The Inertial SubGradient ExtraGradient Method for a Class of PseudoMonotone Equilibrium Problems
Abstract
:1. Introduction
2. Preliminaries
 (i)
 strongly monotone if $f(u,v)+f(v,u)\le {\gamma \parallel uv\parallel}^{2},\phantom{\rule{0.166667em}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}u,v\in K;$
 (ii)
 monotone if $f(u,v)+f(v,u)\le 0,\phantom{\rule{0.166667em}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}u,v\in K;$
 (iii)
 strongly pseudomonotone if $f(u,v)\ge 0\u27f9f(v,u)\le {\gamma \parallel uv\parallel}^{2},\phantom{\rule{0.166667em}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}u,v\in K;$
 (iv)
 pseudomonotone if $f(u,v)\ge 0\u27f9f(v,u)\le 0,\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}u,v\in K;$
 (v)
 satisfying the Lipschitztype condition on K if there are two real numbers ${L}_{1},{L}_{2}>0,$ such that$$f(u,w)\le f(u,v)+f(v,w)+{L}_{1}{\parallel uv\parallel}^{2}+{L}_{2}{\parallel vw\parallel}^{2},\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}u,v,w\in K.$$
 (i)
 For all $u\in K,$ $v\in \mathbb{E},$$$\parallel u{P}_{K}{\left(v\right)\parallel}^{2}+\parallel {P}_{K}{\left(v\right)v\parallel}^{2}\le {\parallel uv\parallel}^{2}.$$
 (ii)
 $w={P}_{K}\left(u\right)$ if and only if$$\langle uw,vw\rangle \le 0.$$
 $\left(i\right)$
 ${\sum}_{n=1}^{+\infty}{[{a}_{n}{a}_{n1}]}_{+}<\infty ,$ with ${\left[q\right]}_{+}:=max\{q,0\};$
 $\left(ii\right)$
 ${lim}_{n\to +\infty}{a}_{n}={a}^{*}\in [0,\infty ).$
 $\left(i\right)$
 For each $\eta \in K,$ ${lim}_{n\to \infty}\parallel {\eta}_{n}\eta \parallel $ exists;
 $\left(ii\right)$
 All sequentially weak cluster point of $\left\{{\eta}_{n}\right\}$ lies in K;
 ${f}_{1}.$
 $f(v,v)=0,$ for all $v\in K$ and f is pseudomontone on a set $K.$
 ${f}_{2}.$
 f satisfy the Lipschitztype condition on $\mathbb{E}$ through positive constants ${L}_{1}$ and ${L}_{2}.$
 ${f}_{3}.$
 $\underset{n\to \infty}{lim\; sup}f({u}_{n},v)\le f({u}^{*},v)$ for each $v\in K$ and $\left\{{u}_{n}\right\}\subset K$ satisfy ${u}_{n}\rightharpoonup {u}^{*}.$
 ${f}_{4}.$
 $f(u,.)$ need to be convex and subdifferentiable on K for arbitrary $u\in K.$
3. An Inertial SubGradient ExtraGradient Method and Its Convergence Analysis
Algorithm 1 Inertial subgradient extragradient method for pseudomontone (EP). 

