論文使用權限 Thesis access permission:自定論文開放時間 user define
開放時間 Available:
校內 Campus: 已公開 available
校外 Off-campus: 已公開 available
論文名稱 Title |
廣義分裂公共固定點問題之迭代方法與應用 Several Iterative Methods to Solve General Split Common Fixed Point Problems with Applications |
||
系所名稱 Department |
|||
畢業學年期 Year, semester |
語文別 Language |
||
學位類別 Degree |
頁數 Number of pages |
103 |
|
研究生 Author |
|||
指導教授 Advisor |
|||
召集委員 Convenor |
|||
口試委員 Advisory Committee |
|||
口試日期 Date of Exam |
2018-09-04 |
繳交日期 Date of Submission |
2018-09-05 |
關鍵字 Keywords |
慣性法、半壓縮映射、分裂可行性問題、近似壓縮映射、S迭代法、分裂公共固定點問題、黏滞法、強擬非擴張映射 S-iterations, Nearly contractive mappings, Inertial methods, Demicontractive mappings, Viscosity methods, Split feasibility problems, Strongly quasi-nonexpansive mappings, Split common fixed point problems |
||
統計 Statistics |
本論文已被瀏覽 5687 次,被下載 58 次 The thesis/dissertation has been browsed 5687 times, has been downloaded 58 times. |
中文摘要 |
令p,r為兩個正整數,H, K_{1}, K_{2},…, K_{r}為實希爾伯特空間,且對所有 j=1, 2, …, r,A_{j}: H→ K_{j}為有界線性算子。我們考慮以下廣義公共固定點問題(GSCFPP): 找一個點 x∈ ∩_{i=1}^{p}Fix(U_{i}) 使得對所有 j=1,…, r,我們有A_{j}x∈ Fix(T_{j}), 其中所有 i=1, 2, …, p,U_{i}: H→ H為α-強擬非擴張映射,所有 j=1, 2,…, r,T_{j}: K_{j}→ K_{j}為β-半壓縮映射。在第二章中,我們簡要介紹了凸分析中的一些基本結果和工具,以幫助我們分析迭代方法的收斂性,我們還介紹了凸最小化問題、變分問題和包含問題。我們構建了幾個迭代法來解決GSCFPP,並在第三章中討論了其弱收斂和強收斂。我們也提供慣性類型的迭代方法以獲得新的收斂結果,這將在第四章中介紹。應用這些結果,在第五章中,我們解決了幾個分裂型逆問題,包括了分裂變分問題、分裂包含問題。 在論文的最後,我們提供了一些人為數值實驗來證實理論結果並觀察迭代法的行為。 |
Abstract |
Let p,r be two positive integers, H, K_{1}, K_{2},…, K_{r} be real Hilbert spaces, and A_{j}: H→ K_{j} be a bounded linear operator for j=1, 2, …, r. We consider the following general split common fixed point problems (GSCFPP): Find a point x∈ ∩_{i=1}^{p}Fix(U_{i}) such that A_{j}x∈ Fix(T_{j}), for all j=1, 2, …, r. where U_{i}: H→ H is an α-strongly quasi-nonexpansive operator for i=1, 2, …, p, and T_{j}: K_{j}→ K_{j} is a β-demicontractive operator for j= j=1, 2, …, r. In Chapter 2, we briefly introduce some fundamental results and tools in the convex analysis to help us analyze the convergence property of iterative methods, we also introduce the convex minimization problems, variational problems, and inclusion problems. We construct several iterations for solving GSCFPP and discuss the weak convergence and strong convergence in Chapter 3. We also provide inertial-type iteration to obtain new results of convergence which is presented in Chapter 4. Applying these results, in Chapter 5, we solve several split-type inverse problems including split variational problems, split inclusion problems. At the end of this thesis, we provide some synthetic experiments to corroborate the theoretical results and observe the behavior of iterations. |
目次 Table of Contents |
1 Introduction 1 1.1 Convex Feasibility Problems . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Split Feasibility Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Multiple-Sets Split Feasibility Problems . . . . . . . . . . . . . . . . . . . 4 1.4 Other Split-type Inverse Problems . . . . . . . . . . . . . . . . . . . . . . 5 2 Preliminaries 7 2.1 Convexity, Lower Semi-continuity and Subdifferentiability . . . . . . . . . 8 2.2 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Nonexpansiveness, Demi-closeness and Fejer Monotonicity . . . . . . . . . 14 2.3.1 Nonexpansiveness . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.2 Demi-closeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.3 Fejer Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Some Technical Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 Iterative algorithms for GSCFPP 23 3.1 Weak Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1.1 Picard Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.2 Krasnosel’ski˘ı-Mann algorithm Iteration . . . . . . . . . . . . . . . 29 3.1.3 S-Iterations and Normal S-Iteration . . . . . . . . . . . . . . . . 31 3.2 Strong Convergence: NC-Viscosity type Algorithm . . . . . . . . . . . . . 35 4 Inertial Algorithms for GSCFPP 45 4.1 Weak Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.1.1 Inertial Picard Iteration . . . . . . . . . . . . . . . . . . . . . . . . 47 4.1.2 Inertial Krasnosel’ski˘ı-Mann Iterations . . . . . . . . . . . . . . . 49 4.1.3 Inertial S |
