Previous |  Up |  Next

Article

Keywords:
split equality; generalized equilibrium problem; variational inclusion problem; variational inequality; quasi-nonexpansive mapping; fixed point problem
Summary:
This paper presents an inertial iterative algorithm for approximating a common solution of split equalities of generalized mixed equilibrium problem, monotone variational inclusion problem, variational inequality problem and common fixed point problem in real Hilbert spaces. The algorithm is designed in such a way that it does not require prior knowledge of the norms of the bounded linear operators. We prove a strong convergence theorem under some mild conditions of the control sequences and also give a numerical example to show the efficiency and accuracy of our algorithm. We see that the inertial algorithm performs better in terms of number of iteration and CPU-time than the non-inertial algorithm. This result improves and generalizes many recent results in the literature.
References:
[1] Attouch H., Bolte J., Redont P., Soubeyran A.: Alternating proximal algorithms for weakly coupled convex minimization problems. Applications to dynamical games and PDE's. J. Convex Anal. 15 (2008), no. 3, 485–506. MR 2431407
[2] Attouch H., Cabot A., Frankel P., Peypouquet J.: Alternating proximal algorithms for linearly constrained variational inequalities: application to domain decomposition for PDE's. Nonlinear Anal. 74 (2011), no. 18, 7455–7473. MR 2833727
[3] Attouch H., Czarnecki M. O.: Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria. J. Differential Equations 179 (2002), no. 1, 278–310. DOI 10.1006/jdeq.2001.4034 | MR 1883745
[4] Attouch H., Goudou X., Redont P.: The heavy ball with friction. I. The continuous dynamical system: global exploration of the local minima of a real-valued function by asymptotic analysis of a dissipative dynamical system. Commun. Contemp. Math. 2 (2000), no. 1, 1–34. DOI 10.1142/S0219199700000025 | MR 1753136
[5] Attouch H., Peypouquet J., Redont P.: A dynamical approach to an inertial forward–backward algorithm for convex minimization. SIAM J. Optim. 24, (2014), no. 1, 232–256. DOI 10.1137/130910294 | MR 3164130
[6] Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Student 63 (1994), no. 1–4, 123–145. MR 1292380
[7] Boţ R. I., Csetnek E. R.: A hybrid proximal-extragradient algorithm with inertial effects. Numer. Funct. Anal. Optim. 36 (2015), no. 8, 951–963. MR 3373745
[8] Boţ R. I., Csetnek E. R.: An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems. Numer. Algorithms 71 (2016), no. 3, 519–540. DOI 10.1007/s11075-015-0007-5 | MR 3463479
[9] Boţ R. I., Csetnek E. R., Hendrich C.: Inertial Douglas–Rachford splitting for monotone inclusion problems. Appl. Math. Comput. 256 (2015), 472–487. MR 3316085
[10] Boţ R. I., Csetnek E. R., László S. C.: An inertial forward-backward algorithm for the minimization of the sum of two nonconvex functions. EURO J. Comput. Optim. 4 (2016), no. 1, 3–25. DOI 10.1007/s13675-015-0045-8 | MR 3500980
[11] Byrne C. L., Moudafi A.: Extensions of the CQ algorithm for the split feasibility and split equality problems. J. Nonlinear Convex Anal. 18 (2017), no. 8, 1485–1496. MR 3716968
[12] Censor Y.: Parallel application of block-iterative methods in medical imaging and radiation therapy. Math. Programming 42 (1988), no. 2, (Ser. B), 307–325. DOI 10.1007/BF01589408 | MR 0976123
[13] Censor Y., Bortfeld T., Martin B., Trofimov A.: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51 (2006), no. 10, 2353–2365. DOI 10.1088/0031-9155/51/10/001
[14] Chang S.-S., Wang L., Wang X. R., Wang G.: General split equality equilibrium problems with application to split optimization problems. J. Optim. Theory Appl. 166 (2015), no. 2, 377–390. DOI 10.1007/s10957-015-0739-3 | MR 3371380
[15] Chen C., Chan R. H., Ma S., Yang J.: Inertial proximal ADMM for linearly constrained separable convex optimization. SIAM J. Imaging Sci. 8 (2015), no. 4, 2239–2267. DOI 10.1137/15100463X | MR 3404682
[16] Cholamjiak W., Pholasa N., Suantai S.: A modified inertial shrinking projection method for solving inclusion problems and quasi-nonexpansive multivalued mappings. Comput. Appl. Math. 37 (2018), no. 5, 5750–5774. DOI 10.1007/s40314-018-0661-z | MR 3885793
[17] Chuang C.-S.: Hybrid inertial proximal algorithm for the split variational inclusion problem in Hilbert spaces with applications. Optimization 66 (2017), no. 5, 777–792. DOI 10.1080/02331934.2017.1306744 | MR 3628551
[18] Dong Q.-L., Lu Y.-Y., Yang J.: The extragradient algorithm with inertial effects for solving the variational inequality. Optimization 65 (2016), no. 12, 2217–2226. DOI 10.1080/02331934.2016.1239266 | MR 3564913