 i.
 Given ${u}_{0}\in \mathbb{E},$ $\varrho \in (0,1),$ $\sigma <min\{1,\frac{1}{2{c}_{1}},\frac{1}{2{c}_{2}}\},$ $\mu \in (0,\sigma )$ and ${\zeta}_{0}>0.$
 ii.
 Compute$$\left\{\begin{array}{c}{v}_{n}=\underset{y\in K}{argmin}\{{\zeta}_{n}f({u}_{n},y)+\frac{1}{2}\parallel {u}_{n}y{\parallel}^{2}\},\hfill \\ {u}_{n+1}=\underset{y\in {\Pi}_{n}}{argmin}\{\mu {\zeta}_{n}f({v}_{n},y)+\frac{1}{2}\parallel {u}_{n}y{\parallel}^{2}\},\hfill \end{array}\right.$$$$\begin{array}{cc}\hfill {\zeta}_{n+1}=\phantom{\rule{1.em}{0ex}}& min\left\{\sigma ,\frac{\mu f({v}_{n},{u}_{n+1})}{f({u}_{n},{u}_{n+1})f({u}_{n},{v}_{n}){c}_{1}\parallel {u}_{n}{v}_{n}{\parallel}^{2}{c}_{2}{\parallel {u}_{n+1}{v}_{n}\parallel}^{2}+1}\right\}.\hfill \end{array}$$
4. Solution for Variational Inequality Problems
 monotone on K if $\langle G\left(u\right)G\left(v\right),uv\rangle \ge 0,\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}u,v\in K;$
 pseudomonotone on K if $\langle G\left(u\right),vu\rangle \ge 0\Rightarrow \langle G\left(v\right),uv\rangle \le 0,\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}u,v\in K;$
 LLipschitz continuous on K if $\parallel G\left(u\right)G\left(v\right)\parallel \le L\parallel uv\parallel ,\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}u,v\in K.$
 ${G}_{1}^{*}$.
 G is monotone on K and $VI(G,K)$ is nonempty;
 ${G}_{1}$.
 G is pseudomonotone on K and $VI(G,K)$ is nonempty;
 ${G}_{2}$.
 G is LLipschitz continuous on K for constant $L>0.$
 ${G}_{3}$.
 $\underset{n\to \infty}{lim\; sup}\langle G\left({u}_{n}\right),v{u}_{n}\rangle \le \langle G\left({u}^{*}\right),v{u}^{*}\rangle $ for every $v\in K$ and $\left\{{u}_{n}\right\}\subset K$ satisfying ${u}_{n}\rightharpoonup {u}^{*}.$
 i.
 Choose ${u}_{1},{u}_{0}\in \mathbb{E},$ $\varrho \in (0,1),$ $\sigma <min\left\{\frac{13\vartheta}{{(1\vartheta )}^{2}},\frac{1}{L}\right\},$ $\mu \in (0,\sigma ),$ ${\zeta}_{0}>0$ and nondecreasing sequence $0\le {\vartheta}_{n}\le \vartheta \in [0,\frac{1}{3}).$
 ii.
 Given ${u}_{n1},{u}_{n}$ and compute$$\left\{\begin{array}{c}{v}_{n}={P}_{K}({t}_{n}{\zeta}_{n}G\left({t}_{n}\right)),\phantom{\rule{1.em}{0ex}}\mathit{where}\phantom{\rule{1.em}{0ex}}{t}_{n}={u}_{n}+{\vartheta}_{n}({u}_{n}{u}_{1}),\hfill \\ {u}_{n+1}={P}_{{\Pi}_{n}}({t}_{n}\mu {\zeta}_{n}G\left({v}_{n}\right)),\hfill \end{array}\right.$$$$\begin{array}{cc}\hfill {\zeta}_{n+1}=\phantom{\rule{1.em}{0ex}}& min\left\{\sigma ,\frac{\mu \langle G{v}_{n},{u}_{n+1}{v}_{n}\rangle}{\langle G{t}_{n},{u}_{n+1}{v}_{n}\rangle \frac{L}{2}\parallel {t}_{n}{v}_{n}{\parallel}^{2}\frac{L}{2}{\parallel {u}_{n+1}{v}_{n}\parallel}^{2}+1}\right\}.\hfill \end{array}$$
 i.
 Choose ${u}_{0}\in \mathbb{E},$ $\varrho \in (0,1),$ $\sigma <min\left\{1,\frac{1}{L}\right\},$ $\mu \in (0,\sigma )$ and ${\zeta}_{0}>0.$
 ii.
 Given ${u}_{n}$ and compute$$\left\{\begin{array}{c}{v}_{n}={P}_{K}({u}_{n}{\zeta}_{n}G\left({u}_{n}\right)),\hfill \\ {u}_{n+1}={P}_{{\Pi}_{n}}({u}_{n}\mu {\zeta}_{n}G\left({v}_{n}\right)),\hfill \end{array}\right.$$$$\begin{array}{cc}\hfill {\zeta}_{n+1}=\phantom{\rule{1.em}{0ex}}& min\left\{\sigma ,\frac{\mu \langle G{v}_{n},{u}_{n+1}{v}_{n}\rangle}{\langle G{u}_{n},{u}_{n+1}{v}_{n}\rangle \frac{L}{2}\parallel {u}_{n}{v}_{n}{\parallel}^{2}\frac{L}{2}{\parallel {u}_{n+1}{v}_{n}\parallel}^{2}+1}\right\}.\hfill \end{array}$$
 i.
 Take ${u}_{1},{u}_{0}\in \mathbb{E},$ $\varrho \in (0,1),$ $\sigma <min\left\{\frac{13\vartheta}{{(1\vartheta )}^{2}},\frac{1}{L}\right\},$ $\mu \in (0,\sigma ),$ ${\zeta}_{0}>0$ and nondecreasing sequence $0\le {\vartheta}_{n}\le \vartheta \in [0,\frac{1}{3}).$
 ii.
 Given ${u}_{n1},{u}_{n}$ and compute$$\left\{\begin{array}{c}{v}_{n}={P}_{K}({t}_{n}{\zeta}_{n}G\left({t}_{n}\right)),\phantom{\rule{1.em}{0ex}}\mathit{where}\phantom{\rule{1.em}{0ex}}{t}_{n}={u}_{n}+{\vartheta}_{n}({u}_{n}{u}_{1}),\hfill \\ {u}_{n+1}={P}_{{\Pi}_{n}}({t}_{n}\mu {\zeta}_{n}G\left({v}_{n}\right)),\hfill \end{array}\right.$$$$\begin{array}{cc}\hfill {\zeta}_{n+1}=\phantom{\rule{1.em}{0ex}}& min\left\{\sigma ,\frac{\mu \langle G{v}_{n},{u}_{n+1}{v}_{n}\rangle}{\langle G{t}_{n},{u}_{n+1}{v}_{n}\rangle \frac{L}{2}\parallel {t}_{n}{v}_{n}{\parallel}^{2}\frac{L}{2}{\parallel {u}_{n+1}{v}_{n}\parallel}^{2}+1}\right\}.\hfill \end{array}$$
 i.
 Choose ${u}_{0}\in \mathbb{E},$ $\varrho \in (0,1),$ $\sigma <min\left\{1,\frac{1}{L}\right\},$ $\mu \in (0,\sigma )$ and ${\zeta}_{0}>0.$
 ii.
 Given ${u}_{n}$ and compute$$\left\{\begin{array}{c}{v}_{n}={P}_{K}({u}_{n}{\zeta}_{n}G\left({u}_{n}\right)),\hfill \\ {u}_{n+1}={P}_{{\Pi}_{n}}({u}_{n}\mu {\zeta}_{n}G\left({v}_{n}\right)),\hfill \end{array}\right.$$$$\begin{array}{cc}\hfill {\zeta}_{n+1}=\phantom{\rule{1.em}{0ex}}& min\left\{\sigma ,\frac{\mu \langle G{v}_{n},{u}_{n+1}{v}_{n}\rangle}{\langle G{u}_{n},{u}_{n+1}{v}_{n}\rangle \frac{L}{2}\parallel {u}_{n}{v}_{n}{\parallel}^{2}\frac{L}{2}{\parallel {u}_{n+1}{v}_{n}\parallel}^{2}+1}\right\}.\hfill \end{array}$$
5. Computational Experiment
5.1. Example 1
5.2. Example 2
5.3. Example 3
5.4. Example 4
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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Tran.EgA  Dadshi.EgA  Int.EgA  

