - T r_{2} ∥^{2} \leq ∥ r_{1} - r_{2} ∥^{2} + κ {∥ (r_{1} - T r_{1}) - (r_{2} - T r_{2}) ∥}^{2}, \forall r_{1}, r_{2} \in Σ;

which is equivalent to

〈T r_{1} - T r_{2}, r_{1} - r_{2}〉 \leq ∥ r_{1} - r_{2} ∥^{2} - \frac{1 - κ}{2} {∥ (r_{1} - T r_{1}) - (r_{2} - T r_{2}) ∥}^{2}, \forall r_{1}, r_{2} \in Σ .

(ii) Weakly sequentially continuous on

Σ

T (s_{k}) ⇀ T (s^{*}) as each sequence in Σ satisfying s_{k} ⇀ s^{*} .

Note: If we take

R (x, y) = 〈 x - T x, y - x 〉, \forall x, y \in Σ,

the equilibrium problem converts into to the fixed-point problem through

2 c_{1} = 2 c_{2} = \frac{3 - 2 κ}{1 - κ} .

The algorithm’s

m_{k}

and

r_{k}

values become (for more information, see [32]):

\{\begin{matrix} m_{k} = \underset{v \in Σ}{arg min} {λ_{k} R (s_{k}, v) + \frac{1}{2} ∥ s_{k} - v ∥^{2}} = (1 - λ_{k}) s_{k} + λ_{k} T (s_{k}), \\ r_{k} = \underset{v \in Π_{k}}{arg min} {λ_{k} R (m_{k}, v) + \frac{1}{2} ∥ s_{k} - v ∥^{2}} = P_{Σ} [s_{k} - λ_{k} (m_{k} - T (m_{k}))] . \end{matrix}

(40)

The following fixed-point theorems are derived from the results in Section 3.

Corollary 1.

Suppose that Σ is a nonempty closed and convex subset of a Hilbert space

Π .

Let

T : Σ \to Σ

is a weakly continuous and κ-strict pseudocontraction with

F i x (T) \neq \emptyset .

Let

s_{1} \in Σ,

λ_{0} > 0

μ \in (0, 1)

δ_{k} \subset [δ, 1)

with

δ > 0

and

ϕ_{k} \subset (0, 1)

lim_{k \to + \infty} ϕ_{k} = 0 a n d \sum_{k = 1}^{+ \infty} ϕ_{k} = + \infty .

Additionally, the sequence

{s_{k}}

is created as follows:

\{\begin{matrix} m_{k} = (1 - λ_{k}) s_{k} + λ_{k} T (s_{k}), \\ r_{k} = P_{Π_{k}} [s_{k} - λ_{k} (m_{k} - T (m_{k}))], \\ s_{k + 1} = P_{Σ} [ϕ_{k} s_{k} + (1 - ϕ_{k}) r_{k} - ϕ_{k} δ_{k} s_{k}], \end{matrix}

where

Π_{k} = {z \in Π : 〈 s_{k} - λ_{k} T (s_{k}) - m_{k}, z - m_{k} 〉 \leq 0} .

The relevant step-size

λ_{k + 1}

is obtained:

λ_{k + 1} = min \{λ_{k}, \frac{μ (∥ s_{k} - m_{k} ∥^{2} + ∥ r_{k} - m_{k} ∥^{2})}{2 {[〈(s_{k} - m_{k}) - (T (s_{k}) - T (m_{k})), r_{k} - m_{k}〉]}_{+}}\} .

Thus, the sequence

{s_{k}}

strongly converges to

s^{*} = P_{F i x (T)} (θ) .

Corollary 2.

Suppose that Σ is a nonempty closed and convex subset of a Hilbert space

Π .

Let

T : Σ \to Σ

is a weakly continuous and κ-strict pseudocontraction with

F i x (T) \neq \emptyset .

Let

s_{1} \in Π,

λ_{0} > 0

μ \in (0, 1)

δ_{k} \subset [δ, 1)

with

δ > 0

and

ϕ_{k}, φ_{k} \subset (0, 1)

such that

lim_{k \to + \infty} ϕ_{k} = 0, \sum_{k = 1}^{+ \infty} ϕ_{k} = + \infty a n d \underset{k \to + \infty}{lim inf} φ_{k} (1 - φ_{k}) > 0 .

