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doi: 10.3934/naco.2021039
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Modified inertial algorithm for solving mixed equilibrium problems in Hadamard spaces

1. 

Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

2. 

Department of Mathematics, The Technion-Israel Institute of Technology, 32000 Haifa, Israel

3. 

Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria, P.O. Box 60, 0204, South Africa

4. 

Department of Mathematics, University of Eswatini, Private Bag 4 Kwaluseni, Eswatini

* Corresponding author: Godwin Chidi Ugwunnadi

Received  April 2021 Revised  August 2021 Early access September 2021

The main purpose of this paper is to introduce the concept of modified inertial algorithm in Hadamard spaces. We emphasize that, as far as we know, this is the first time that this concept is being considered in this setting. Under some weak assumptions, we prove that the modified inertial algorithm converges strongly to a common solution of a finite family of mixed equilibrium problems and fixed point problem of a nonexpansive mapping. We also give a primary numerical illustration in the framework of Hadamard spaces, to show the efficiency of the modified inertial term in our proposed algorithm.

Citation: Abdul Rahim Khan, Chinedu Izuchukwu, Maggie Aphane, Godwin Chidi Ugwunnadi. Modified inertial algorithm for solving mixed equilibrium problems in Hadamard spaces. Numerical Algebra, Control and Optimization, doi: 10.3934/naco.2021039
References:
[1]

F. Alvarez and H. Attouch, An Inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal., 9 (2001), 3-11.  doi: 10.1023/A:1011253113155.

[2]

M. Bač$\acute{a}$k, Convex Analysis and Optimization in Hadamard Spaces, De Gruyter Series in Nonlinear Analysis and Applications, De Gruyter, Berlin, 22 (2014). doi: 10.1515/9783110361629.

[3]

M. Bač$\acute{a}$k, Old and new challenges in Hadamard spaces, 2018, arXiv: math/01355v2.

[4]

M. Bač$\acute{a}$k, The proximal point algorithm in metric spaces, Israel J. Math., 194 (2013), 689-701.  doi: 10.1007/s11856-012-0091-3.

[5]

I. D. Berg and I. G. Nikolaev, Quasilinearization and curvature of Alexandrov spaces, Geom. Dedicata, 133 (2008), 195-218.  doi: 10.1007/s10711-008-9243-3.

[6]

M. Bianchi and S. Schaible, Generalized monotone bifunctions and equilibrium problems, J. Optim Theory Appl., 90 (1996), 31-43.  doi: 10.1007/BF02192244.

[7]

E. Blum and W. Oettli, From optimization and variational inequality to equilibrium problems, Math. Stud., 63 (1994) 123–145.

[8]

M. Bridson and A. Haefliger, Metric Spaces of Nonpositive Curvature, Springer-Verlag, Berlin, Heidelberg, New York, 1999. doi: 10.1007/978-3-662-12494-9.

[9]

P. Chaoha and A. Phon-on, A note on fixed point sets in CAT(0) spaces, J. Math. Anal. Appl., 320 (2006), 983-987.  doi: 10.1016/j.jmaa.2005.08.006.

[10]

P. CholamjiakD. V. Thong and Y. J. Cho, A novel inertial projection and contraction method for solving pseudomonotone variational inequality problems, Act. Appl. Math., 169 (2020), 217-245.  doi: 10.1007/s10440-019-00297-7.

[11]

V. ColaoG. LopezG. Marino and V. Martn-Marquez, Equilibrium Problems in Hadamard manifolds, J. Math. Anal. Appl., 388 (2012), 61-77.  doi: 10.1016/j.jmaa.2011.11.001.

[12]

P. L. Combetes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117-136. 

[13]

S. S. Chang, J. C. Yao, C. F. Wen, L. Yang and L. J. Qin, Common zero for a finite family of monotone mappings in Hadamard spaces with applications, Mediterr. J. Math., (2018), Article number: 160. doi: 10.1007/s00009-018-1205-x.

[14]

B. J. Choi and U. C. Ji, The proximal point algorithm in uniformly convex metric spaces, Commun. Korean Math. Soc., 31 (2016), 845-855.  doi: 10.4134/CKMS.c150114.

[15]

H. Dehghan, C. Izuchukwu, O. T. Mewomo, D. A. Taba and G. C. Ugwunnadi, Iterative algorithm for a family of monotone inclusion problems in CAT(0) spaces, Quaest. Math., (2019), 1–24. doi: 10.2989/16073606.2019.1593255.

