Strong convergence of a new hybrid algorithm for fixed point problems and equilibrium problems
Abstract
The paper considers the problem of finding a common solution of a pseudomonotone and Lipschitz-type equilibrium problem and a fixed point problem for a quasi nonexpansive mapping in a Hilbert space. A new hybrid algorithm is introduced for approximating a solution of this problem. The presented algorithm can be considered as a combination of the extragradient method (two-step proximal-like method) and a modified version of the normal Mann iteration. It is well known that the normal Mann iteration has the weak convergence, but in this paper we has obtained the strong convergence of the new algorithm under some mild conditions on parameters. Several numerical experiments are reported to illustrate the convergence of the algorithm and also to show the advantages of it over existing methods.
First Published Online: 21 Nov 2018
Keyword : proximal-like method, extragradient method, equilibrium problem, fixed pointproblem
How to Cite
Hieu, D. V. (2019). Strong convergence of a new hybrid algorithm for fixed point problems and equilibrium problems. Mathematical Modelling and Analysis, 24(1), 1-19. https://doi.org/10.3846/mma.2019.001
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
P.N. Anh. A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optimization, 62(2):271–283, 2013. https://doi.org/10.1080/02331934.2011.607497.
A.S. Antipin. On convergence of proximal methods to fixed points of extremal mappings and estimates of their rate of convergence. Comp. Maths. Math. Phys., 35:539–551, 1995.
H.H. Bauschke and J.A. Chen. Projection method for approximating fixed points of quasi - non expansive mappings without the usual demiclosedness condition. J. Nonlinear Convex Anal.,15:129–135, 2014.
H.H. Bauschke and P.L. Combettes. Combettes. Convex analysis and monotone operator theory in Hilbert Spaces. New York: Springer, 2001.
E. Blum and W. Oettli. From optimization and variational inequalities to equi-librium problems. Math. Student,63:123–145, 1994.
P L. Combettes and S.A. Hirstoaga. Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal., 6:117–136, 2005.
Y. Dang and Y. Gao. The strong convergence of a KM-CQ-like algo-rithm for a split feasibility problem. Inverse Problems, 27(1):015007, 2011. https://doi.org/10.1088/0266-5611/27/1/015007.
F. Facchinei and J.S. Pang. Finite-dimensional variational inequalities and com-plementarity problems. Berlin: Springer, 2002.
K. Fan. A minimax inequality and applications. In: Shisha, O. (ed.) Inequality,III, Academic Press, New York, pp. 103–113, 1972.
S.D. Flam and A.S. Antipin. Equilibrium programming and proximal-like algo-rithms. Math. Program., 78:29–41, 1997.https://doi.org/10.1007/BF02614504.
D.V. Hieu. An extension of hybrid method without extrapolation step to equi-librium problems. J. Ind. Manag. Optim., 13:1723–1741, 2017.
D.V. Hieu. Hybrid projection methods for equilibrium problems with non-Lipschitz type bifunctions. Math. Meth. Appl. Sci., 40:4065–4079, 2017. https://doi.org/10.1002/mma.4286.
D.V. Hieu.New subgradient extragradient methods for common solu-tions to equilibrium problems. Comput. Optim. Appl., 67(3):571–594, 2017. https://doi.org/10.1007/s10589-017-9899-4.
D.V. Hieu.Convergence analysis of a new algorithm for strongly pseu-domontone equilibrium problems. Numer. Algorithms., 77(4):983–1001, 2018. https://doi.org/10.1007/s11075-017-0350-9.
D.V. Hieu, L.D. Muu and P.K. Anh. Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings. Numer. Algorithms., 73(1):197–217, 2016. https://doi.org/10.1007/s11075-015-0092-5.
H. Iiduka and I. Yamada. A subgradient-type method for the equilibrium prob-lem over the fixed point set and its applications. Optimization,58(2):251–261,2009. https://doi.org/10.1080/02331930701762829.
I.V. Konnov. Application of the proximal point method to nonmonotone equilib-rium problems. Journal of Optimization Theory and Applications,119(2):317–333, 2003. https://doi.org/10.1023/B:JOTA.0000005448.12716.24.