Additionally, the sequence

{s_{k}}

is created as follows:

\{\begin{matrix} m_{k} = (1 - λ_{k}) P_{Σ} (s_{k}) + λ_{k} T (P_{Σ} (s_{k})), \\ r_{k} = P_{Π_{k}} [s_{k} - λ_{k} (m_{k} - T (m_{k}))], \\ s_{k + 1} = ϕ_{k} (1 - δ_{k}) s_{k} + (1 - ϕ_{k}) [φ_{k} r_{k} + (1 - φ_{k}) s_{k})], \end{matrix}

where

Π_{k} = {z \in Π : 〈 P_{Σ} (s_{k}) - λ_{k} T (P_{Σ} (s_{k})) - m_{k}, z - m_{k} 〉 \leq 0} .

The relevant step size

λ_{k + 1}

is obtained as follows:

λ_{k + 1} = min \{λ_{k}, \frac{μ (∥ P_{Σ} (s_{k}) - m_{k} ∥^{2} + ∥ r_{k} - m_{k} ∥^{2})}{2 {[〈(P_{Σ} (s_{k}) - m_{k}) - (T (P_{Σ} (s_{k})) - T (m_{k})), r_{k} - m_{k}〉]}_{+}}\} .

Thus, the sequence

{s_{k}}

strongly converges to

s^{*} = P_{F i x (T)} (θ) .

The variational inequality problem is presented as follows:

Find s^{*} \in Σ such that 〈G (s^{*}), y - s^{*}〉 \geq 0, \forall y \in Σ .

An operator

G : Π \to Π

is said to be

(i): L-Lipschitz continuous on $Σ$ if

$∥ G (r_{1}) - G (r_{2}) ∥ \leq L ∥ r_{1} - r_{2} ∥, \forall r_{1}, r_{2} \in Σ;$
(ii): pseudomonotone on $Σ$ if

$〈G (r_{1}), r_{2} - r_{1}〉 \geq 0 ⟹ 〈G (r_{2}), r_{1} - r_{2}〉 \leq 0, \forall r_{1}, r_{2} \in Σ .$

Note: If

R (x, y) : = 〈G (x), y - x〉

for all

x, y \in Σ,

the equilibrium problem converts into a variational inequality problem via

L = 2 c_{1} = 2 c_{2}

(for more information, see [44]). By the value of

m_{k}

and

r_{k}

in Algorithm 1, we derived

\{\begin{matrix} m_{k} = \underset{v \in Σ}{arg min} {λ_{k} R (s_{k}, v) + \frac{1}{2} ∥ s_{k} - v ∥^{2}} = P_{Σ} [s_{k} - λ_{k} G (s_{k})], \\ r_{k} = \underset{v \in Π_{k}}{arg min} {λ_{k} R (m_{k}, v) + \frac{1}{2} ∥ s_{k} - v ∥^{2}} = P_{Π_{k}} [s_{k} - λ_{k} G (m_{k})] . \end{matrix}

(41)

Due to

ω_{k} \in \partial R (s_{k}, m_{k})

, we obtain

\begin{matrix} 〈 ω_{k}, z - m_{k} 〉 & \leq 〈 G (s_{k}), z - s_{k} 〉 - 〈 G (s_{k}), m_{k} - s_{k} 〉, \forall z \in Π \\ = 〈 G (s_{k}), z - m_{k} 〉, \forall z \in Π, \end{matrix}

(42)

and consequently

0 \leq 〈 G (s_{k}) - ω_{k}, z - m_{k} 〉,

\forall z \in Π .

It implies that

\begin{matrix} 〈 s_{k} - λ_{k} G (s_{k}) - m_{k}, z - m_{k} 〉 \\ \leq 〈 s_{k} - λ_{k} G (s_{k}) - m_{k}, z - m_{k} 〉 + λ_{k} 〈 G (s_{k}) - ω_{k}, z - m_{k} 〉 \\ = 〈 s_{k} - λ_{k} ω_{k} - m_{k}, z - m_{k} 〉 . \end{matrix}

(43)

Assumption 1.