[16]

H. Dehghan and J. Rooin, Metric projection and convergence theorems for nonexpansive mappings in Hadamard spaces, 2014, arXiv: math/1137v1.

[17]

S. DhompongsaW. A. Kirk and B. Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal., 64 (2006), 762-772.  doi: 10.1016/j.na.2005.09.044.

[18]

S. Dhompongsa and B. Panyanak, On $\triangle$-convergence theorems in CAT(0) spaces, Comput. Math. Appl., 56 (2008), 2572-2579.  doi: 10.1016/j.camwa.2008.05.036.

[19]

A. Feragen, S. Hauberg, M. Nielsen and F. Lauze, Means in spaces of tree-like shapes, in Proceedings of the IEEE International Conference on Computer Vision (ICCV), 2011, IEEE, Piscataway, NJ, (2011), 736–746.

[20]

A. FeragenP. LoM. de BruijneM. Nielsen and F. Lauze, Toward a theory of statistical tree-shape analysis, IEEE Trans. Pattern Anal. Mach. Intell., 35 (2013), 2008-2021. 

[21]

K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings, Marcel Dekker, New York, 1984.

[22]

A. N. IusemG. Kassay and W. Sosa, On certain conditions for the existence of solutions of equilibrium problems, Math. Program., Ser. B, 116 (2009), 259-273.  doi: 10.1007/s10107-007-0125-5.

[23]

C. IzuchukwuG. C. UgwunnadiO. T. MewomoA. R. Khan and M. Abbas, Proximal-type algorithms for split minimization problem in P-uniformly convex metric spaces, Numer. Algorithms, 82 (2019), 909-935.  doi: 10.1007/s11075-018-0633-9.

[24]

C. Izuchukwu, K. O. Aremu, O. K. Oyewole and O. T. Mewomo, On mixed equilibrium problems in Hadamard spaces, J. Math., (2019), Article ID 3210649, 13 pages. doi: 10.1155/2019/3210649.

[25]

C. IzuchukwuK. O. AremuA. A. Mebawondu and O. T. Mewomo, A viscosity iterative technique for equilibrium and fixed point problems in a Hadamard space, Appl. Gen. Topol., 20 (2019), 193-210.  doi: 10.4995/agt.2019.10635.

[26]

A. R. KhanG. C. UgwunnadiZ. G. Makukula and M. Abbas, Strong convergence of inertial subgradient extragradient method for solving variational inequality in Banach space, Carpathian J. Math., 35 (2019), 327-338. 

[27]

W. A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008), 3689-3696.  doi: 10.1016/j.na.2007.04.011.

[28]

P. Kumam and P. Chaipunya, Equilibrium problems and proximal algorithms in Hadamard spaces, 2018, arXiv: math/10900v1.

[29]

L. Leustean, A quadratic rate of asymptotic regularity for CAT(0)-spaces, J. Math. Anal. Appl., 325 (2007), 386-399.  doi: 10.1016/j.jmaa.2006.01.081.

[30]

T. C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc., 60 (1976), 179-182.  doi: 10.2307/2041136.

[31]

B. Martinet, R$\acute{e}$gularisation d'in$\acute{e}$quations variationnelles par approximations successives, Rev.Fran$\acute{c}$aise dnform. et de Rech. Op$\acute{e}$rationnelle, 3 (1970), 154–158.

[32]

C. C. Okeke and C. Izuchukwu, A strong convergence theorem for monotone inclusion and minimization problems in complete CAT(0) spaces, Optim. Methods Softw., 34 (2019), 1168-1183.  doi: 10.1080/10556788.2018.1472259.

[33]

W. Phuengrattana, N. Onjai-uea and P. Cholamjiak, Modified proximal algorithms for solving constrained minimization and fixed point problems in complete CAT(0) spaces, Mediterr. J. Math., (2018), Article Number: 97. doi: 10.1007/s00009-018-1144-6.

[34]

B. T. Polyak, Some methods of speeding up the convergence of iterates methods, U.S.S.R Comput. Math. Phys., 4 (1994), 1-17. 

[35]

S. Ranjbar and H. Khatibzadeh, Strong and delta convergence to a zero of a monotone operator in CAT(0) spaces, Mediterr. J. Math., 14 (2017), 15 pp. doi: 10.1007/s00009-017-0885-y.

