# Shrinking Extragradient Method for Pseudomonotone Equilibrium Problems and Quasi-Nonexpansive Mappings

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition 1**

- (i)
- pseudocontractive if$${\parallel Tx-Ty\parallel}^{2}\le {\parallel x-y\parallel}^{2}+{\parallel (I-T)x-(I-T)y\parallel}^{2},\phantom{\rule{7.11317pt}{0ex}}\forall x,y\in C,$$
- (ii)
- Lipschitzian if there exists $L\ge 0$ such that$$\parallel Tx-Ty\parallel \le L\parallel x-y\parallel ,\phantom{\rule{7.11317pt}{0ex}}\forall x,y\in C.$$
- (iii)
- quasi-nonexpansive if $Fix\left(T\right)$ is nonempty and$$\parallel Tx-p\parallel \le \parallel x-p\parallel ,\phantom{\rule{7.11317pt}{0ex}}\forall x\in C,p\in Fix\left(T\right).$$
- (iv)
- $(\alpha ,\beta ,\gamma ,\delta )$-symmetric generalized hybrid if there exists $\alpha ,\beta ,\gamma ,\delta \in (-\infty ,+\infty )$ such that$${\alpha \parallel Tx-Ty\parallel}^{2}{+\beta (\parallel x-Ty\parallel}^{2}{+\parallel y-Tx\parallel}^{2}{)+\gamma \parallel x-y\parallel}^{2}\phantom{\rule{0ex}{0ex}}+\delta (\parallel x-Tx{\parallel}^{2}+\parallel y-Ty{\parallel}^{2})\le 0,\forall x,y\in C.$$

**Definition 2.**

**Definition 3**

**.**Let C be a nonempty closed convex subset of H and $f:C\times C\to (-\infty ,+\infty )$ be a bifunction. The bifunction f is said to be:

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

**Remark**

**1.**

- (A1)
- f is weakly continuous on $C\times C$ in the sense that, if $x,y\in C$ and $\{{x}_{k}\}$, $\{{y}_{k}\}$ are two sequences in C converge weakly to x and y respectively, then $f({x}_{k},{y}_{k})$ converges to $f(x,y)$;
- (A2)
- $f(x,\xb7)$ is convex and subdifferentiable on C for each fixed $x\in C$;
- (A3)
- f is psuedomonotone on C;
- (A4)
- f is Lipshitz-type continuous on C with constants ${L}_{1}>0$ and ${L}_{2}>0$.

**Lemma 1**

**.**Let $f:C\times C\to (-\infty ,+\infty )$ be satisfied $\left(A2\right)-\left(A4\right)$. If $EP(f,C)$ is nonempty set and $0<{\rho}_{0}<min\{\frac{1}{2{L}_{1}},\frac{1}{2{L}_{2}}\}$. Let ${x}_{0}\in C$. If ${y}_{0}$ and ${z}_{0}$ are defined by

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

**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)
- $\parallel {P}_{C}\left(x\right)-{P}_{C}{\left(y\right)\parallel}^{2}\le {\parallel x-y\parallel}^{2}-{\parallel {P}_{C}\left(x\right)-x+y-{P}_{C}\left(y\right)\parallel}^{2}$, $\forall x,y\in C$.

## 3. Main Result

**CSEM Algorithm**(Cyclic Shrinking Extragradient Method)

**Initialization.**Pick ${x}_{0}\in C=:{C}_{0}$, choose parameters $\{{\rho}_{k}\}$ with $0<inf{\rho}_{k}\le sup{\rho}_{k}<min\{\frac{1}{2{L}_{1}},\frac{1}{2{L}_{2}}\}$, $\{{\alpha}_{k}\}\subset [0,1]$ such that ${lim}_{k\to \infty}{\alpha}_{k}=1$, and $\{{\beta}_{k}\}$ with $0\le inf{\beta}_{k}\le sup{\beta}_{k}<1$.