Assume that G fulfills the following conditions:

(i): An operator G is pseudomonotone upon Σ and $V I (G, Σ)$ is nonempty;
(ii): G is L-Lipschitz continuous on Σ with $L > 0;$
(iii): $\underset{k \to + \infty}{lim sup} 〈 G (s_{k}), y - s_{k} 〉 \leq 〈 G (s^{*}, y - s^{*} 〉$ for any $y \in Σ$ and ${s_{k}} \subset Σ$ meet $s_{k} ⇀ s^{*} .$

Corollary 3.

Let

G : Σ \to Π

be an operator and satisfies Assumption 1. Assume that sequence

{s_{k}}

is generated as follows: Let

s_{1} \in Π,

λ_{0} > 0

μ \in (0, 1)

δ_{k} \subset [δ, 1)

with

δ > 0

and

ϕ_{k} \subset (0, 1)

such that

lim_{k \to + \infty} ϕ_{k} = 0 a n d \sum_{k = 1}^{+ \infty} ϕ_{k} = + \infty .

Moreover, sequence

{s_{k}}

is generated as follows:

\{\begin{matrix} m_{k} = P_{Σ} [s_{k} - λ_{k} G (s_{k})], \\ r_{k} = P_{Π_{k}} [s_{k} - λ_{k} G (m_{k})], \\ s_{k + 1} = P_{Σ} [ϕ_{k} s_{k} + (1 - ϕ_{k}) r_{k} - ϕ_{k} δ_{k} s_{k}], \end{matrix}

where

Π_{k} = {z \in Π : 〈 s_{k} - λ_{k} G (s_{k}) - m_{k}, z - m_{k} 〉 \leq 0} .

Next, step size

λ_{k + 1}

is obtained as follows:

λ_{k + 1} = min \{λ_{k}, \frac{μ (∥ s_{k} - m_{k} ∥^{2} + ∥ r_{k} - m_{k} ∥^{2})}{2 {[〈F (s_{k}) - F (m_{k}), r_{k} - m_{k}〉]}_{+}}\} .

Then, sequence

{s_{k}}

strongly converges to the solution

s^{*} \in V I (G, Σ) .

Corollary 4.

Let

G : Σ \to Π

be an operator that satisfies Assumption 1. Assume that

{s_{k}},

is generated as follows: Let

s_{1} \in Π,

λ_{0} > 0

μ \in (0, 1)

δ_{k} \subset [δ, 1)

with

δ > 0

and

ϕ_{k}, φ_{k} \subset (0, 1)

such that

lim_{k \to + \infty} ϕ_{k} = 0, \sum_{k = 1}^{+ \infty} ϕ_{k} = + \infty a n d \underset{k \to + \infty}{lim inf} φ_{k} (1 - φ_{k}) > 0 .

Moreover, the sequence

{s_{k}}

generated as follows:

\{\begin{matrix} m_{k} = P_{Σ} [P_{Σ} (s_{k}) - λ_{k} G (P_{Σ} (s_{k}))], \\ r_{k} = P_{Π_{k}} [P_{Σ} (s_{k}) - λ_{k} G (m_{k})], \\ s_{k + 1} = ϕ_{k} (1 - δ_{k}) s_{k} + (1 - ϕ_{k}) [φ_{k} r_{k} + (1 - φ_{k}) s_{k})], \end{matrix}

where

Π_{k} = {z \in Π : 〈 P_{Σ} (s_{k}) - λ_{k} G (P_{Σ} (s_{k})) - m_{k}, z - m_{k} 〉 \leq 0} .

Next step-size

λ_{k + 1}

is obtained as follows:

λ_{k + 1} = min \{λ_{k}, \frac{μ (∥ P_{Σ} (s_{k}) - m_{k} ∥^{2} + ∥ r_{k} - m_{k} ∥^{2})}{2 {[〈F (P_{Σ} (s_{k})) - F (m_{k}), r_{k} - m_{k}〉]}_{+}}\} .

Then, sequence

{s_{k}}

strongly converges to the solution

s^{*} \in V I (G, Σ) .

5. Numerical Illustration

The computational results in this section show that our proposed algorithms are more efficient than Algorithms 3.1 and 3.2 in [32]. The MATLAB program was executed in MATLAB version 9.5 on a PC (with Intel(R) Core(TM)i3-4010U CPU @ 1.70 GHz 1.70 GHz, RAM 4.00 GB) (R2018b). In all our algorithms, we used the built-in MATLAB fmincon function to solve the minimization problems. (i) The setting for design variables for Algorithm 3.1 (Algo. 3.1) and Algorithm 3.2 (Algo. 3.2) in [32] possess different values that are given in all examples.