[36]

S. Reich and I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal., 15 (1990), 537-558.  doi: 10.1016/0362-546X(90)90058-O.

[37]

R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.  doi: 10.1137/0314056.

[38]

Y. Shehu and P. Cholamjiak, Iterative method with inertial for variational inequalities in Hilbert spaces, Calcolo, 51 (2019), Article number: 4. doi: 10.1007/s10092-018-0300-5.

[39]

R. SuparatulatornP. Cholamjiak and S. Suantai, On solving the minimization problem and the fixed-point problem for nonexpansive mappings in CAT(0) spaces, Optim. Methods Softw., 32 (2017), 182-192.  doi: 10.1007/s10092-018-0300-5.

[40]

R. Suparatulatorn, P. Cholamjiak and S. Suantai, Self-adaptive algorithms with inertial effects for solving the split problem of the demicontractive operators, RACSAM, 114 (2019), Article number: 40. doi: 10.1007/s13398-019-00737-x.

[41]

T. Suzuki, Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces, Fixed Point Theory Appl., 1 (2005), 103-123.  doi: 10.1155/fpta.2005.103.

[42]

W. Takahashi and K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal., 70 (2009), 45-57.  doi: 10.1016/j.na.2007.11.031.

[43]

J. Tang, Viscosity approximation methods for a family of nonexpansive mappings in CAT(0) spaces, Abstr. Appl. Anal., (2014), Article ID 389804, 9 pages. doi: 10.1155/2014/389804.

[44]

D. V. Thong and D. V. Hieu, Weak and strong convergence theorems for variational inequality problems, Numer. Algorithms, 78 (2018), 1045-1060.  doi: 10.1007/s11075-017-0412-z.

[45]

D. V. Thong and D. V. Hieu, Inertial subgradient extragradient algorithms with line-search process for solving variational inequality problems and fixed point problems, Numer. Algorithms, 80 (2018), 1283-1307.  doi: 10.1007/s11075-018-0527-x.

[46]

D. V. Thong and D. V. Hieu, New extragradient methods for solving variational inequality problems and fixed point problems, J. Fixed Point Theory Appl., (2018), Article number: 129. doi: 10.1007/s11784-018-0610-x.

[47]

G. C. UgwunnadiC. Izuchukwu and O. T. Mewomo, Strong convergence theorem for monotone inclusion problem in CAT(0) spaces, Afr. Mat., 30 (2019), 151-169.  doi: 10.1007/s13370-018-0633-x.

[48]

H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240-256.  doi: 10.1112/S0024610702003332.

[49]

G. Zamani Eskandani and M. Raeisi, On the zero point problem of monotone operators in Hadamard spaces, Numer. Algorithms, 80 (2019), 1155-1179.  doi: 10.1007/s11075-018-0521-3.

show all references

References:
[1]

F. Alvarez and H. Attouch, An Inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal., 9 (2001), 3-11.  doi: 10.1023/A:1011253113155.

[2]

M. Bač$\acute{a}$k, Convex Analysis and Optimization in Hadamard Spaces, De Gruyter Series in Nonlinear Analysis and Applications, De Gruyter, Berlin, 22 (2014). doi: 10.1515/9783110361629.

[3]

M. Bač$\acute{a}$k, Old and new challenges in Hadamard spaces, 2018, arXiv: math/01355v2.

[4]

M. Bač$\acute{a}$k, The proximal point algorithm in metric spaces, Israel J. Math., 194 (2013), 689-701.  doi: 10.1007/s11856-012-0091-3.

[5]

I. D. Berg and I. G. Nikolaev, Quasilinearization and curvature of Alexandrov spaces, Geom. Dedicata, 133 (2008), 195-218.  doi: 10.1007/s10711-008-9243-3.

[6]

M. Bianchi and S. Schaible, Generalized monotone bifunctions and equilibrium problems, J. Optim Theory Appl., 90 (1996), 31-43.  doi: 10.1007/BF02192244.

[7]

E. Blum and W. Oettli, From optimization and variational inequality to equilibrium problems, Math. Stud., 63 (1994) 123–145.

[8]

M. Bridson and A. Haefliger, Metric Spaces of Nonpositive Curvature, Springer-Verlag, Berlin, Heidelberg, New York, 1999. doi: 10.1007/978-3-662-12494-9.

