| Citation: |
Liya Liu, Xiaolong Qin. ON THE STRONG CONVERGENCE OF A PROJECTION-BASED ALGORITHM IN HILBERT SPACES[J]. Journal of Applied Analysis & Computation, 2020, 10(1): 104-117. doi: 10.11948/20190004
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ON THE STRONG CONVERGENCE OF A PROJECTION-BASED ALGORITHM IN HILBERT SPACES
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Abstract
In this paper, we introduce a new projection-based algorithm for solving variational inequality problems with a Lipschitz continuous pseudo-monotone mapping in Hilbert spaces. We prove a strong convergence of the generated sequences. The numerical behaviors of the proposed algorithm on test problems are illustrated and compared with previously known algorithms. -
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References
[1] H. Attouch, J. Peypouquet and P. Redont, A dynamical approach to an inertial forward-backward algorithm for convex minimization, SIAM J. Optim., 2014, 24, 232–256. doi: 10.1137/130910294 [2] H. Attouch and A. Cabot, Convergence rates of inertial forward-backward algorithms, SIAM J. Optim., 2018, 28, 849–874. doi: 10.1137/17M1114739 [3] H. Ansari, F. Babu and J.C. Yao, Regularization of proximal point algorithms in Hadamard manifolds, J. Fixed Point Theory Appl., 2019, 21(1), 25. doi: 10.1007/s11784-019-0658-2 [4] F. Alvarez, Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space, SIAM J. Optim., 2004, 14, 773–782. doi: 10.1137/S1052623403427859 [5] A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2009, 2, 183–202. doi: 10.1137/080716542 [6] I. Boţ and E.R. Csetnek, A hybrid proximal-extragradient algorithm with inertial effects, Numer. Funct. Anal. Optim., 2015, 36, 951–963. doi: 10.1080/01630563.2015.1042113 [7] I. Boţ and E.R. Csetnek, An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems, Numer. Algo., 2016, 71, 519–540. doi: 10.1007/s11075-015-0007-5 [8] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Amsterdam. 1976. [9] Y. Cho, Strong convergence analysis of a hybrid algorithm for nonlinear operators in a Banach space, J. Appl. Anal. Comput., 2018, 8, 19–31. [10] Y. Cho and S.M. Kang, Approximation of common solutions of variational inequalities via strict pseudocontractions, Acta Math. Sci., 2012, 32, 1607–1618. doi: 10.1016/S0252-9602(12)60127-1 [11] Y. Cho, W. Li and S.M. Kang, Convergence analysis of an iterative algorithm for monotone operators, J. Inequal. Appl., 2013, 1, 199. [12] Y. Cho and S.M. Kang, Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process, Appl. Math. Lett., 2011, 24, 224–228. doi: 10.1016/j.aml.2010.09.008 [13] S. Chang, C.F. Wen and J.C. Yao, Common zero point for a finite family of inclusion problems of accretive mappings in Banach spaces, Optimization, 2018, 67, 1183–1196. doi: 10.1080/02331934.2018.1470176 [14] S. Chang, C.F. Wen and J.C. Yao, Zero point problem of accretive operators in Banach spaces, Bull. Malaysian Math. Sci. Soc. 2019, 42, 105–118. doi: 10.1007/s40840-017-0470-3 [15] L.C. Ceng, Q.H. Ansari and J.C. Yao, An extragradient method for solving split feasibility and fixed point problems, Comput. Math. Appl., 2012, 64, 633–642. doi: 10.1016/j.camwa.2011.12.074 [16] Y. Censor, A. Gibali and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl., 2011, 148, 318–335. doi: 10.1007/s10957-010-9757-3 [17] Y. Censor, A. Gibali and S. Reich, Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space, Optim. Methods Softw., 2011, 26, 827–845. doi: 10.1080/10556788.2010.551536 [18] W. Cottle and J.C. Yao, Pseudo-monotone complementarity problems in Hilbert space, J. Optim. Theory Appl., 1992, 75, 281–295. doi: 10.1007/BF00941468 [19] J. Chen, E. Kobis, J.C. Yao, Optimality conditions and duality for robust nonsmooth multiobjective optimization problems with constraints, J. Optim. Theory Appl., 2019, 181, 411-436. doi: 10.1007/s10957-018-1437-8 [20] V. Hieu, P.K. Anh and