Inertial ExtraGradient Method for Solving a Family of Strongly Pseudomonotone Equilibrium Problems in Real Hilbert Spaces with Application in Variational Inequality Problem
Abstract
:1. Introduction
2. Background
 (i).
 Strongly monotone if $f(u,v)+f(v,u)\le {\gamma \parallel uv\parallel}^{2},\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}u,v\in C;$
 (ii).
 Monotone if $f(u,v)+f(v,u)\le 0,\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}u,v\in C;$
 (iii).
 Strongly pseudomonotone if $f(u,v)\ge 0\u27f9f(v,u)\le {\gamma \parallel uv\parallel}^{2},\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}u,v\in C;$
 (iv).
 Pseudomonotone if $f(u,v)\ge 0\u27f9f(v,u)\le 0,\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}u,v\in C;$
 (v).
 Satisfying the Lipschitztype condition on C if there are two positive real numbers ${c}_{1},{c}_{2}$ such that$$f(u,w)\le f(u,v)+f(v,w)+{c}_{1}{\parallel uv\parallel}^{2}+{c}_{2}{\parallel vw\parallel}^{2},\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}u,v,w\in C.$$
3. Convergence Analysis for an Algorithm
 f_{1}.
 $f(u,u)=0$, for all $u\in C$ and f is strongly pseudomonotone on $C$.
 f_{2}.
 f satisfy the Lipschitztype condition through two positive constants ${c}_{1}$ and ${c}_{2}$.
 f_{3}.
 $f(u,.)$ is convex and subdifferentiable on C for each fixed $u\in C$.
Algorithm 1 (Inertial extragradient algorithm for strongly pseudomonotone equilibrium problems). 

 (i).
 Notice that if $\theta =0$, in the above method then it is equivalent to the default extragradient method in, e.g., [30].
 (ii).
 Evidently, from the expression (3) and (5) we have$$\sum _{n=1}^{\infty}{\vartheta}_{n}\parallel {u}_{n}{u}_{n1}\parallel \le \sum _{n=1}^{\infty}{\beta}_{n}\parallel {u}_{n}{u}_{n1}\parallel <\infty ,$$which implies that$$\underset{n\to \infty}{\mathrm{lim}}{\beta}_{n}\parallel {u}_{n}{u}_{n1}\parallel =0.$$
 i.
 Let ${u}_{0}\in C$ and compute$$\left\{\begin{array}{c}{v}_{n}=\underset{y\in C}{\mathrm{arg}\mathrm{min}}\{{\xi}_{n}f({u}_{n},y)+\frac{1}{2}\parallel {u}_{n}y{\parallel}^{2}\},\hfill \\ {u}_{n+1}=\underset{y\in C}{\mathrm{arg}\mathrm{min}}\{{\xi}_{n}f({v}_{n},y)+\frac{1}{2}\parallel {u}_{n}y{\parallel}^{2}\},\hfill \end{array}\right.$$$${T}_{1}:\phantom{\rule{1.em}{0ex}}\underset{n\to \infty}{\mathrm{lim}}{\xi}_{n}=0\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\mathit{and}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{T}_{2}:\phantom{\rule{1.em}{0ex}}\sum _{n=0}^{\infty}{\xi}_{n}=+\infty .$$
4. Application to Variational Inequality Problems
 strongly pseudomonotone upon C for $\gamma >0$ if$$\langle G\left(u\right),vu\rangle \ge 0\u27f9\langle G\left(v\right),uv\rangle \le {\gamma \parallel uv\parallel}^{2},\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}u,v\in C;$$
 LLipschitz continuous upon C if $\parallel G\left(u\right)G\left(v\right)\parallel \le L\parallel uv\parallel ,\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}u,v\in C.$
 G_{1}.
 G is strongly pseudomonotone on C and $VI(G,C)\ne \varnothing $;
 G_{2}.
 G is LLipschitz continuous upon C for some constant $L>0.$
 i.