[9]

P. Chaoha and A. Phon-on, A note on fixed point sets in CAT(0) spaces, J. Math. Anal. Appl., 320 (2006), 983-987.  doi: 10.1016/j.jmaa.2005.08.006.

[10]

P. CholamjiakD. V. Thong and Y. J. Cho, A novel inertial projection and contraction method for solving pseudomonotone variational inequality problems, Act. Appl. Math., 169 (2020), 217-245.  doi: 10.1007/s10440-019-00297-7.

[11]

V. ColaoG. LopezG. Marino and V. Martn-Marquez, Equilibrium Problems in Hadamard manifolds, J. Math. Anal. Appl., 388 (2012), 61-77.  doi: 10.1016/j.jmaa.2011.11.001.

[12]

P. L. Combetes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117-136. 

[13]

S. S. Chang, J. C. Yao, C. F. Wen, L. Yang and L. J. Qin, Common zero for a finite family of monotone mappings in Hadamard spaces with applications, Mediterr. J. Math., (2018), Article number: 160. doi: 10.1007/s00009-018-1205-x.

[14]

B. J. Choi and U. C. Ji, The proximal point algorithm in uniformly convex metric spaces, Commun. Korean Math. Soc., 31 (2016), 845-855.  doi: 10.4134/CKMS.c150114.

[15]

H. Dehghan, C. Izuchukwu, O. T. Mewomo, D. A. Taba and G. C. Ugwunnadi, Iterative algorithm for a family of monotone inclusion problems in CAT(0) spaces, Quaest. Math., (2019), 1–24. doi: 10.2989/16073606.2019.1593255.

[16]

H. Dehghan and J. Rooin, Metric projection and convergence theorems for nonexpansive mappings in Hadamard spaces, 2014, arXiv: math/1137v1.

[17]

S. DhompongsaW. A. Kirk and B. Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal., 64 (2006), 762-772.  doi: 10.1016/j.na.2005.09.044.

[18]

S. Dhompongsa and B. Panyanak, On $\triangle$-convergence theorems in CAT(0) spaces, Comput. Math. Appl., 56 (2008), 2572-2579.  doi: 10.1016/j.camwa.2008.05.036.

[19]

A. Feragen, S. Hauberg, M. Nielsen and F. Lauze, Means in spaces of tree-like shapes, in Proceedings of the IEEE International Conference on Computer Vision (ICCV), 2011, IEEE, Piscataway, NJ, (2011), 736–746.

[20]

A. FeragenP. LoM. de BruijneM. Nielsen and F. Lauze, Toward a theory of statistical tree-shape analysis, IEEE Trans. Pattern Anal. Mach. Intell., 35 (2013), 2008-2021. 

[21]

K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings, Marcel Dekker, New York, 1984.

[22]

A. N. IusemG. Kassay and W. Sosa, On certain conditions for the existence of solutions of equilibrium problems, Math. Program., Ser. B, 116 (2009), 259-273.  doi: 10.1007/s10107-007-0125-5.

[23]

C. IzuchukwuG. C. UgwunnadiO. T. MewomoA. R. Khan and M. Abbas, Proximal-type algorithms for split minimization problem in P-uniformly convex metric spaces, Numer. Algorithms, 82 (2019), 909-935.  doi: 10.1007/s11075-018-0633-9.

[24]

C. Izuchukwu, K. O. Aremu, O. K. Oyewole and O. T. Mewomo, On mixed equilibrium problems in Hadamard spaces, J. Math., (2019), Article ID 3210649, 13 pages. doi: 10.1155/2019/3210649.

[25]

C. IzuchukwuK. O. AremuA. A. Mebawondu and O. T. Mewomo, A viscosity iterative technique for equilibrium and fixed point problems in a Hadamard space, Appl. Gen. Topol., 20 (2019), 193-210.  doi: 10.4995/agt.2019.10635.

[26]

A. R. KhanG. C. UgwunnadiZ. G. Makukula and M. Abbas, Strong convergence of inertial subgradient extragradient method for solving variational inequality in Banach space, Carpathian J. Math., 35 (2019), 327-338. 

[27]

W. A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008), 3689-3696.  doi: 10.1016/j.na.2007.04.011.

[28]

P. Kumam and P. Chaipunya, Equilibrium problems and proximal algorithms in Hadamard spaces, 2018, arXiv: math/10900v1.

[29]

L. Leustean, A quadratic rate of asymptotic regularity for CAT(0)-spaces, J. Math. Anal. Appl., 325 (2007), 386-399.  doi: 10.1016/j.jmaa.2006.01.081.