[19] Guo H., He H., Chen R.: Strong convergence theorems for the split equality variational inclusion problem and fixed point problem in Hilbert spaces. Fixed Point Theory Appl. (2015), 2015:223, 18 pages. MR 3430301
[20] He Z.: The split equilibrium problem and its convergence algorithms. J. Inequal. Appl. (2012), 2012:162, 15 pages. MR 2972645
[21] Jolaoso L. O., Abass H. A., Mewomo O. T.: A viscosity-proximal gradient method with inertial extrapolation for solving certain minimization problems in Hilbert space. Arch. Math. (Brno) 55 (2019), no. 3, 167–194. DOI 10.5817/AM2019-3-167 | MR 3994324
[22] Jolaoso L. O., Alakoya T. O., Taiwo A., Mewomo O. T.: A parallel combination extragradient method with Armijo line searching for finding common solutions of finite families of equilibrium and fixed point problems. Rend. Circ. Mat. Palermo (2) 69 (2019), 711–735. MR 4168136
[23] Jolaoso L. O., Ogbuisi F. U., Mewomo O. T.: An iterative method for solving minimization, variational inequality and fixed point problems in reflexive Banach spaces. Adv. Pure Appl. Math. 9 (2018), no. 3, 167–184. DOI 10.1515/apam-2017-0037 | MR 3819533
[24] Jolaoso L. O., Oyewole K. O., Okeke C. C., Mewomo O. T.: A unified algorithm for solving split generalized mixed equilibrium problem, and for finding fixed point of nonspreading mapping in Hilbert spaces. Demonstr. Math. 51 (2018), no. 1, 211–232. DOI 10.1515/dema-2018-0015 | MR 3856588
[25] Jolaoso L. O., Taiwo A., Alakoya T. O., Mewomo O. T.: A self adaptive inertial subgradient extragradient algorithm for variational inequality and common fixed point of multivalued mappings in Hilbert spaces. Demonstr. Math. 52 (2019), no. 1, 183–203. DOI 10.1515/dema-2019-0013 | MR 3938331
[26] Kazmi K. R., Rizvi S. H.: An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping. Optim. Lett. 8 (2014), no. 3, 1113–1124. DOI 10.1007/s11590-013-0629-2 | MR 3170590
[27] Latif A., Eslamian M.: Split equality problem with equilibrium problem, variational inequality problem, and fixed point problem of nonexpansive semigroups. J. Nonlinear Sci. Appl. 10 (2017), no. 6, 3217–3230. DOI 10.22436/jnsa.010.06.34 | MR 3670500
[28] Lemaire B.: Which fixed point does the iteration method select?. Recent Advances in Optimization, Trier, 1996, Lecture Notes in Econom. and Math. Systems, 452, Springer, Berlin, 1997, pages 154–157. DOI 10.1007/978-3-642-59073-3_11 | MR 1467027
[29] Li S., Li L., Cao L., He X., Yue X.: Hybrid extragradient method for generalized mixed equilibrium problem and fixed point problems in Hilbert space. Fixed Point Theory Appl. (2013), 2013:240, 13 pages. MR 3261016
[30] Lin L.-J., Chen Y.-D., Chuang C.-S.: Solutions for a variational inclusion problem with applications to multiple sets split feasibility problems. Fixed Point Theory Appl. (2013), 2013:333, 21 pages. MR 3338266
[31] López G., Martín-Márquez V., Wang F., Xu H.-K.: Solving the split feasibility problem without prior knowledge of matrix norm. Inverse Problems 28 (2012), no. 8, 085004, 18 pages. MR 2948743
[32] Ma Z., Wang L., Chang S.-S., Duan W.: Convergence theorems for split equality mixed equilibrium problems with applications. Fixed Point Theory Appl. (2015), 2015:31, 18 pages. MR 3316770
[33] Maingé P.-E.: Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 325 (2007), no. 1, 469–479. DOI 10.1016/j.jmaa.2005.12.066 | MR 2273538
[34] Maingé P.-E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 16 (2008), no. 7–8, 899–912. DOI 10.1007/s11228-008-0102-z | MR 2466027
[35] Marino G., Xu H.-K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 329 (2007), no. 1, 336–346. DOI 10.1016/j.jmaa.2006.06.055 | MR 2306805
[36] Matinez-Yanes C., Xu H.-K.: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal. 64 (2006), no. 11, 2400–2411. DOI 10.1016/j.na.2005.08.018 | MR 2215815
[37] Mewomo O. T., Ogbuisi F. U.: Convergence analysis of an iterative method for solving multiple-set split feasibility problems in certain Banach spaces. Quaest. Math. 41 (2018), no. 1, 129–148. DOI 10.2989/16073606.2017.1375569 | MR 3761493
[38] Moudafi A.: A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonlinear Anal. 74 (2011), no. 12, 4083–4087. DOI 10.1016/j.na.2011.03.041 | MR 2802988
[39] Moudafi A.: Split monotone variational inclusions. J. Optim. Theory Appl. 150 (2011), no. 2, 275–283. DOI 10.1007/s10957-011-9814-6 | MR 2818920
[40] Moudafi A.: Alternating CQ-algorithms for convex feasibility and split fixed-point problems. J. Nonlinear Convex Anal. 15 (2014), no. 4, 809–818. MR 3222909
[41] Moudafi A., Al-Shemas E.: Simultaneous iterative methods for split equality problems and applications. Trans. Math. Program. Appl. 1 (2013), 1–11.