 Choose ${u}_{1},{u}_{0}\in C,$$\theta \in [0,1)$ and a sequence $\{{\u03f5}_{n}\}\subset [0,+\infty )$ such that$$\sum _{n=0}^{+\infty}{\u03f5}_{n}<+\infty ,$$$${T}_{1}:\phantom{\rule{1.em}{0ex}}\underset{n\to \infty}{\mathrm{lim}}{\xi}_{n}=0\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\mathit{and}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{T}_{2}:\phantom{\rule{1.em}{0ex}}\sum _{n=0}^{\infty}{\xi}_{n}=+\infty .$$
 ii.
 Choose ${\vartheta}_{n}$ such that $0\le {\vartheta}_{n}\le {\beta}_{n}$ where$${\beta}_{n}=\left\{\begin{array}{cc}\mathrm{min}\left\{\theta ,\frac{{\u03f5}_{n}}{\parallel {u}_{n}{u}_{n1}\parallel}\right\}\hfill & \mathrm{if}\phantom{\rule{1.em}{0ex}}{u}_{n}\ne {u}_{n1},\hfill \\ \theta \hfill & \mathit{otherwise}.\hfill \end{array}\right.$$
 iii.
 Compute$$\left\{\begin{array}{c}{w}_{n}={u}_{n}+{\vartheta}_{n}({u}_{n}{u}_{n1}),\hfill \\ {v}_{n}={P}_{C}({w}_{n}{\xi}_{n}G\left({w}_{n}\right)),\hfill \\ {u}_{n+1}={P}_{C}({w}_{n}{\xi}_{n}G\left({v}_{n}\right)).\hfill \end{array}\right.$$
 i.
 Choose ${u}_{0}\in C$ and compute$$\left\{\begin{array}{c}{v}_{n}={P}_{C}({u}_{n}{\xi}_{n}G\left({u}_{n}\right)),\hfill \\ {u}_{n+1}={P}_{C}({u}_{n}{\xi}_{n}G\left({v}_{n}\right)),\hfill \end{array}\right.$$$${T}_{1}:\phantom{\rule{1.em}{0ex}}\underset{n\to \infty}{\mathrm{lim}}{\xi}_{n}=0\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\mathit{and}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{T}_{2}:\phantom{\rule{1.em}{0ex}}\sum _{n=0}^{\infty}{\xi}_{n}=+\infty .$$
5. Computational Experiment
 (i).
 For Hieu et al. [30] (shortly, Algo1), we use$${D}_{n}={\parallel {u}_{n}{v}_{n}\parallel}^{2}.$$
 (ii).
 For Hieu et al. [39] (shortly, Algo2), we use$${D}_{n}=\mathrm{max}\left\{\parallel {u}_{n+1}{v}_{n}{\parallel}^{2},{\parallel {u}_{n+1}{u}_{n}\parallel}^{2}\right\}.$$
 (iii).
 For Hieu et al. [40] (shortly, Algo3), we use ${\theta}_{n}=0.50$ and$${D}_{n}=\mathrm{max}\left\{\parallel {u}_{n+1}{v}_{n}{\parallel}^{2},{\parallel {u}_{n+1}{w}_{n}\parallel}^{2}\right\}.$$
 (iv).
 For Algorithm 1 (shortly, Algo4) we use ${\theta}_{n}=0.50,$${\u03f5}_{n}=\frac{1}{{n}^{2}}$ and$${D}_{n}={\parallel {w}_{n}{v}_{n}\parallel}^{2}.$$
5.1. Example 1
5.2. Example 2
 1.
 There is no need to have prior knowledge of Lipschitzconstant for running algorithms on Matlab.
 2.
 The convergence rate of the iterative sequence is based on the convergence rate of the stepsize sequence.
 3.
 The convergence rate of the iterative sequence also depends on the nature of the problem and the size of the problem.
 4.
 Due to the variable stepsize sequence, a particular value of the stepsize that is not suited to the current iteration of the algorithm often causes disturbance and hump in the behaviour of an iterative sequence.