**Step 1.**Solve the strongly convex program

**Step 2.**Solve the strongly convex program

**Step 3.**Compute

**Step 4.**Construct closed convex subset of C:

**Step 5.**The next approximation ${x}_{k+1}$ is defined as the projection of ${x}_{0}$ onto ${C}_{k+1}$, i.e.,

**Step 6.**Put $k=k+1$ and go to

**Step 1**.

**Lemma**

**3.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**PSEM Algorithm**(Parallel Shrinking Extragradient Method)

**Initialization.**Pick ${x}_{0}\in C=:{C}_{0}$, choose parameters $\left\{{\rho}_{k}^{i}\right\}$ with $0<inf{\rho}_{k}^{i}\le sup{\rho}_{k}^{i}<min\{\frac{1}{2{L}_{1}},\frac{1}{2{L}_{2}}\},i=1,2,\dots ,N$, $\left\{{\alpha}_{k}\right\}\subset [0,1]$ such that ${lim}_{k\to \infty}{\alpha}_{k}=1$, and $\left\{{\beta}_{k}\right\}$ with $0\le inf{\beta}_{k}\le sup{\beta}_{k}<1$.

**Step 1.**Solve N strongly convex programs

**Step 2.**Solve N strongly convex programs

**Step 3.**Find the farthest element from ${x}_{k}$ among ${z}_{k}^{i}$, $i=1,2,\dots ,N$, i.e.,

**Step 4.**Compute

**Step 5.**Find the farthest element from ${x}_{k}$ among ${u}_{k}^{j}$, $j=1,2,\dots ,M$, i.e.,

**Step 6.**Construct closed convex subset of C:

**Step 7.**The next approximation ${x}_{k+1}$ is defined as the projection of ${x}_{0}$ onto ${C}_{k+1}$, i.e.,

**Step 8.**Put $k=k+1$ and go to

**Step 1**.

**Theorem**

**2.**

**Proof.**

**Remark**

**2.**

## 4. A Numerical Experiment

**Remark**

**3.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Average Times (sec) | Average Iterations | |||||
---|---|---|---|---|---|---|

Cases | CSEM | PSEM | PHMEM | CSEM | PSEM | PHMEM |

1 | 4.905197 | 165.099794 | 173.347257 | 14.25 | 13.75 | 14.25 |

2 | 7.326055 | 287.918141 | 345.025914 | 25.25 | 24.25 | 28.25 |

3 | 20.371064 | 834.001035 | 2004.693844 | 91.25 | 74.25 | 177 |

4 | 5.079676 | 173.091716 | 173.347257 | 14.75 | 14.25 | 14.25 |

5 | 8.016109 | 342.870819 | 345.025914 | 28.75 | 28.25 | 28.25 |

6 | 42.035240 | 1986.147273 | 2004.693844 | 200 | 177 | 177 |

Average Times (sec) | Average Iterations | ||
---|---|---|---|

CSEM | PSEM | CSEM | PSEM |

4.657696 | 137.200812 | 12.50 | 11.50 |

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**MDPI and ACS Style**

Khonchaliew, M.; Farajzadeh, A.; Petrot, N.
Shrinking Extragradient Method for Pseudomonotone Equilibrium Problems and Quasi-Nonexpansive Mappings. *Symmetry* **2019**, *11*, 480.
https://doi.org/10.3390/sym11040480

**AMA Style**

Khonchaliew M, Farajzadeh A, Petrot N.
Shrinking Extragradient Method for Pseudomonotone Equilibrium Problems and Quasi-Nonexpansive Mappings. *Symmetry*. 2019; 11(4):480.
https://doi.org/10.3390/sym11040480

**Chicago/Turabian Style**

Khonchaliew, Manatchanok, Ali Farajzadeh, and Narin Petrot.
2019. "Shrinking Extragradient Method for Pseudomonotone Equilibrium Problems and Quasi-Nonexpansive Mappings" *Symmetry* 11, no. 4: 480.
https://doi.org/10.3390/sym11040480