ϕ_{k} = \frac{1}{40 k}, δ_{k} = \frac{1}{10} + \frac{1}{10 k}, λ_{k} = \frac{k}{3 + 2 c_{1}}, φ_{k} = \frac{1}{4} + \frac{1}{4 n} and D_{k} = ∥ s_{k} - m_{k} ∥ \leq ϵ .

(ii) The settings for the design variables for Algorithm 1 (Algo. 1 ) and Algorithm 2 (Algo. 2) are

ϕ_{k} = \frac{1}{40 k}, δ_{k} = \frac{1}{10} + \frac{1}{10 k}, φ_{k} = \frac{1}{4} + \frac{1}{4 k}, D_{k} = ∥ s_{k} - m_{k} ∥ \leq ϵ and for different λ_{0} .

Example 1.

Let us consider a bifunction

R : Σ \times Σ \to R,

which is represented as follows:

R (s, m) = \sum_{i = 2}^{5} (m_{i} - s_{i}) ∥ s ∥, \forall s, m \in R^{5} .

In addition, the convex set is defined as follows:

Σ = \{(s_{1}, \dots, s_{5}) : s_{1} \geq - 1, s_{i} \geq 1, i = 2, \dots, 5\} .

Consequently,

R

is Lipschitz-type continuous across

c_{1} = c_{2} = 2

and meets the condition(

R

1)–(

R

4). The obtained simulations are shown in Figure 1 and Figure 2 and Table 1 and Table 2 by using

s_{1} = (2, 3, 2, 5, 5)

and

ϵ = 10^{- 4} .

Example 2.

According to the articles [29], the bifunction

R

might be written as follows:

R (s, m) = 〈 A s + B m + c, m - s 〉,

where

c \in R^{5}

and A, B are

A = (\begin{matrix} 3.1 & 2 & 0 & 0 & 0 \\ 2 & 3.6 & 0 & 0 & 0 \\ 0 & 0 & 3.5 & 2 & 0 \\ 0 & 0 & 2 & 3.3 & 0 \\ 0 & 0 & 0 & 0 & 3 \end{matrix}) B = (\begin{matrix} 1.6 & 1 & 0 & 0 & 0 \\ 1 & 1.6 & 0 & 0 & 0 \\ 0 & 0 & 1.5 & 1 & 0 \\ 0 & 0 & 1 & 1.5 & 0 \\ 0 & 0 & 0 & 0 & 2 \end{matrix}) c = (\begin{matrix} 1 \\ - 2 \\ - 1 \\ 2 \\ - 1 \end{matrix}) .

The Lipschitz parameters are also

c_{1} = c_{2} = \frac{1}{2} ∥ A - B ∥

(see [29]). The possible set Σ and its subset

R^{5}

are given as

Σ : = {s \in R^{5} : - 5 \leq s_{i} \leq 5} .

Figure 3 and Figure 4 and Table 3 and Table 4 display the numeric effects with

s_{1} = (1, \dots, 1)

and

ϵ = 10^{- 6} .

Example 3.

Consider that

Π = L^{2} ([0, 1])

is indeed a Hilbert space with

∥ s ∥ = \sqrt{\int_{0}^{1} {| s (t) |}^{2} d t},

where the internal product

〈 s, m 〉 = \int_{0}^{1} s (t) m (t) d t, \forall s, m \in Π .

Suppose that unit ball is

Σ : = {s \in L^{2} ([0, 1]) : ∥ s ∥ \leq 1} .

Let us begin by defining an operator

G (s) (t) = \int_{0}^{1} (s (t) - H (t, s) R (s (s))) d s + g (t),

where

H (t, s) = \frac{2 t s e^{(t + s)}}{e \sqrt{e^{2} - 1}}, R (s) = c o s x, g (t) = \frac{2 t e^{t}}{e \sqrt{e^{2} - 1}} .

As illustrated in [45], G is monotone and L-Lipschitz-continuous via

L = 2 .