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
 Blum, E. From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63, 123–145. [Google Scholar]
 Dafermos, S. Traffic equilibrium and variational inequalities. Transp. Sci. 1980, 14, 42–54. [Google Scholar] [CrossRef] [Green Version]
 Ferris, M.C.; Pang, J.S. Engineering and economic applications of complementarity problems. Siam Rev. 1997, 39, 669–713. [Google Scholar] [CrossRef] [Green Version]
 Patriksson, M. The Traffic Assignment Problem: Models and Methods; Courier Dover Publications: Mineola, NY, USA, 2015. [Google Scholar]
 Facchinei, F.; Pang, J.S. FiniteDimensional Variational Inequalities and Complementarity Problems; Springer Science & Business Media: Berlin, Germany, 2007. [Google Scholar]
 Konnov, I. Equilibrium Models and Variational Inequalities; Elsevier: Amsterdam, The Netherlands, 2007; Volume 210. [Google Scholar]
 Giannessi, F.; Maugeri, A.; Pardalos, P.M. Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models; Springer Science & Business Media: Berlin, Germany, 2006; Volume 58. [Google Scholar]
 Moudafi, A. Proximal point algorithm extended to equilibrium problems. J. Nat. Geom. 1999, 15, 91–100. [Google Scholar]
 Mastroeni, G. On auxiliary principle for equilibrium problems. In Equilibrium Problems and Variational Models; Springer: Berlin, Germany, 2003; pp. 289–298. [Google Scholar]
 Martinet, B. Brève communication. Régularisation d’inéquations variationnelles par approximations successives. ESAIM: Mathematical Modelling and Numerical Analysis—Modélisation Mathématique et Analyse Numérique 1970, 4, 154–158. [Google Scholar] [CrossRef] [Green Version]
 Rockafellar, R.T. Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 1976, 14, 877–898. [Google Scholar] [CrossRef] [Green Version]
 Konnov, I. Application of the proximal point method to nonmonotone equilibrium problems. J. Optim. Theory Appl. 2003, 119, 317–333. [Google Scholar] [CrossRef]
 Antipin, A.S. The convergence of proximal methods to fixed points of extremal mappings and estimates of their rate of convergence. Comput. Math. Math. Phys. 1995, 35, 539–552. [Google Scholar]
 Combettes, P.L.; Hirstoaga, S.A. Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 2005, 6, 117–136. [Google Scholar]
 Flåm, S.D.; Antipin, A.S. Equilibrium programming using proximallike algorithms. Math. Prog. 1996, 78, 29–41. [Google Scholar] [CrossRef]
 Van Hieu, D.; Anh, P.K.; Muu, L.D. Modified hybrid projection methods for finding common solutions to variational inequality problems. Comput. Optim. Appl. 2017, 66, 75–96. [Google Scholar] [CrossRef]
 Van Hieu, D. Halpern subgradient extragradient method extended to equilibrium problems. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales—Serie A: Matematicas 2017, 111, 823–840. [Google Scholar] [CrossRef]
 Argyros, I.K.; d Hilout, S. Computational Methods in Nonlinear Analysis: Efficient Algorithms, Fixed Point Theory and Applications; World Scientific: Singapore, 2013. [Google Scholar]
 Hieua, D.V. Parallel extragradientproximal methods for split equilibrium problems. Math. Model. Anal. 2016, 21, 478–501. [Google Scholar] [CrossRef]
 Iusem, A.N.; Sosa, W. Iterative algorithms for equilibrium problems. Optimization 2003, 52, 301–316. [Google Scholar] [CrossRef]
