# Inertial Extragradient Methods for Solving Split Equilibrium Problems

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

- (i)
- monotone on C if$$f(x,y)+f(y,x)\le 0,\forall x,y\in C;$$
- (ii)
- pseudomonotone on C if$$f(x,y)\ge 0\Rightarrow f(y,x)\le 0,\forall x,y\in C;$$
- (iii)
- Lipshitz-type continuous on H with constants ${L}_{1}>0$ and ${L}_{2}>0$ if$$f(x,y)+f(y,z)\ge f(x,z)-{L}_{1}{\parallel x-y\parallel}^{2}-{L}_{2}{\parallel y-z\parallel}^{2},\forall x,y,z\in H.$$

**Remark**

**1.**

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

**Assumption**

**4.**

**Lemma**

**1.**

- (i)
- ${\lambda}_{0}$$[f({x}_{0},w)-f({x}_{0},{y}_{0})]\ge \langle {y}_{0}-{x}_{0},{y}_{0}-w\rangle $, $\forall w\in C$;
- (ii)
- $\parallel {z}_{0}{-q\parallel}^{2}$$\le \parallel {x}_{0}{-q\parallel}^{2}-(1-2{\lambda}_{0}{L}_{1})\parallel {x}_{0}-{y}_{0}{\parallel}^{2}-(1-2{\lambda}_{0}{L}_{2}){\parallel {y}_{0}-{z}_{0}\parallel}^{2}$, $\forall q\in $$EP(f,C)$.

**Lemma**

**2.**

- (i)
- ${P}_{C}\left(x\right)$ is singleton and well-defined for each $x\in H$;
- (ii)
- $z={P}_{C}\left(x\right)$ if and only if $\langle x-z,y-z\rangle $$\le 0$, $\forall y\in C$;
- (iii)
- ${P}_{C}$ is a nonexpansive operator, that is,$$\parallel {P}_{C}\left(x\right)-{P}_{C}\left(y\right)\parallel \le \parallel x-y\parallel ,\forall x,y\in H.$$

**Lemma**

**3.**

**Lemma**

**4.**

- (i)
- For each $z\in S$, $\underset{k\to \infty}{lim}\parallel {x}_{k}-z\parallel $ exists;
- (ii)
- ${\omega}_{w}\left({x}_{k}\right)\subset S$, where ${\omega}_{w}\left({x}_{k}\right)=\{x\in H:\mathit{there}\phantom{\rule{4.pt}{0ex}}\mathit{is}\phantom{\rule{4.pt}{0ex}}a\phantom{\rule{4.pt}{0ex}}\mathit{subsequence}\phantom{\rule{3.33333pt}{0ex}}\left\{{x}_{{k}_{n}}\right\}\phantom{\rule{3.33333pt}{0ex}}\mathit{of}\phantom{\rule{3.33333pt}{0ex}}\left\{{x}_{k}\right\}\phantom{\rule{3.33333pt}{0ex}}\mathit{such}\phantom{\rule{4.pt}{0ex}}\mathit{that}{x}_{{k}_{n}}\rightharpoonup x\}$.

**Lemma**

**5.**

**Lemma**

**6.**

- (i)
- If there is $M>0$ such that ${b}_{k}\le {\delta}_{k}M$, for all $k\in \mathbb{N}\cup \left\{0\right\}$, then $\left\{{a}_{k}\right\}$ is a bounded sequence;
- (ii)
- If $\sum _{k=0}^{\infty}}{\delta}_{k}=\infty $ and $\underset{k\to \infty}{lim\; sup}({b}_{k}/{\delta}_{k})\le 0$, then $\underset{k\to \infty}{lim}{a}_{k}=0$.