I.V. Konnov. Equilibrium models and variational inequalities. Amsterdam: El-sevier, 2007.
G.M. Korpelevich. The extragradient method for finding saddle points and otherproblems. Ekonomikai Matematicheskie Metody, 12:747–756, 1976.
P. Kumam and P. Katchang. A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point prob-lems for nonexpansive mappings. Nonlinear Analysis: Hybrid Systems, 3(4):475–486, 2009. https://doi.org/10.1016/j.nahs.2009.03.006.
W. Kumam, H. Piri and P. Kumam. Solutions of system of equilibrium andvariational inequality problems on fixed points of infinite family of nonexpan-sive mappings. Applied Mathematics and Computation, 248(1):441–455, 2014. https://doi.org/10.1016/j.amc.2014.09.118.
M. Li and Y. Yao. Strong convergence of an iterative algorithm forλ-strictlypseudo-contractive mappings in hilbert spaces. An. St. Univ. Ovidius Constanta., 18(1):219–228, 2010.
S.I. Lyashko and V.V. Semenov. A new two-step proximal algorithm of solving the problem of equilibrium programming. Optimization and its applications in control and data sciences. Springer, Switzerland,115:315–325, 2016.
P.E. Maing ́e. A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim., 47(3):1499–1515, 2008. https://doi.org/10.1137/060675319.
P.E. Maing ́e and A. Moudafi. Coupling viscosity methods with the extragradient algorithm for solving equilibrium problems. J. Nonlinear Convex Anal.,9:283–294, 2008.
G. Mastroeni. On auxiliary principle for equilibrium problems. Publicatione del Dipartimento di Mathematica dell, Universita di Pisa,3:1244–1258, 2000.
G. Mastroeni. Gap function for equilibrium problems. J. Global. Optim., 27(4):411–426, 2003.https://doi.org/10.1023/A:1026050425030.
A. Moudafi. Proximal point algorithm extended to equilibrum problem. J. Nat. Geometry,15:91–100, 1999.
A. Moudafi. Viscosity approximation methods for fixed point problems. J. Math.Anal. Appl., 241(1):46–55, 2000.https://doi.org/10.1006/jmaa.1999.6615.
L.D. Muu and W. Oettli. Convergence of an adative penalty scheme forfinding constrained equilibria. Nonlinear Anal. TMA,18(2):1159–1166, 1992. https://doi.org/10.1016/0362-546X(92)90159-C.
T.T.V. Nguyen, J.J. Strodiot and V.H. Nguyen. Hybrid methods for solv-ing simultaneously an equilibrium problem and countably many fixed point problems in a Hilbert space .J. Optim. Theory Appl.,160(3):809–831, 2014. https://doi.org/10.1007/s10957-013-0400-y.
T.D. Quoc, L.D. Muu and V.H. Nguyen. Extragradient algorithms extended to mequilibrium problems. Optimization,57(6):749–776, 2008. https://doi.org/10.1080/02331930601122876.
S. Reich. Constructive techniques for accretive and monotone operators. Applied Nonlinear Analysis, Academic Press, New York, pp. 335–345, 1979.
J.J. Strodiot, P.T. Vuong and T.T.V. Nguyen. A class of shrinking pro-jection extragradient methods for solving non-monotone equilibrium prob-lems in hilbert spaces. Journal of Global Optimization, 64(1):159–178, 2016.https://doi.org/10.1007/s10898-015-0365-5.
S. Takahashi and W. Takahashi. Viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces. Jour-nal of Mathematical Analysis and Applications,331(1):506–515, 2007.https://doi.org/10.1016/j.jmaa.2006.08.036.
P.T. Vuong, J.J. Strodiot and V.H. Nguyen. Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point prob-lems. Journal of Optimization Theory and Applications,155(2):605–627,2012.https://doi.org/10.1007/s10957-012-0085-7.