 Quoc, T.D.; Anh, P.N.; Muu, L.D. Dual extragradient algorithms extended to equilibrium problems. J. Glob. Optim. 2012, 52, 139–159. [Google Scholar] [CrossRef]
 Quoc Tran, D.; Le Dung, M.; Nguyen, V.H. Extragradient algorithms extended to equilibrium problems. Optimization 2008, 57, 749–776. [Google Scholar] [CrossRef]
 Santos, P.; Scheimberg, S. An inexact subgradient algorithm for equilibrium problems. Comput. Appl. Math. 2011, 30, 91–107. [Google Scholar]
 Takahashi, S.; Takahashi, W. Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 2007, 331, 506–515. [Google Scholar] [CrossRef] [Green Version]
 Ur Rehman, H.; Kumam, P.; Cho, Y.J.; Yordsorn, P. Weak convergence of explicit extragradient algorithms for solving equilibirum problems. J. Inequalities Appl. 2019, 2019, 1–25. [Google Scholar] [CrossRef]
 Argyros, I.K.; Cho, Y.J.; Hilout, S. Numerical Methods for Equations and Its Applications; CRC Press: Boca Raton, FL, USA, 2012. [Google Scholar]
 Rehman, H.U.; Kumam, P.; Abubakar, A.B.; Cho, Y.J. The extragradient algorithm with inertial effects extended to equilibrium problems. Comput. Appl. Math. 2020, 39, 100. [Google Scholar]
 Rehman, H.U.; Kumam, P.; Cho, Y.J.; Suleiman, Y.I.; Kumam, W. Modified Popov’s explicit iterative algorithms for solving pseudomonotone equilibrium problems. Optim. Methods. Softw. 2020, 1–32. [Google Scholar] [CrossRef]
 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. [Google Scholar] [CrossRef] [Green Version]
 Hieu, D.V. New extragradient method for a class of equilibrium problems in Hilbert spaces. Appl. Anal. 2017, 1–14. [Google Scholar] [CrossRef]
 Polyak, B.T. Some methods of speeding up the convergence of iteration methods. USSR Comput. Math. Math. Phys. 1964, 4, 1–17. [Google Scholar] [CrossRef]
 Beck, A.; Teboulle, M. A fast iterative shrinkagethresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2009, 2, 183–202. [Google Scholar] [CrossRef] [Green Version]
 Dong, Q.L.; Lu, Y.Y.; Yang, J. The extragradient algorithm with inertial effects for solving the variational inequality. Optimization 2016, 65, 2217–2226. [Google Scholar] [CrossRef]
 Thong, D.V.; Van Hieu, D. Modified subgradient extragradient method for variational inequality problems. Numer. Algorithms 2018, 79, 597–610. [Google Scholar] [CrossRef]
 Dong, Q.; Cho, Y.; Zhong, L.; Rassias, T.M. Inertial projection and contraction algorithms for variational inequalities. J. Glob. Optim. 2018, 70, 687–704. [Google Scholar] [CrossRef]
 Yang, J. Selfadaptive inertial subgradient extragradient algorithm for solving pseudomonotone variational inequalities. Appl. Anal. 2019, 1–12. [Google Scholar] [CrossRef]
 Thong, D.V.; Van Hieu, D.; Rassias, T.M. Self adaptive inertial subgradient extragradient algorithms for solving pseudomonotone variational inequality problems. Optim. Lett. 2020, 14, 115–144. [Google Scholar] [CrossRef]
 Vinh, N.T.; Muu, L.D. Inertial Extragradient Algorithms for Solving Equilibrium Problems. Acta Math. Vietnam. 2019, 44, 639–663. [Google Scholar] [CrossRef]
 Van Hieu, D. Convergence analysis of a new algorithm for strongly pseudomontone equilibrium problems. Numer. Algorithms 2018, 77, 983–1001. [Google Scholar] [CrossRef]
 Hieu, D.V.; Cho, Y.J.; bin Xiao, Y. Modified extragradient algorithms for solving equilibrium problems. Optimization 2018, 67, 2003–2029. [Google Scholar] [CrossRef]