Figure 5 and Figure 6 and Table 5 and Table 6 illustrate the numerical results with

s_{1} = t

and

ϵ = 10^{- 6} .

Discussion About Numerical Experiments: The following conclusions may be drawn from the numerical experiments outlined above: (i) Examples 1–3 have reported data for numerous methods in both finite- and infinite-dimensional domains. It is apparent that the given algorithms outperformed in terms of number of iterations and elapsed time in practically all circumstances. All trials demonstrate that the suggested algorithms outperform the previously available techniques. (ii) Examples 1–3 have reported results for several methods in finite and infinite-dimensional domains. In most cases, we can observe that the scale of the problem and the relative standard deviation used impact the algorithm’s effectiveness. (iii) The development of an inappropriate variable step size generates a hump in the graph of algorithms in all examples. It has no impact on the effectiveness of the algorithms. (iv) For large-dimensional problems, all approaches typically took longer and showed significant variation in execution time. The number of iterations, on the other hand, changes slightly less.

6. Conclusions

The paper provides two explicit extragradient-like approaches for solving an equilibrium problem involving a pseudomonotone and a Lipschitz-type bifunction in a real Hilbert space. A new step-size rule has been presented that does not rely on Lipschitz-type constant information. The algorithm’s convergence has been established. Several tests are presented to show the numerical behavior of our two algorithms and to compare them to others that are well known in the literature.

Author Contributions

Conceptualization, M.S., W.K. and H.u.R.; methodology, M.S., H.u.R. and W.K.; software, M.S., W.K. and K.S.; validation, H.u.R., K.S. and W.K.; formal analysis, W.K., H.u.R. and K.S.; investigation, H.u.R., K.S. and W.K.; writing—original draft preparation, M.S., W.K. and H.u.R.; writing—review and editing, W.K., H.u.R. and K.S.; visualization, M.S., W.K. and H.u.R.; supervision and funding, W.K. and K.S. All authors have read and agree to the published version of the manuscript.

Funding

Thailand Science Research and Innovation (TSRI) and Rajamangala University of Technology Thanyaburi (RMUTT) under National Science, Research and Innovation Fund (NSRF); Basic Research Fund: Fiscal year 2022 (Contract No. FRB650070/0168 and under project number FRB65E0632M.1).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

The authors acknowledge the financial support provided by Thailand Science Research and Innovation (TSRI) and Rajamangala University of Technology Thanyaburi (RMUTT) under National Science, Research and Innovation Fund (NSRF); Basic Research Fund: Fiscal year 2022 (Contract No. FRB650070/0168 and under project number FRB65E0632M.1).

Conflicts of Interest

The authors declare that they have no competing interests.