P.T. Vuong, J.J. Strodiot and V.H. Nguyen. On extragradient-viscosity methods for solving equilibrium and fixed point problems in a hilbert space. Optimization, 64(2):429–451, 2015. https://doi.org/10.1080/02331934.2012.759327.
H. K. Xu. Iterative algorithms for nonlinear operators. Journal of London Mathematical Society,66(1):240–256,2002. https://doi.org/10.1112/S0024610702003332.
I. Yamada and N. Ogura. Hybrid steepest descent method for the variational inequality problem over the the fixed point set of certain quasi-nonexpansive mappings. Numerical Functional Analysis and Optimization,25(7–8):619–655,2004. https://doi.org/10.1081/NFA-200045815.
A.S. Antipin. On convergence of proximal methods to fixed points of extremal mappings and estimates of their rate of convergence. Comp. Maths. Math. Phys., 35:539–551, 1995.
H.H. Bauschke and J.A. Chen. Projection method for approximating fixed points of quasi - non expansive mappings without the usual demiclosedness condition. J. Nonlinear Convex Anal.,15:129–135, 2014.
H.H. Bauschke and P.L. Combettes. Combettes. Convex analysis and monotone operator theory in Hilbert Spaces. New York: Springer, 2001.
E. Blum and W. Oettli. From optimization and variational inequalities to equi-librium problems. Math. Student,63:123–145, 1994.
P L. Combettes and S.A. Hirstoaga. Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal., 6:117–136, 2005.
Y. Dang and Y. Gao. The strong convergence of a KM-CQ-like algo-rithm for a split feasibility problem. Inverse Problems, 27(1):015007, 2011. https://doi.org/10.1088/0266-5611/27/1/015007.
F. Facchinei and J.S. Pang. Finite-dimensional variational inequalities and com-plementarity problems. Berlin: Springer, 2002.
K. Fan. A minimax inequality and applications. In: Shisha, O. (ed.) Inequality,III, Academic Press, New York, pp. 103–113, 1972.
S.D. Flam and A.S. Antipin. Equilibrium programming and proximal-like algo-rithms. Math. Program., 78:29–41, 1997.https://doi.org/10.1007/BF02614504.
D.V. Hieu. An extension of hybrid method without extrapolation step to equi-librium problems. J. Ind. Manag. Optim., 13:1723–1741, 2017.
D.V. Hieu. Hybrid projection methods for equilibrium problems with non-Lipschitz type bifunctions. Math. Meth. Appl. Sci., 40:4065–4079, 2017. https://doi.org/10.1002/mma.4286.
D.V. Hieu.New subgradient extragradient methods for common solu-tions to equilibrium problems. Comput. Optim. Appl., 67(3):571–594, 2017. https://doi.org/10.1007/s10589-017-9899-4.
D.V. Hieu.Convergence analysis of a new algorithm for strongly pseu-domontone equilibrium problems. Numer. Algorithms., 77(4):983–1001, 2018. https://doi.org/10.1007/s11075-017-0350-9.
D.V. Hieu, L.D. Muu and P.K. Anh. Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings. Numer. Algorithms., 73(1):197–217, 2016. https://doi.org/10.1007/s11075-015-0092-5.
H. Iiduka and I. Yamada. A subgradient-type method for the equilibrium prob-lem over the fixed point set and its applications. Optimization,58(2):251–261,2009. https://doi.org/10.1080/02331930701762829.
I.V. Konnov. Application of the proximal point method to nonmonotone equilib-rium problems. Journal of Optimization Theory and Applications,119(2):317–333, 2003. https://doi.org/10.1023/B:JOTA.0000005448.12716.24.
I.V. Konnov. Equilibrium models and variational inequalities. Amsterdam: El-sevier, 2007.
G.M. Korpelevich. The extragradient method for finding saddle points and otherproblems. Ekonomikai Matematicheskie Metody, 12:747–756, 1976.
P. Kumam and P. Katchang. A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point prob-lems for nonexpansive mappings. Nonlinear Analysis: Hybrid Systems, 3(4):475–486, 2009. https://doi.org/10.1016/j.nahs.2009.03.006.
