A viscosity-proximal gradient method with inertial extrapolation for solving certain minimization problems in Hilbert space

Jolaoso, L.O.   ; Abass, H.A.   ; Mewomo, O.T.

Archivum mathematicum, Tome 55 (2019) no. 3, pp. 167-194

Abstract (VO)

In this paper, we study the strong convergence of the proximal gradient algorithm with inertial extrapolation term for solving classical minimization problem and finding the fixed points of $\delta $-demimetric mapping in a real Hilbert space. Our algorithm is inspired by the inertial proximal point algorithm and the viscosity approximation method of Moudafi. A strong convergence result is achieved in our result without necessarily imposing the summation condition $\sum _{n=1}^\infty \beta _n\Vert x_{n-1} -x_n\Vert + \infty $ on the inertial term. Finally, we provide some applications and numerical example to show the efficiency and accuracy of our algorithm. Our results improve and complement many other related results in the literature.

Détail

DOI : 10.5817/AM2019-3-167

Classification : 46N10, 47H10, 47J25, 65K10, 65K15
Keywords: proximal gradient algorithm; proximal operator; demimetric mappings; inertial algorithm; viscosity approximation; Meir Keeler contraction; fixed point theory

Bibliographie

[1] Abass, H.A., Ogbuisi, F.U., Mewomo, O.T.: Common solution of split equilibrium problem and fixed point problem with no prior knowledge of operator norm. U.P.B. Sci. Bull., Series A 80 (1) (2018), 175–190. | MR

[2] Alvarez, F., Attouch, H.: An inertial proximal method for monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9 (2001), 3–11. | DOI | MR

[3] Beck, A., Teboull, M.: Gradient-based algorithms with applications to signal-recovery problems. Convex optimization in signal processing and communications (Palomar, D., Elder, Y., eds.), Cambridge Univ. Press, Cambridge, 2010, pp. 42–88. | MR

[4] Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problem. SIAM J. Imaging Sci. 2 (1) (2009), 183–202. | DOI | MR

[5] Bot, R.I., Csetnek, E.R.: An inerial Tseng’s type proximal point algorithm for nonsmooth and nonconvex optimization problem. J. Optim. Theory Appl. 171 (2016), 600–616. | DOI | MR

[6] Bot, R.I., Csetnek, E.R., Laszlo, S.C.: An inertial forward-backward algorithm for the minimization of the sum of two nonconvex functions. EJCO 4 (2016), 3–25. | MR

[7] Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems 20 (2004), 103–120. | DOI | MR

[8] Cai, G., Shehu, Y.: An iterative algorithm for fixed point problem and convex minimization problem with applications. Fixed Point Theory and Appl. 2015 123 (2015), 18 pp. | MR

[9] Ceng, L.-C., Ansari, Q.H., Ya, J.-C.: Some iterative methods for finding fixed points and for solving constrained convex minimization problems. Nonlinear Anal. 74 (2011), 5286–5302. | DOI | MR

[10] Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8 (2–4) (1994), 221–239. | DOI | MR | Zbl

[11] Chambolle, A., Dossal, C.: On the convergence of the iterates of the fast iterative shrinkage thresholding algorithm. J. Optim. Theory Appl. 166 (2015), 968–982. | DOI | MR

[12] Chan, R.H., Ma, S., Jang, J.F.: Inertial proximal ADMM for linearly constrained separable convex optimization. SIAM J. Imaging Sci. 8 (4) (2015), 2239–2267. | DOI | MR

[13] Combettes, P.L.: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 53 (2004), 475–504. | DOI | MR

[14] Combettes, P.L., Pesquet, J.-C.: Proximal Splitting Methods in Signal Processing. Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer, New York, 2011, pp. 185–212. | MR

[15] Fichera, G.: Problemi elastostatic con vincoli unilaterli: II Problema di signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I (8) 7 (1963/1964), 91–140. | MR

[16] Geobel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, Cambridge, 1990. | MR

[17] Guo, Y., Cui, W.: Strong convergence and bounded perturbation resilence of a modified proximal gradient algorithm. J. Ineq. Appl. 2018 (2018). | DOI | MR

[18] Izuchukwu, C., Ugwunnadi, G.C., Mewomo, O.T., Khan, A.R., Abbas, M.: Proximal-type algorithms for split minimization problem in P-uniformly convex metric spaces. Numer. Algorithms (2018), | DOI | MR

[19] 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 (3) (2018), 167–183. | DOI | MR

[20] Jolaoso, L.O., Oyewole, K.O., Okeke, C.C., Mewomo, O.T.: A unified algorithm for solving split generalized mixed equilibrium problem and fixed point of nonspreading mapping in Hilbert space. Demonstratio Math. 51 (2018), 211–232. | DOI | MR

[21] Jolaoso, L.O., Taiwo, A., Alakoya, T.O., Mewomo, O.T.: A strong convergence theorem for solving variational inequalities using an inertial viscosity subgradient extragradient algorithm with self adaptive stepsize. Demonstratio Math. 52 (1) (2019), 183–203. | MR