References

Blum, E. From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63, 123–145. [Google Scholar]
Fan, K. A Minimax Inequality and Applications, Inequalities III; Shisha, O., Ed.; Academic Press: New York, NY, USA, 1972. [Google Scholar]
Bigi, G.; Castellani, M.; Pappalardo, M.; Passacantando, M. Existence and solution methods for equilibria. Eur. J. Oper. Res. 2013, 227, 1–11. [Google Scholar] [CrossRef] [Green Version]
Muu, L.; Oettli, W. Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal. Theory Methods Appl. 1992, 18, 1159–1166. [Google Scholar] [CrossRef]
Hung, P.G.; Muu, L.D. The Tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions. Nonlinear Anal. Theory Methods Appl. 2011, 74, 6121–6129. [Google Scholar] [CrossRef]
Konnov, I. Application of the Proximal Point Method to Nonmonotone Equilibrium Problems. J. Optim. Theory Appl. 2003, 119, 317–333. [Google Scholar] [CrossRef]
Moudafi, A. Proximal point algorithm extended to equilibrium problems. J. Nat. Geom. 1999, 15, 91–100. [Google Scholar]
Oliveira, P.; Santos, P.; Silva, A. A Tikhonov-type regularization for equilibrium problems in Hilbert spaces. J. Math. Anal. Appl. 2013, 401, 336–342. [Google Scholar] [CrossRef]
Rehman, H.U.; Kumam, P.; Shutaywi, M.; Pakkaranang, N.; Wairojjana, N. An Inertial Extragradient Method for Iteratively Solving Equilibrium Problems in Real Hilbert Spaces. Int. J. Comput. Math. 2021, 99, 1081–1104. [Google Scholar] [CrossRef]
Rehman, H.u.; Pakkaranang, N.; Hussain, A.; Wairojjana, N. A modified extra-gradient method for a family of strongly pseudomonotone equilibrium problems in real Hilbert spaces. J. Math. Comput. Sci. 2021, 22, 38–48. [Google Scholar] [CrossRef]
Wairojjana, N.; Pakkaranang, N.; Pholasa, N. Strong convergence inertial projection algorithm with self-adaptive step size rule for pseudomonotone variational inequalities in Hilbert spaces. Demonstr. Math. 2021, 54, 110–128. [Google Scholar] [CrossRef]
Rehman, H.U.; Kumam, P.; Abubakar, A.B.; Cho, Y.J. The extragradient algorithm with inertial effects extended to equilibrium problems. Comput. Appl. Math. 2020, 39, 100. [Google Scholar] [CrossRef]
Rehman, H.U.; Kumam, P.; Kumam, W.; Shutaywi, M.; Jirakitpuwapat, W. The Inertial Sub-Gradient Extra-Gradient Method for a Class of Pseudo-Monotone Equilibrium Problems. Symmetry 2020, 12, 463. [Google Scholar] [CrossRef] [Green Version]
Konnov, I.V. Regularization method for nonmonotone equilibrium problems. J. Nonlinear Convex Anal. 2009, 10, 93–101. [Google Scholar]
Konnov, I.V. Partial proximal point method for nonmonotone equilibrium problems. Optim. Methods Softw. 2006, 21, 373–384. [Google Scholar] [CrossRef]
Singh, A.; Shukla, A.; Vijayakumar, V.; Udhayakumar, R. Asymptotic stability of fractional order (1,2] stochastic delay differential equations in Banach spaces. Chaos Solitons Fractals 2021, 150, 111095. [Google Scholar] [CrossRef]
Patel, R.; Shukla, A.; Jadon, S.S. Existence and optimal control problem for semilinear fractional order (1,2] control system. Math. Methods Appl. Sci. 2020. [Google Scholar] [CrossRef]
Shukla, A.; Sukavanam, N.; Pandey, D. Controllability of Semilinear Stochastic System with Multiple Delays in Control. IFAC Proc. Vol. 2014, 47, 306–312. [Google Scholar] [CrossRef] [Green Version]
Shukla, A.; Patel, R. Existence and Optimal Control Results for Second-Order Semilinear System in Hilbert Spaces. Circuits Syst. Signal Process. 2021, 40, 4246–4258. [Google Scholar] [CrossRef]
Van Hieu, D.; Duong, H.N.; Thai, B. Convergence of relaxed inertial methods for equilibrium problems. J. Appl. Numer. Optim. 2021, 3, 215–229. [Google Scholar]
Ogbuisi, F.U. The projection method with inertial extrapolation for solving split equilibrium problems in Hilbert spaces. Appl. Set-Valued Anal. Optim. 2021, 3, 239–255. [Google Scholar]
Liu, L.; Cho, S.Y.; Yao, J.C. Convergence Analysis of an Inertial Tseng’s Extragradient Algorithm for Solving Pseudomonotone Variational Inequalities and Applications. J. Nonlinear Var. Anal. 2021, 5, 627–644. [Google Scholar]