n  Iter.  CPU(s)  Iter.  CPU(s)  Iter.  CPU(s) 
5  69  0.5508  28  0.2356  13  0.1145 
10  124  1.2234  101  0.8967  51  0.4338 
20  283  3.4558  223  2.5063  155  1.4874 
40  379  5.1930  259  3.0970  177  1.7652 
Tran.EgA  Dadshi.EgA  Int.EgA  

u_{0} = v_{0}  Iter.  CPU(s)  Iter.  CPU(s)  Iter.  CPU(s) 
$(1.5,1.7)$  86  2.9587  62  1.9962  40  1.6405 
$(2.0,3.0)$  89  3.5329  71  2.0817  46  1.4633 
$(1.0,2.0)$  99  3.4713  73  2.2057  52  1.5730 
$(2.7,2.6)$  71  2.7353  55  1.9161  36  1.2266 
Tran.EgA  Dadshi.EgA  Int.EgA  

u_{0} = v_{0}  Iter.  CPU(s)  Iter.  CPU(s)  Iter.  CPU(s) 
5  198  4.6833  136  2.0156  78  1.1475 
10  498  13.9149  190  2.3003  94  0.8930 
20  1471  35.1972  119  1.5241  65  0.8603 
Tran.EgA  Dadshi.EgA  Int.EgA  

u_{0} = v_{0}  Iter.  CPU(s)  Iter.  CPU(s)  Iter.  CPU(s) 
$(1,1,\cdots ,{1}_{5000},0,0,\cdots )$  69  0.5508  28  0.2356  13  0.1145 
$(1,2,\cdots ,5000,0,0,\cdots )$  124  1.2234  101  0.8967  51  0.4338 
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Rehman, H.u.; Kumam, P.; Kumam, W.; Shutaywi, M.; Jirakitpuwapat, W. The Inertial SubGradient ExtraGradient Method for a Class of PseudoMonotone Equilibrium Problems. Symmetry 2020, 12, 463. https://doi.org/10.3390/sym12030463
Rehman Hu, Kumam P, Kumam W, Shutaywi M, Jirakitpuwapat W. The Inertial SubGradient ExtraGradient Method for a Class of PseudoMonotone Equilibrium Problems. Symmetry. 2020; 12(3):463. https://doi.org/10.3390/sym12030463
Chicago/Turabian StyleRehman, Habib ur, Poom Kumam, Wiyada Kumam, Meshal Shutaywi, and Wachirapong Jirakitpuwapat. 2020. "The Inertial SubGradient ExtraGradient Method for a Class of PseudoMonotone Equilibrium Problems" Symmetry 12, no. 3: 463. https://doi.org/10.3390/sym12030463