L.D. Muu, Modified hybrid projection methods for finding common solutions to variational inequality problems, Comput. Optim. Appl., 2017, 66, 75–96. doi: 10.1007/s10589-016-9857-6 [21] X. Hu and J. Wang, Solving pseudomonotone variational inequalities and pseudoconvex optimization problems using the projection neural network, IEEE Trans. Neural Networks, 2006, 17, 1487–1499. doi: 10.1109/TNN.2006.879774 [22] M. Korpelevich, The extragradient method for finding saddle points and other problems, Matecon, 1976, 12, 747–756. [23] S. Karamardian and S. Scchaible, Seven kinds of monotone maps, J. Optim. Theory Appl., 1990, 66, 37–46. doi: 10.1007/BF00940531 [24] L. Liu, A hybrid steepest descent method for solving split feasibility problems involving nonexpansive mappings, J. Nonlinear Convex Anal. 20 (2019), 471-488. [25] A. Lorenz and T. Pock, An inertial forward-backward algorithm for monotone inclusions, J. Math. Imaging Vision., 2015, 51, 311–325. doi: 10.1007/s10851-014-0523-2 [26] E. Maingé, Strong convergence of projected reflected gradient methods for variational inequalities, Fixed Point Theory, 2018, 19, 659–680. doi: 10.24193/fpt-ro.2018.2.52 [27] E. Maingé, Regularized and inertial algorithms for common fixed points of nonlinear operators, J. Math. Anal. Appl., 2008, 34, 876–887. [28] A. Moudafi and M. Oliny, Convergence of a splitting inertial proximal method for monotone operators, J. Comput. Appl. Math., 2003, 155, 447–454. doi: 10.1016/S0377-0427(02)00906-8 [29] V. Malitsky and V.V. Semenov, A hybrid method without extrapolation step for solving variational inequality problems, J. Global Optim., 2015, 61, 193–202. doi: 10.1007/s10898-014-0150-x [30] N. Nadezhkina and W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM J. Optim., 2006, 16, 1230–1241. doi: 10.1137/050624315 [31] T. Polyak, Some methods of speeding up the convergence of iteration methods, Comput. Math. Math. Phys., 1964, 4, 1–17. [32] X. Qin and N.T. An, Smoothing algorithms for computing the projection onto a Minkowski sum of convex sets, Comput. Optim. Appl., 2019, https://doi.org/10.1007/s10589-019-00124-7. doi: 10.1007/s10589-019-00124-7 [33] W. Takahashi, C.F. Wen and J.C. Yao, The shrinking projection method for a finite family of demimetric mappings with variational inequality problems in a Hilbert space, Fixed Point Theory, 2018, 19, 407–419. doi: 10.24193/fpt-ro.2018.1.32 [34] V. Thong and D.V. Hieu, An inertial method for solving split common fixed point problems, J. Fixed Point Theory Appl., 2017, 19, 3029–3051. doi: 10.1007/s11784-017-0464-7 [35] X. Zhao, K.F. Ng, C. Li and J.C. Yao, Linear regularity and linear convergence of projection-based methods for solving convex feasibility problems, Appl. Math. Optim., 2018, 78, 613–641. doi: 10.1007/s00245-017-9417-1 [36] A. Voitova, S.V. Denisov and V.V. Semenov, Strongly convergent modification of Korpelevich's method for equilibrium programming problems, J. Comput. Appl. Math., 2011, 104, 10–23. -
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Liya Liu, Xiaolong Qin. ON THE STRONG CONVERGENCE OF A PROJECTION-BASED ALGORITHM IN HILBERT SPACES[J]. Journal of Applied Analysis & Computation, 2020, 10(1): 104-117. doi: 10.11948/20190004
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Figure 1. Iterative steps of Algorithm 2.1. The number of the projection onto the feasible set
$ C $ is$ 1 $ . -
Figure 2. Behaviors of elements of
$ (x_{n})_{10\times 1} $ and the value of$ \| D_{n}\| $ with the number of iterations$ n = 500 $ . Numerical results for Algorithms 2.1, 3.1. -
Figure 3. Behaviors of elements of
$ (x_{n})_{10\times 1} $ and the value of$ \| D_{n}\| $ with the number of iterations$ n = 200 $ . Numerical results for Algorithms 2.1, 3.2. -
Figure 4. Behaviors of elements of
$ (D_{n})_{4\times 1} $ and$ (E_{n})_{4\times 1} $ with the number of iterations$ n = 50 $ . Numerical results for Algorithm 2.1.