Rehman, H.u.; Kumam, P.; Argyros, I.K.; Shutaywi, M.; Shah, Z. Optimization Based Methods for Solving the Equilibrium Problems with Applications in Variational Inequality Problems and Solution of Nash Equilibrium Models. Mathematics 2020, 8, 822. [Google Scholar] [CrossRef]
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. [Google Scholar] [CrossRef]
Rehman, H.; Kumam, P.; Dong, Q.-L.; Peng, Y.; Deebani, W. A new Popov’s subgradient extragradient method for two classes of equilibrium programming in a real Hilbert space. Optimization 2020, 70, 2675–2710. [Google Scholar] [CrossRef]
Rehman, H.U.; Pakkaranang, N.; Kumam, P.; Cho, Y.J. Modified subgradient extragradient method for a family of pseudomonotone equilibrium problems in real a Hilbert space. J. Nonlinear Convex Anal. 2020, 21, 2011–2025. [Google Scholar]
Flåm, S.D.; Antipin, A.S. Equilibrium programming using proximal-like algorithms. Math. Program. 1996, 78, 29–41. [Google Scholar] [CrossRef]
Korpelevich, G. The extragradient method for finding saddle points and other problems. Matecon 1976, 12, 747–756. [Google Scholar]
Tran, D.Q.; Dung, M.L.; Nguyen, V.H. Extragradient algorithms extended to equilibrium problems. Optimization 2008, 57, 749–776. [Google Scholar] [CrossRef]
Hieu, D.V.; Quy, P.K.; Vy, L.V. Explicit iterative algorithms for solving equilibrium problems. Calcolo 2019, 56, 11. [Google Scholar] [CrossRef]
Yang, J.; Liu, H.; Liu, Z. Modified subgradient extragradient algorithms for solving monotone variational inequalities. Optimization 2018, 67, 2247–2258. [Google Scholar] [CrossRef]
Wang, S.; Zhang, Y.; Ping, P.; Cho, Y.; Guo, H. New extragradient methods with non-convex combination for pseudomonotone equilibrium problems with applications in Hilbert spaces. Filomat 2019, 33, 1677–1693. [Google Scholar] [CrossRef] [Green Version]
Censor, Y.; Gibali, A.; Reich, S. The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space. J. Optim. Theory Appl. 2010, 148, 318–335. [Google Scholar] [CrossRef] [PubMed] [Green Version]
Tiel, J.V. Convex Analysis: An Introductory Text, 1st ed.; Wiley: New York, NY, USA, 1984. [Google Scholar]
Bushell, P.J. UNIFORM CONVEXITY, HYPERBOLIC GEOMETRY, AND NONEXPANSIVE MAPPINGS (Pure and Applied Mathematics: A Series of Monographs & Textbooks, 83) By K. Goebel and S. Reich: Pp. 192. SFr.96.-. (Marcel Dekker Inc, U.S.A., 1984). Bull. Lond. Math. Soc. 1985, 17, 293–294. [Google Scholar] [CrossRef]
Kreyszig, E. Introductory Functional Analysis with Applications, 1st ed.; Wiley Classics Library, Wiley: Hoboken, NJ, USA, 1989. [Google Scholar]
Bauschke, H.H.; Combettes, P.L. Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd ed.; CMS Books in Mathematics; Springer International Publishing: Berlin/Heidelberg, Germany, 2017. [Google Scholar]
Xu, H.K. Another control condition in an iterative method for nonexpansive mappings. Bulletin of the Australian Mathematical Society 2002, 65, 109–113. [Google Scholar] [CrossRef] [Green Version]
Maingé, P.E. Strong Convergence of Projected Subgradient Methods for Nonsmooth and Nonstrictly Convex Minimization. Set-Valued Anal. 2008, 16, 899–912. [Google Scholar] [CrossRef]
Bianchi, M.; Schaible, S. Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 1996, 90, 31–43. [Google Scholar] [CrossRef]
Mastroeni, G. On Auxiliary Principle for Equilibrium Problems. In Nonconvex Optimization and Its Applications; Springer: Boston, MA, USA, 2003; pp. 289–298. [Google Scholar] [CrossRef]
Rehman, H.U.; Kumam, P.; Cho, Y.J.; Yordsorn, P. Weak convergence of explicit extragradient algorithms for solving equilibirum problems. J. Inequalities Appl. 2019, 2019, 282. [Google Scholar] [CrossRef]
Browder, F.; Petryshyn, W. Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl. 1967, 20, 197–228. [Google Scholar] [CrossRef] [Green Version]
Rehman, H.U.; Kumam, P.; Je Cho, Y.; Suleiman, Y.I.; Kumam, W. Modified Popov’s explicit iterative algorithms for solving pseudomonotone equilibrium problems. Optim. Methods Softw. 2020, 36, 82–113. [Google Scholar] [CrossRef]
Van Hieu, D.; Anh, P.K.; Muu, L.D. Modified hybrid projection methods for finding common solutions to variational inequality problems. Comput. Optim. Appl. 2017, 66, 75–96. [Google Scholar] [CrossRef]