**Lemma**

**7.**

## 3. Main Results

#### 3.1. Inertial Extragradient Method

Algorithm 1: Inertial Extragradient Method (IEM) |

Initialization. Choose parameters $\alpha \in [0,1)$, $\eta \in \left(\right)open="("\; close=")">0,\frac{1}{{\parallel A\parallel}^{2}}$, $\left\{{\lambda}_{k}\right\}$ with $0<inf{\lambda}_{k}\le sup{\lambda}_{k}<min\left(\right)open="\{"\; close="\}">\frac{1}{2{c}_{1}},\frac{1}{2{c}_{2}}$, $\left\{{\mu}_{k}\right\}$ with $0<inf{\mu}_{k}\le sup{\mu}_{k}<min\left(\right)open="\{"\; close="\}">\frac{1}{2{d}_{1}},\frac{1}{2{d}_{2}}$, and $\left\{{\u03f5}_{k}\right\}\subset [0,\infty )$ such that $\sum _{k=0}^{\infty}}{\u03f5}_{k}<\infty $. Pick ${x}_{0},{x}_{1}\in C$ and set $k=1$.Step 1. Choose ${\theta}_{k}$ such that $0\le {\theta}_{k}\le {\overline{\theta}}_{k}$, where
$${\overline{\theta}}_{k}=\left(\right)open="\{"\; close>\begin{array}{c}min\left(\right)open="\{"\; close="\}">\alpha ,\frac{{\u03f5}_{k}}{\parallel {x}_{k}-{x}_{k-1}\parallel},\phantom{\rule{1.em}{0ex}}\mathrm{if}{x}_{k}\ne {x}_{k-1},\hfill \end{array}\alpha ,\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\mathrm{otherwise},\hfill $$
$${w}_{k}={x}_{k}+{\theta}_{k}({x}_{k}-{x}_{k-1}).$$
Step 2. Solve the strongly convex program
$$\begin{array}{ccc}\hfill & & {y}_{k}=argmin\left(\right)open="\{"\; close="\}">{\lambda}_{k}f({w}_{k},y)+\frac{1}{2}{\parallel y-{w}_{k}\parallel}^{2}:y\in C.\hfill \end{array}$$
Step 3. Solve the strongly convex program
$$\begin{array}{ccc}\hfill & & {z}_{k}=argmin\left(\right)open="\{"\; close="\}">{\lambda}_{k}f({y}_{k},y)+\frac{1}{2}{\parallel y-{w}_{k}\parallel}^{2}:y\in C.\hfill \end{array}$$
Step 4. Solve the strongly convex program
$$\begin{array}{ccc}\hfill & & {u}_{k}=argmin\left(\right)open="\{"\; close="\}">{\mu}_{k}g(A{z}_{k},u)+\frac{1}{2}{\parallel u-A{z}_{k}\parallel}^{2}:u\in Q.\hfill \end{array}$$
Step 5. Solve the strongly convex program
$$\begin{array}{ccc}\hfill & & {v}_{k}=argmin\left(\right)open="\{"\; close="\}">{\mu}_{k}g({u}_{k},u)+\frac{1}{2}{\parallel u-A{z}_{k}\parallel}^{2}:u\in Q.\hfill \end{array}$$
Step 6. The next approximation ${x}_{k+1}$ is defined by
$${x}_{k+1}={P}_{C}({z}_{k}+\eta {A}^{*}({v}_{k}-A{z}_{k})).$$
Step 7. Put $k:=k+1$ and go to Step 1. |

**Remark**

**3.**

**Theorem**

**1.**

**Proof.**

#### 3.2. Mann-Type Inertial Extragradient Method

Algorithm 2: Mann-type Inertial Extragradient Method (MIEM) |

Initialization. Choose parameters $\alpha \in [0,1)$, $\eta \in \left(\right)open="("\; close=")">0,\frac{1}{{\parallel A\parallel}^{2}}$, $\left\{{\lambda}_{k}\right\}$ with $0<inf{\lambda}_{k}\le sup{\lambda}_{k}<min\left(\right)open="\{"\; close="\}">\frac{1}{2{c}_{1}},\frac{1}{2{c}_{2}}$, $\left\{{\mu}_{k}\right\}$ with $0<inf{\mu}_{k}\le sup{\mu}_{k}<min\left(\right)open="\{"\; close="\}">\frac{1}{2{d}_{1}},\frac{1}{2{d}_{2}}$, and $\left\{{\u03f5}_{k}\right\}\subset [0,\infty )$, $\left\{{\beta}_{k}\right\}\subset (0,1)$, $\left\{{\gamma}_{k}\right\}\subset (0,1)$ such that $\underset{k\to \infty}{inf}{\beta}_{k}(1-{\beta}_{k}-{\gamma}_{k})>0$, $\sum _{k=0}^{\infty}}{\gamma}_{k}=\infty $, $\underset{k\to \infty}{lim}{\gamma}_{k}=0$, $\sum _{k=0}^{\infty}}{\u03f5}_{k}<\infty $, and ${\u03f5}_{k}=o\left({\gamma}_{k}\right)$, where ${\u03f5}_{k}=o\left({\gamma}_{k}\right)$ means that the sequence $\left\{{\u03f5}_{k}\right\}$ is an infinitesimal of higher order than $\left\{{\gamma}_{n}\right\}$. Pick ${x}_{0},{x}_{1}\in C$ and set $k=1$.Step 1. Choose ${\theta}_{k}$ such that $0\le {\theta}_{k}\le {\overline{\theta}}_{k}$, where
$${\overline{\theta}}_{k}=\left(\right)open="\{"\; close>\begin{array}{c}min\left(\right)open="\{"\; close="\}">\alpha ,\frac{{\u03f5}_{k}}{\parallel {x}_{k}-{x}_{k-1}\parallel},\phantom{\rule{1.em}{0ex}}\mathrm{if}{x}_{k}\ne {x}_{k-1},\hfill \end{array}\alpha ,\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\mathrm{otherwise},\hfill $$
$${w}_{k}={x}_{k}+{\theta}_{k}({x}_{k}-{x}_{k-1}).$$
Step 2. Solve the strongly convex program
Step 3. Solve the strongly convex program
Step 4. Solve the strongly convex program
Step 5. Solve the strongly convex program
Step 6. Compute
$${t}_{k}={P}_{C}({z}_{k}+\eta {A}^{*}({v}_{k}-A{z}_{k})).$$
Step 7. The next approximation ${x}_{k+1}$ is defined by
$${x}_{k+1}=(1-{\beta}_{k}-{\gamma}_{k}){w}_{k}+{\beta}_{k}{t}_{k}.$$
Step 8. Put $k:=k+1$ and go to Step 1. |