W. Kumam, H. Piri and P. Kumam. Solutions of system of equilibrium andvariational inequality problems on fixed points of infinite family of nonexpan-sive mappings. Applied Mathematics and Computation, 248(1):441–455, 2014. https://doi.org/10.1016/j.amc.2014.09.118.
M. Li and Y. Yao. Strong convergence of an iterative algorithm forλ-strictlypseudo-contractive mappings in hilbert spaces. An. St. Univ. Ovidius Constanta., 18(1):219–228, 2010.
S.I. Lyashko and V.V. Semenov. A new two-step proximal algorithm of solving the problem of equilibrium programming. Optimization and its applications in control and data sciences. Springer, Switzerland,115:315–325, 2016.
P.E. Maing ́e. A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim., 47(3):1499–1515, 2008. https://doi.org/10.1137/060675319.
P.E. Maing ́e and A. Moudafi. Coupling viscosity methods with the extragradient algorithm for solving equilibrium problems. J. Nonlinear Convex Anal.,9:283–294, 2008.
G. Mastroeni. On auxiliary principle for equilibrium problems. Publicatione del Dipartimento di Mathematica dell, Universita di Pisa,3:1244–1258, 2000.
G. Mastroeni. Gap function for equilibrium problems. J. Global. Optim., 27(4):411–426, 2003.https://doi.org/10.1023/A:1026050425030.
A. Moudafi. Proximal point algorithm extended to equilibrum problem. J. Nat. Geometry,15:91–100, 1999.
A. Moudafi. Viscosity approximation methods for fixed point problems. J. Math.Anal. Appl., 241(1):46–55, 2000.https://doi.org/10.1006/jmaa.1999.6615.
L.D. Muu and W. Oettli. Convergence of an adative penalty scheme forfinding constrained equilibria. Nonlinear Anal. TMA,18(2):1159–1166, 1992. https://doi.org/10.1016/0362-546X(92)90159-C.
T.T.V. Nguyen, J.J. Strodiot and V.H. Nguyen. Hybrid methods for solv-ing simultaneously an equilibrium problem and countably many fixed point problems in a Hilbert space .J. Optim. Theory Appl.,160(3):809–831, 2014. https://doi.org/10.1007/s10957-013-0400-y.
T.D. Quoc, L.D. Muu and V.H. Nguyen. Extragradient algorithms extended to mequilibrium problems. Optimization,57(6):749–776, 2008. https://doi.org/10.1080/02331930601122876.
S. Reich. Constructive techniques for accretive and monotone operators. Applied Nonlinear Analysis, Academic Press, New York, pp. 335–345, 1979.
J.J. Strodiot, P.T. Vuong and T.T.V. Nguyen. A class of shrinking pro-jection extragradient methods for solving non-monotone equilibrium prob-lems in hilbert spaces. Journal of Global Optimization, 64(1):159–178, 2016.https://doi.org/10.1007/s10898-015-0365-5.
S. Takahashi and W. Takahashi. Viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces. Jour-nal of Mathematical Analysis and Applications,331(1):506–515, 2007.https://doi.org/10.1016/j.jmaa.2006.08.036.
P.T. Vuong, J.J. Strodiot and V.H. Nguyen. Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point prob-lems. Journal of Optimization Theory and Applications,155(2):605–627,2012.https://doi.org/10.1007/s10957-012-0085-7.
P.T. Vuong, J.J. Strodiot and V.H. Nguyen. On extragradient-viscosity methods for solving equilibrium and fixed point problems in a hilbert space. Optimization, 64(2):429–451, 2015. https://doi.org/10.1080/02331934.2012.759327.
H. K. Xu. Iterative algorithms for nonlinear operators. Journal of London Mathematical Society,66(1):240–256,2002. https://doi.org/10.1112/S0024610702003332.
I. Yamada and N. Ogura. Hybrid steepest descent method for the variational inequality problem over the the fixed point set of certain quasi-nonexpansive mappings. Numerical Functional Analysis and Optimization,25(7–8):619–655,2004. https://doi.org/10.1081/NFA-200045815.