# A Mean Extragradient Method for Solving Variational Inequalities

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. Mann’s Type Mean Extragradient Algorithm

- (A1)
- for all $k,j\ge 1$, ${\alpha}_{k,j}\ge 0$;
- (A2)
- for all $k\ge 1$, if $j>k$, then ${\alpha}_{k,j}=0$;
- (A3)
- for all $k\ge 1$, ${\sum}_{j=1}^{k}{\alpha}_{k,j}=1$;
- (A4)
- for all $j\ge 1$, ${lim}_{k\to +\infty}{\alpha}_{k,j}=0$.

Algorithm 1: Mann’s type mean extragradient method (Mann-MEM). |

Initialization: Select a point ${x}_{1}\in \mathcal{H}$, a parameter $\tau >0$ and an averaging matrix ${\left[{\alpha}_{k,j}\right]}_{k,j=1}^{\infty}$. Step 1: Given a current iterate ${x}_{k}\in \mathcal{H}$, compute the mean iterate
$${\overline{x}}_{k}=\sum _{j=1}^{k}{\alpha}_{k,j}{x}_{j}.$$
Compute
$${y}_{k}={P}_{C}({\overline{x}}_{k}-\tau F\left({\overline{x}}_{k}\right)).$$
Step 2: If ${y}_{k}={\overline{x}}_{k}$, then ${\overline{x}}_{k}\in \mathrm{VIP}(F,C)$ and STOP. If not, construct the half-space ${T}_{k}$ defined by
$${T}_{k}:=\{w\in \mathcal{H}:\langle ({\overline{x}}_{k}-\tau F\left({\overline{x}}_{k}\right))-{y}_{k},w-{y}_{k}\rangle \le 0\},$$
and calculate the next iterate
$${x}_{k+1}={P}_{{T}_{k}}({\overline{x}}_{k}-\tau F\left({y}_{k}\right)).$$
Update $k=k+1$ and go to Step 1. |

**Remark**

**1.**

**Proposition**

**1.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Proposition**

**2.**

**Theorem**

**1.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Example**

**1.**

## 4. Numerical Result

## 5. Some Concluding Remarks

- (i)
- Let us observe that the convergence of Mann-MEM requires us to know the Lipschitz constant L of the operator F, nevertheless, it is sometimes difficult to indicate exactly the Lipschitz constant, so that Mann-MEM and its convergence result can not be practically applicable. It is very interesting to consider a variant Mann-MEM with a variable stepsize ${\left\{{\tau}_{k}\right\}}_{k=1}^{\infty}$ in place of the fixed stepsize $\tau >0$ and the prior knowledge of L is not necessarily known.
- (ii)
- It can be noted that the superiority of Mann-MEM with respect to SEM is depended on the optimal choice of the averaging matrix ${\left[{\alpha}_{k,j}\right]}_{k,j=1}^{\infty}$. It is also very interesting to find more possible examples of averaging matrices satisfying the M-concentrating condition.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Influences of the stepsize ${\lambda}_{k}=\lambda /(k+1)$ for several paramters $\lambda >0$ when performing subgradient–extragradient method (SEM) and Mann mean extragradient method (Mann-MEM).

Method | $\mathit{\lambda}$ | Time | #(Iters) | #(Inner) |
---|---|---|---|---|

SEM | 1.3 | 0.1826 | 14 | 47,630 |

1.4 | 0.1501 | 15 | 37,476 | |

1.5 | 0.1177 | 15 | 28,591 | |

1.6 | 0.0906 | 16 | 23,691 | |

1.7 | >0.2871 | >100 | >84,555 | |

1.8 | 0.0899 | 30 | 23,648 | |

1.9 | 0.0699 | 28 | 17,749 | |

Mann-MEM | 1.3 | >1.0595 | >100 | >319,322 |

1.4 | >0.7589 | >100 | >224,422 | |

1.5 | 0.1180 | 18 | 33,285 | |

1.6 | 0.0924 | 18 | 25,846 | |

1.7 | 0.0906 | 19 | 21,495 | |

1.8 | 0.0851 | 30 | 23,508 | |

1.9 | 0.0607 | 23 | 15,925 |

**Table 2.**Behavior of SEM, Mann-MEM for different dimensions (n) and different number of constraints (m).

n | m | SEM | Mann-MEM | ||
---|---|---|---|---|---|

Time | #(Iters) | Time | #(Iters) | ||

500 | 50 | 38.7986 | 51.0 | 36.3368 | 51.2 |

100 | 94.3647 | 51.0 | 88.4383 | 51.0 | |

200 | 248.4960 | 50.6 | 239.0405 | 51.0 | |

1000 | 50 | 61.8089 | 52.0 | 58.6253 | 53.0 |

100 | 143.5451 | 52.0 | 137.0350 | 53.0 | |

200 | 368.2668 | 52.0 | 344.8198 | 52.7 | |

2000 | 50 | 123.3731 | 53.1 | 118.4089 | 54.0 |

100 | 257.4444 | 53.0 | 245.7529 | 54.0 | |

200 | 604.0555 | 53.0 | 576.3775 | 54.0 | |

3000 | 50 | 247.8855 | 54.0 | 242.2706 | 55.0 |

100 | 452.0647 | 54.0 | 440.8821 | 55.0 | |

200 | 1070.5699 | 54.0 | 1031.5349 | 55.0 |

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Buakird, A.; Nimana, N.; Petrot, N.
A Mean Extragradient Method for Solving Variational Inequalities. *Symmetry* **2021**, *13*, 462.
https://doi.org/10.3390/sym13030462

**AMA Style**

Buakird A, Nimana N, Petrot N.
A Mean Extragradient Method for Solving Variational Inequalities. *Symmetry*. 2021; 13(3):462.
https://doi.org/10.3390/sym13030462

**Chicago/Turabian Style**

Buakird, Apichit, Nimit Nimana, and Narin Petrot.
2021. "A Mean Extragradient Method for Solving Variational Inequalities" *Symmetry* 13, no. 3: 462.
https://doi.org/10.3390/sym13030462