 Goebel, K.; Reich, S. Uniform Convexity. Hyperbolic Geometry, and Nonexpansive; CRC Press: Boca Raton, FL, USA, 1984. [Google Scholar]
 Bianchi, M.; Schaible, S. Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 1996, 90, 31–43. [Google Scholar] [CrossRef]
 Tiel, J.V. Convex Analysis; John Wiley: Hoboken, NJ, USA, 1984. [Google Scholar]
 Bauschke, H.H.; Combettes, P.L. Convex Analysis and Monotone Operator Theory in Hilbert Spaces; Springer: Berlin, Germany, 2011; Volume 408. [Google Scholar]
 Tan, K.K.; Xu, H.K. Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 1993, 178, 301. [Google Scholar] [CrossRef] [Green Version]
 Ofoedu, E. Strong convergence theorem for uniformly LLipschitzian asymptotically pseudocontractive mapping in real Banach space. J. Math. Anal. Appl. 2006, 321, 722–728. [Google Scholar] [CrossRef] [Green Version]
Algo1 [30]  Algo2 [39]  Algo3 [40]  Algo4  

n  N  ${\xi}_{n}$  iter.  time  iter.  time  iter.  time  iter.  time 
10  10  $\frac{1}{n+1}$  83  0.8633  56  0.4295  35  0.2929  19  0.1319 
10  10  $\frac{\mathrm{log}(n+3)}{n+1}$  52  0.4297  64  0.4862  40  0.3040  23  0.1896 
10  10  $\frac{1}{\mathrm{log}(n+3)}$  94  0.8761  400  5.0501  305  3.4549  82  0.6732 
50  10  $\frac{1}{n+1}$  136  1.2545  107  0.9765  69  0.7521  54  0.4691 
50  10  $\frac{\mathrm{log}(n+3)}{n+1}$  86  0.6913  80  0.7453  55  0.4792  38  0.3128 
50  10  $\frac{1}{\mathrm{log}(n+3)}$  100  0.8427  205  2.2437  175  1.7925  86  0.7685 
100  10  $\frac{1}{n+1}$  222  3.0913  150  1.8105  105  1.1990  76  0.8656 
100  10  $\frac{\mathrm{log}(n+3)}{n+1}$  100  1.1624  92  1.0639  69  0.7964  36  0.4207 
100  10  $\frac{1}{\mathrm{log}(n+3)}$  113  1.3110  211  2.7524  188  2.4022  98  1.1311 
Algo1 [30]  Algo2 [39]  Algo3 [40]  Algo4  

n  N  ${\xi}_{n}$  iter.  time  iter.  time  iter.  time  iter.  time 
5  10  $\frac{1}{(n+1)\mathrm{log}(n+3)}$  212  2.5360  225  2.6580  179  3.3746  122  1.3161 
5  10  $\frac{1}{n+1}$  200  2.1717  254  3.4295  164  1.8637  137  1.7299 
5  10  $\frac{\mathrm{log}(n+3)}{n+1}$  181  2.6688  194  2.3646  158  1.8703  106  1.1469 
5  10  $\frac{1}{\sqrt{n+1}}$  89  0.9550  132  1.5186  72  0.7889  52  0.5644 
5  10  $\frac{1}{\mathrm{log}(n+3)}$  137  1.5127  152  1.8906  89  0.9427  80  0.8514 
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Rehman, H.u.; Kumam, P.; Argyros, I.K.; Deebani, W.; Kumam, W. Inertial ExtraGradient Method for Solving a Family of Strongly Pseudomonotone Equilibrium Problems in Real Hilbert Spaces with Application in Variational Inequality Problem. Symmetry 2020, 12, 503. https://doi.org/10.3390/sym12040503
Rehman Hu, Kumam P, Argyros IK, Deebani W, Kumam W. Inertial ExtraGradient Method for Solving a Family of Strongly Pseudomonotone Equilibrium Problems in Real Hilbert Spaces with Application in Variational Inequality Problem. Symmetry. 2020; 12(4):503. https://doi.org/10.3390/sym12040503
Chicago/Turabian StyleRehman, Habib ur, Poom Kumam, Ioannis K. Argyros, Wejdan Deebani, and Wiyada Kumam. 2020. "Inertial ExtraGradient Method for Solving a Family of Strongly Pseudomonotone Equilibrium Problems in Real Hilbert Spaces with Application in Variational Inequality Problem" Symmetry 12, no. 4: 503. https://doi.org/10.3390/sym12040503