[42] Ochs P., Brox T., Pock T.: iPiasco: inertial proximal algorithm for strongly convex optimization. J. Math. Imaging Vision 53 (2015), no. 2, 171–181. DOI 10.1007/s10851-015-0565-0 | MR 3372139
[43] Rahaman M., Liou Y.-C., Ahmad R., Ahmad I.: Convergence theorems for split equality generalized mixed equilibrium problems for demi-contractive mappings. J. Inequal. Appl. (2015), 2015:418, 25 pages. MR 3438079
[44] Rockafellar R. T.: Monotone operators and the proximal point algorithm. SIAM J. Control. Optim. 14 (1976), no. 5, 877–898. DOI 10.1137/0314056 | MR 0410483
[45] Shehu Y., Mewomo O. T.: Further investigation into split common fixed point problem for demicontractive operators. Acta Math. Sin. (Engl. Ser.) 32 (2016), no. 11, 1357–1376. DOI 10.1007/s10114-016-5548-6 | MR 3557403
[46] Shehu Y., Mewomo O. T., Ogbuisi F. U.: Further investigation into approximation of a common solution of fixed point problems and split feasibility problems. Acta Math. Sci. Ser. B (Engl. Ed.) 36 (2016), no. 3, 913–930. MR 3479264
[47] Shukla R., Pant R.: Approximating solution of split equality and equilibrium problems by viscosity approximation algorithms. Comput. Appl. Math. 37 (2018), no. 4, 5293–5314. DOI 10.1007/s40314-018-0637-z | MR 3848595
[48] Taiwo A., Jolaoso L. O., Mewomo O. T.: A modified Halpern algorithm for approximating a common solution of split equality convex minimization problem and fixed point problem in uniformly convex Banach spaces. Comput. Appl. Math. 38 (2019), no. 2, Paper No. 77, 28 pages. DOI 10.1007/s40314-019-0841-5 | MR 3933584
[49] Taiwo A., Jolaoso L. O., Mewomo O. T.: Parallel hybrid algorithm for solving pseudomonotone equilibrium and split common fixed point problems. Bull. Malays. Math. Sci. Soc. 43 (2020), no. 2, 1893–1918. DOI 10.1007/s40840-019-00781-1 | MR 4061458
[50] Thong D. V., Hieu D. V.: An inertial method for solving split common fixed point problems. J. Fixed Point Theory Appl. 19 (2017), no. 4, 3029–3051. DOI 10.1007/s11784-017-0464-7 | MR 3720493
[51] Thong D. V., Hieu D. V.: Inertial subgradient extragradient algorithms with line-search process for solving variational inequality problems and fixed point problems. Numer. Algorithms 80 (2019), no. 4, 1283–1307. DOI 10.1007/s11075-018-0527-x | MR 3927234
[52] Zegeye H., Shahzad N.: Convergence of Mann's type iteration method for generalized asymptotically nonexpansive mappings. Comput. Math. Appl. 62 (2011), no. 11, 4007–4014. DOI 10.1016/j.camwa.2011.09.018 | MR 2859956
[53] Zhao J.: Solving split equality fixed-point problem of quasi-nonexpanive mappings without prior knowledge of operators norms. Optimization 64 (2015), no. 12, 2619–2630. DOI 10.1080/02331934.2014.883515 | MR 3411824
[54] Zhao J., He S.: Strong convergence of the viscosity approximation process for the split common fixed-point problem of quasi-nonexpansive mappings. J. Appl. Math. 2012 (2012), Art. ID 438023, 12 pages. MR 2904520
[55] Zhao J., Wang S.: Viscosity approximation methods for the split equality common fixed point problem of quasi-nonexpansive operators. Acta Math. Sci. Ser. B (Engl. Ed.) 36 (2016), no. 5, 1474–1486. MR 3529631
[56] Zhao J., Yang Q.: A simple projection method for solving the multiple-sets split feasibility problem. Inverse Probl. Sci. Eng. 21 (2013), no. 3, 537–546. DOI 10.1080/17415977.2012.712521 | MR 3022436
Partner of
EuDML logo