[30]

T. C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc., 60 (1976), 179-182.  doi: 10.2307/2041136.

[31]

B. Martinet, R$\acute{e}$gularisation d'in$\acute{e}$quations variationnelles par approximations successives, Rev.Fran$\acute{c}$aise dnform. et de Rech. Op$\acute{e}$rationnelle, 3 (1970), 154–158.

[32]

C. C. Okeke and C. Izuchukwu, A strong convergence theorem for monotone inclusion and minimization problems in complete CAT(0) spaces, Optim. Methods Softw., 34 (2019), 1168-1183.  doi: 10.1080/10556788.2018.1472259.

[33]

W. Phuengrattana, N. Onjai-uea and P. Cholamjiak, Modified proximal algorithms for solving constrained minimization and fixed point problems in complete CAT(0) spaces, Mediterr. J. Math., (2018), Article Number: 97. doi: 10.1007/s00009-018-1144-6.

[34]

B. T. Polyak, Some methods of speeding up the convergence of iterates methods, U.S.S.R Comput. Math. Phys., 4 (1994), 1-17. 

[35]

S. Ranjbar and H. Khatibzadeh, Strong and delta convergence to a zero of a monotone operator in CAT(0) spaces, Mediterr. J. Math., 14 (2017), 15 pp. doi: 10.1007/s00009-017-0885-y.

[36]

S. Reich and I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal., 15 (1990), 537-558.  doi: 10.1016/0362-546X(90)90058-O.

[37]

R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.  doi: 10.1137/0314056.

[38]

Y. Shehu and P. Cholamjiak, Iterative method with inertial for variational inequalities in Hilbert spaces, Calcolo, 51 (2019), Article number: 4. doi: 10.1007/s10092-018-0300-5.

[39]

R. SuparatulatornP. Cholamjiak and S. Suantai, On solving the minimization problem and the fixed-point problem for nonexpansive mappings in CAT(0) spaces, Optim. Methods Softw., 32 (2017), 182-192.  doi: 10.1007/s10092-018-0300-5.

[40]

R. Suparatulatorn, P. Cholamjiak and S. Suantai, Self-adaptive algorithms with inertial effects for solving the split problem of the demicontractive operators, RACSAM, 114 (2019), Article number: 40. doi: 10.1007/s13398-019-00737-x.

[41]

T. Suzuki, Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces, Fixed Point Theory Appl., 1 (2005), 103-123.  doi: 10.1155/fpta.2005.103.

[42]

W. Takahashi and K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal., 70 (2009), 45-57.  doi: 10.1016/j.na.2007.11.031.

[43]

J. Tang, Viscosity approximation methods for a family of nonexpansive mappings in CAT(0) spaces, Abstr. Appl. Anal., (2014), Article ID 389804, 9 pages. doi: 10.1155/2014/389804.

[44]

D. V. Thong and D. V. Hieu, Weak and strong convergence theorems for variational inequality problems, Numer. Algorithms, 78 (2018), 1045-1060.  doi: 10.1007/s11075-017-0412-z.

[45]

D. V. Thong and D. V. Hieu, Inertial subgradient extragradient algorithms with line-search process for solving variational inequality problems and fixed point problems, Numer. Algorithms, 80 (2018), 1283-1307.  doi: 10.1007/s11075-018-0527-x.

[46]

D. V. Thong and D. V. Hieu, New extragradient methods for solving variational inequality problems and fixed point problems, J. Fixed Point Theory Appl., (2018), Article number: 129. doi: 10.1007/s11784-018-0610-x.

[47]

G. C. UgwunnadiC. Izuchukwu and O. T. Mewomo, Strong convergence theorem for monotone inclusion problem in CAT(0) spaces, Afr. Mat., 30 (2019), 151-169.  doi: 10.1007/s13370-018-0633-x.

[48]

H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240-256.  doi: 10.1112/S0024610702003332.

[49]

G. Zamani Eskandani and M. Raeisi, On the zero point problem of monotone operators in Hadamard spaces, Numer. Algorithms, 80 (2019), 1155-1179.  doi: 10.1007/s11075-018-0521-3.

Figure 1.  Errors vs Iteration numbers(n): Case 1 (top left); Case 2 (top right); Case 3 (bottom left); Case 4 (bottom right)
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