**Theorem**

**2.**

**Proof.**

**Case 1.**Suppose that there exists ${k}_{0}\in \mathbb{N}$ such that $\parallel {x}_{k+1}-\tilde{p}\parallel \le \parallel {x}_{k}-\tilde{p}\parallel $, for all $k\ge {k}_{0}$. This means that $\{\parallel {x}_{k}-\tilde{p}{\parallel \}}_{k\ge {k}_{0}}$ is a nonincreasing sequence. Consequently, by using this one together with the boundness property of $\{\parallel {x}_{k}-\tilde{p}\parallel \}$, we have that the limit of $\parallel {x}_{k}-\tilde{p}\parallel $ exists. Since $\underset{k\to \infty}{lim}{\theta}_{k}\parallel {x}_{k}-{x}_{k-1}\parallel =0$ and the properties of the control sequences $\left\{{\beta}_{k}\right\}$, $\left\{{\gamma}_{k}\right\}$, $\left\{{\lambda}_{k}\right\}$, $\left\{{\theta}_{k}\right\}$, it follows from (45) that

**Case 2.**Suppose that there exists a subsequence $\{\parallel {x}_{{k}_{i}}-\tilde{p}\parallel \}$ of $\{\parallel {x}_{k}-\tilde{p}\parallel \}$ such that

## 4. Numerical Experiments

**Example**

**1.**

**Example**

**2.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Average CPU Times (s) | Average Iterations | ||||||
---|---|---|---|---|---|---|---|

Cases | IEM | MIEM | PPA | IEM | MIEM | PPA | |

1 | 3.1609 | 4.1125 | 1.7781 | 179.6 | 237.5 | 177.6 | |

2 | 2.7578 | 3.4859 | 164.6 | 207.9 | |||

3 | 2.5641 | 2.9828 | 148.7 | 177.9 | |||

4 | 2.3875 | 2.4531 | 131.8 | 146.7 | |||

5 | 2.0031 | 1.8766 | 110.9 | 112.6 |

Average CPU Times (s) | Average Iterations | |||||||
---|---|---|---|---|---|---|---|---|

Cases | IEM | MIEM | PPA | IEM | MIEM | PPA | ||

1 | 1.9609 | 2.5922 | 2.1891 | 79.7 | 111.4 | 88.2 | ||

2 | 1.7250 | 2.2891 | 71.5 | 94.6 | ||||

3 | 1.5281 | 1.8844 | 64.4 | 78.4 | ||||

4 | 1.4094 | 1.5313 | 59.2 | 63.6 |

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**MDPI and ACS Style**

Suantai, S.; Petrot, N.; Khonchaliew, M.
Inertial Extragradient Methods for Solving Split Equilibrium Problems. *Mathematics* **2021**, *9*, 1884.
https://doi.org/10.3390/math9161884

**AMA Style**

Suantai S, Petrot N, Khonchaliew M.
Inertial Extragradient Methods for Solving Split Equilibrium Problems. *Mathematics*. 2021; 9(16):1884.
https://doi.org/10.3390/math9161884

**Chicago/Turabian Style**

Suantai, Suthep, Narin Petrot, and Manatchanok Khonchaliew.
2021. "Inertial Extragradient Methods for Solving Split Equilibrium Problems" *Mathematics* 9, no. 16: 1884.
https://doi.org/10.3390/math9161884