[22] Lions, J.L., Stampacchia, G.: Variational inequalities. Comm. Pure Appl. Math. 20 (1967), 493–519. | DOI | MR | Zbl

[23] Lorenz, D., Pock, T.: An inertial forward-backward algorithm for monotone inclusions. J. Math. Imaging Vision 51 (2) (2015), 311–325. | DOI | MR

[24] Maingé, P.E.: Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 325 (2007), 469–479. | DOI | MR

[25] Maingé, P.E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 16 (2008), 899–912. | DOI | MR

[26] Martinez-Yanes, C., Xu, H.K.: Strong convergence of the CQ method for fixed-point iteration processes. Nonlinear Anal. 64 (2006), 2400–2411. | DOI | MR

[27] Meir, A., Keeler, E.: A theorem on contraction mappings. J. Math. Anal. Appl. 28 (1969), 326–329. | DOI | MR

[28] Mewomo, O.T., Ogbuisi, F.U.: Convergence analysis of iterative method for multiple set split feasibility problems in certain Banach spaces. Quaestiones Math. 41 (1) (2018), 129–148. | DOI | MR

[29] Moudafi, A.: Viscosity approximation method for fixed-points problems. J. Math. Anal. Appl. 241 (1) (2000), 46–55. | DOI | MR

[30] Moudafi, A., Oliny, M.: Convergence of a splitting inertial proximal method for monotone operators. J. Comput. Appl. Math. 155 (2003), 447–454. | DOI | MR

[31] Moudafi, A., Thakur, B.S.: Solving proximal split feasibility problems without prior knowledge of operator norms. Optim. Lett. 8 (7) (2014), 2099–2110. | DOI | MR

[32] Nesterov, Y.: A method for solving the convex programming problem with convergence rate $0(\frac{1}{k^2})$. Dokl. Akad. Nauk SSSR 269 (3) (1983), 543–547. | MR

[33] Ogbuisi, F.U., Mewomo, O.T.: On split generalized mixed equilibrium problems and fixed point problems with no prior knowledge of operator norm. J. Fixed Point Theory Appl. 19 (3) (2016), 2109–2128. | DOI | MR

[34] Ogbuisi, F.U., Mewomo, O.T.: Iterative solution of split variational inclusion problem in a real Banach space. Afrika Mat. (3) 28 (1–2) (2017), 295–309. | DOI | MR

[35] Ogbuisi, F.U., Mewomo, O.T.: Convergence analysis of common solution of certain nonlinear problems. Fixed Point Theory 19 (1) (2018), 335–358. | DOI | MR

[36] Okeke, C.C., Mewomo, O.T.: On split equilibrim problem, variational inequality problem and fixed point problem for multi-valued mappings. Ann. Acad. Rom. Sci. Ser. Math. Appl. 9 (2) (2017), 255–280. | MR

[37] Parith, N., Boyd, S.: Proximal algorithms. Foundations and Trends in Optimization 1 (3) (2013), 123–231.

[38] Pesquet, J.-C., Putselnik, N.: A parallel inertial proximal optimization method. Pacific J. Optim. 8 (2) (2012), 273–306. | MR

[39] Polyak, B.T.: Some methods of speeding up the convergence of iteration methods. U.S.S.R. Comput. Math. Math. Phys. 4 (5) (1964), 1–17. | DOI | MR

[40] Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Amer. Math. Soc. 149 (1970), 75–88. | DOI | MR

[41] Rockafellar, R.T., Wets, R.: Variational Analysis. Springer, Berlin, 1988.

[42] Shehu, Y.: Approximation of solutions to constrained convex minimization problem in Hilbert spaces. Vietnam J. Math. (2014), DOI 10.1007/s10013-014-0091-1. | DOI | MR

[43] Stampacchia, G.: Formes bilinearies coercitives sur les ensembles convexes. Comput. Rend. Acad. Sci. Paris 258 (1964), 4413–4416. | MR

[44] Su, M., Xu, H.K.: Remarks on the gradient-projection algorithm. J. Nonlinear Anal. Optim. 1 (1) (2010), 35–43. | MR

[45] Suzuki, T.: Moudai’s viscosity approximations with Meir-Keeler contractions. J. Math. Anal. Appl. 325 (2007), 342–352. | DOI | MR

[46] Takahashi, W., Wen, C.-F., Yao, J.-C.: The shrinking projection method for a finite family of demimetric mappings with variational inequality problems in a Hilbert space. Fixed Point Theory 19 (1) (2018), 407–420. | DOI | MR

[47] Tian, M., Huang, L.H.: A general approximation method for a kind of convex optimization problems in Hilbert spaces. J. Appl. Math. 2014 (2014), 9 pages, Article ID 156073. | MR

[48] Xu, H.K.: Viscosity approximation method for nonexpansive mappings. J. Math. Anal. Appl. 298 (1) (2004), 279–291. | DOI | MR

[49] Xu, H.K.: Average mappings and the gradient projection algorithm. J. Optim. Theory Appl. 150 rm (2) (2011), 360–378. | DOI | MR