Figure 1. Algorithm 1 is compared to Algorithm 3.1 in [32].

Figure 2. Algorithm 2 is compared to Algorithm 3.2 in [32].

Figure 3. Algorithm 1 is compared to Algorithm 3.1 in [32].

Figure 4. Algorithm 2 is compared to Algorithm 3.2 in [32].

Figure 5. Algorithm 1 is compared to Algorithm 3.1 in [32].

Figure 6. Algorithm 2 is compared to Algorithm 3.2 in [32].

Table 1. Algorithm 1 is compared to Algorithm 3.1 in [32].

	Number of Iterations		CPU Time in Seconds
$λ_{0}$	Algo. 3.1	Algo. 1	Algo. 3.1	Algo. 1
0.20	35	33	1.6799	1.6119
0.50	-	30	-	1.4787
0.70	-	25	-	1.1520
1.00	-	22	-	1.0100

Table 2. Algorithm 2 is compared to Algorithm 3.2 in [32].

	Number of Iterations		CPU Time in Seconds
$λ_{0}$	Algo. 3.2	Algo. 2	Algo. 3.2	Algo. 2
0.20	99	109	4.9391	5.3081
0.50	-	84	-	4.0511
0.70	-	72	-	3.2269
1.00	-	64	-	2.9225

Table 3. Algorithm 1 is compared to Algorithm 3.1 in [32].

	Number of Iterations		CPU Time in Seconds
$λ_{0}$	Algo. 3.1	Algo. 1	Algo. 3.1	Algo. 1
0.20	149	126	7.0363	5.7321
0.50	-	114	-	5.0527
0.70	-	107	-	4.8159
1.00	-	101	-	4.5495

Table 4. Algorithm 2 is compared to Algorithm 3.2 in [32].

	Number of Iterations		CPU Time in Seconds
$λ_{0}$	Algo. 3.2	Algo. 2	Algo. 3.2	Algo. 2
0.20	300	326	14.7791	14.2233
0.50	-	273	-	13.7107
0.70	-	236	-	11.7754
1.00	-	211	-	11.2054

Table 5. Algorithm 1 is compared to Algorithm 3.1 in [32].

	Number of Iterations		CPU Time in Seconds
$λ_{0}$	Algo. 3.1	Algo. 1	Algo. 3.1	Algo. 1
0.20	64	72	0.0174	0.0331
0.50	–	61	–	0.0295
0.70	–	53	–	0.0273
1.00	–	47	–	0.0265

Table 6. Algorithm 2 is compared to Algorithm 3.2 in [32].

	Number of Iterations		CPU Time in Seconds
$λ_{0}$	Algo. 3.2	Algo. 2	Algo. 3.2	Algo. 2
0.20	213	200	0.0313	0.0500
0.50	–	181	–	0.0460
0.70	–	177	–	0.0352
1.00	–	155	–	0.0260

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

Share and Cite

MDPI and ACS Style

Shutaywi, M.; Kumam, W.; Rehman, H.u.; Sombut, K. On Strengthened Extragradient Methods Non-Convex Combination with Adaptive Step Sizes Rule for Equilibrium Problems. Symmetry 2022, 14, 1045. https://doi.org/10.3390/sym14051045

AMA Style

Shutaywi M, Kumam W, Rehman Hu, Sombut K. On Strengthened Extragradient Methods Non-Convex Combination with Adaptive Step Sizes Rule for Equilibrium Problems. Symmetry. 2022; 14(5):1045. https://doi.org/10.3390/sym14051045

Chicago/Turabian Style

Shutaywi, Meshal, Wiyada Kumam, Habib ur Rehman, and Kamonrat Sombut. 2022. "On Strengthened Extragradient Methods Non-Convex Combination with Adaptive Step Sizes Rule for Equilibrium Problems" Symmetry 14, no. 5: 1045. https://doi.org/10.3390/sym14051045

APA Style

Shutaywi, M., Kumam, W., Rehman, H. u., & Sombut, K. (2022). On Strengthened Extragradient Methods Non-Convex Combination with Adaptive Step Sizes Rule for Equilibrium Problems. Symmetry, 14(5), 1045. https://doi.org/10.3390/sym14051045

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

On Strengthened Extragradient Methods Non-Convex Combination with Adaptive Step Sizes Rule for Equilibrium Problems

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Applications

5. Numerical Illustration

6. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI