An Alternated Inertial Projection Algorithm for MultiValued Variational Inequality and Fixed Point Problems
Abstract
:1. Introduction and Preliminaries
2. Main Results
Algorithm 1 Choose parameters $\sigma ,l\in (0,1)$, ${x}_{0},{x}_{1}\in C$ as initial points. 
Step 1 Take arbitrarily ${\xi}_{n}\in F\left({\omega}_{n}\right)$, if ${\omega}_{n}{P}_{C}({\omega}_{n}{\xi}_{n})=0$ and $T\left({\omega}_{n}\right)={\omega}_{n}$, then stop, otherwise, go to step 2, where
$$\begin{array}{c}\hfill {\omega}_{n}=\left\{\begin{array}{ccc}& {x}_{n},\hspace{1em}\hspace{1em}n\hspace{1em}\hspace{1em}\mathrm{is}\hspace{1em}\hspace{1em}\mathrm{even};\hfill & \\ & {x}_{n}+\theta ({x}_{n}{x}_{n1}),\hspace{1em}\hspace{1em}n\hspace{1em}\hspace{1em}\mathrm{is}\hspace{1em}\hspace{1em}\mathrm{odd}.\hfill & \end{array}\right.\end{array}$$

Step 2 Let ${m}_{n}$ is the smallest nonegative integer m such that
$${l}^{m}\langle {\xi}_{n}{t}_{n}\left(m\right),{\omega}_{n}{y}_{n}\left(m\right)\rangle \le \sigma {\parallel {\omega}_{n}{y}_{n}\left(m\right)\parallel}^{2},$$

where ${y}_{n}\left(m\right)={P}_{C}({\omega}_{n}{l}^{m}{\xi}_{n})$, ${t}_{n}\left(m\right)={P}_{F\left({y}_{n}\left(m\right)\right)}\left({\xi}_{n}\right)$, ${t}_{n}={t}_{n}\left({m}_{n}\right)$ and ${y}_{n}={y}_{n}\left({m}_{n}\right)$, ${\alpha}_{n}={l}^{{m}_{n}}.$ 
Step 3 Compute ${d}_{n}={P}_{{H}_{n}}\left({\omega}_{n}\right)$, where
$${h}_{n}\left(v\right)=\langle {\omega}_{n}{y}_{n}{\alpha}_{n}({\xi}_{n}{t}_{n}),v{y}_{n}\rangle ,$$
$${H}_{n}=\{v:{h}_{n}\left(v\right)\le 0\}.$$

Step 4 Compute ${x}_{n+1}=(1{\beta}_{n}){d}_{n}+{\beta}_{n}T{d}_{n}$. Set $n=n+1$ and go to Step 1. 
3. Numerical Examples
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Algorithm 1  

e  iter  Cpu  ${x}^{*}$ 
${10}^{1}$  7  2.1875  (0.8854; 0.0465; 0.0215; 0.0463) 
${10}^{2}$  17  2.9218  (0.9891; 0.0052; 0.0004; 0.0051) 
${10}^{3}$  26  3.4218  (0.9989; 0.0005; 7.6653$\times {10}^{6}$; 0.0005) 
${10}^{3.5}$  30  3.7500  (0.9994; 0.0002; 9.3523$\times {10}^{5}$; 0.0002) 
Algorithm 1  Algorithm 1 [1]  Algorithm 3.1 [31]  

Case I  iter  15  23  29 
Cpu  2.7340  3.1093  3.3656  
Case II  iter  16  21  28 
Cpu  2.7968  3.0781  3.9218 
e  Algorithm 1  Algorithm 1 [1]  Algorithm 3.1 [31]  

${10}^{1}$  iter  5  6  9 
cpu  1.5625  1.7031  1.8750  
${10}^{2}$  iter  8  9  15 
cpu  1.7812  1.8281  2.0000  
${10}^{3}$  iter  12  12  20 
cpu  1.8750  2.3125  2.2343  
${10}^{3}$  iter  15  16  33 
cpu  1.9375  2.4218  2.4843  
${10}^{5}$  iter  18  19  52 
cpu  2.1406  2.4687  3.0468  
${10}^{6}$  iter  21  22   
cpu  2.2343  2.5156    
${10}^{7}$  iter  24  25   
cpu  2.4062  2.5468   
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Zhang, H.; Liu, X.; Sun, Y.; Hu, J. An Alternated Inertial Projection Algorithm for MultiValued Variational Inequality and Fixed Point Problems. Mathematics 2023, 11, 1850. https://doi.org/10.3390/math11081850
Zhang H, Liu X, Sun Y, Hu J. An Alternated Inertial Projection Algorithm for MultiValued Variational Inequality and Fixed Point Problems. Mathematics. 2023; 11(8):1850. https://doi.org/10.3390/math11081850
Chicago/Turabian StyleZhang, Huan, Xiaolan Liu, Yan Sun, and Ju Hu. 2023. "An Alternated Inertial Projection Algorithm for MultiValued Variational Inequality and Fixed Point Problems" Mathematics 11, no. 8: 1850. https://doi.org/10.3390/math11081850