參考文獻 References |
[1] R. P. Agarwal, D. O. Regan, D. R. Sahu, Fixed Point Theory for Lipschitzian-type Mappings with Applications, Topological Fixed Point Theory and Its Applications, 6, Springer, New York, 2009. [2] R. P. Agarwal, D. O. Regan, D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal. 8 (2007), 61–79. [3] F. Alvarez, 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. [4] H. Attouch, A. Cabot, Convergence Rates of Inertial Forward-Backward Algorithms, SIAM J. Optim., 28(1) (2018), 849–874. [5] H. H. Bauschke, J. M. Borwein, On Projection Algorithms for Solving Convex Feasibility Problems, SIAM Rev. 38 (1996), 367–426. [6] H. H. Bauschke, J.V. Burke, F.R. Deutsch, H.S. Hundal, J.D. Vanderwerff, A new proximal point iteration that converges weakly but not in norm. Proceedings of the American Mathematical Society 133 (2005) 1829–1835. [7] H. H. Bauschke, P. L. Combettes, A weak-to-strong convergence principle for Fejermonotone methods in Hilbert spaces. Math. Oper. Res. 26 (2001), 248–264. [8] H. H. Bauschke, P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, CMS Books in Mathematics, Springer, New York, 2011. [9] A. Beck, M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183–202. [10] J. M. Borwein, A. S. Lewis, Convex Analysis and Nonlinear Optimization, Theory and Examples, Springer, 2006. [11] F. E. Browder, Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. USA 54 (1965), 1041–1044. [12] C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems 18 (2002), 441–453. [13] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems 20 (2004), 103–120. [14] C. Byrne, Y. Censor, A. Gibali, S. Reich, The split common null point problem, J. Nonlinear Convex Anal. 13 (2012) 759–775. [15] A. Cegielski, Iterative Methods for Fixed Point Problems in Hilbert Spaces, Lecture Notes in Mathematics, vol. 2057. Springer, Heidelberg, 2012. [16] L. C. Ceng and J. C. Yao, Convergence and certain control conditions for hybrid viscosity approximation methods, Nonlinear Anal. 73 (2010), 2078–2087. [17] L. C. Ceng, Q. H. Ansari, and J. C. Yao, Mann-type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces. Numer. Funct. Anal. Optim. 29 (2008), no. 9-10, 987–1033. [18] L. C. Ceng, H. K. Xu, and J. C. Yao, A hybrid steepest-descent method for variational inequalities in Hilbert spaces, Appl. Anal. 87 (2008), no. 5, 575–589. [19] Y. Censor, Row-action methodsfor huge and sparse systems and their applications, SIAM Rev., 23 (1981), 444-466. [20] Y. Censor, Iterative methods for the convexfeasibility problem, in Convexity and Graph Theory, M. Rosenfeld and J. Zaks, eds., North-Holland, Amsterdam, 1984, 83-91. Proceedings of the Conference on Convexity and Graph Theory, Israel, March 1981. Ann. Discrete Math., Vol. 20. [21] Y. Censor, Parallel application of block-iterative methods in medical imaging and radiation therapy, Math. Programming 42 (1988) 307–325. [22] Y. Censor, On variable block algebraic reconstruction techniques, in Mathematical Methods in Tomography, G. Herman, A. Louis, and E Natterer, eds., Springer, New York, 1990, 133-140. Proceedings of a Conference held in Oberwolfach, Germany, June 11-15. Lecture Notes in Math., Vol. 1497. [23] Y. Censor, M.D. Altschuler and W.D. Powlis, On the use of Cimmino’s simultaneous projections method for computing a solution of the inverse problem in radiation therapy treatment planning, Inverse Problems 4 (1988) 607–623. [24] Y. Censor, T. Bortfeld, B. Martin, A. Trofimov, A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. Med. Biol. 51 (2006), 2353–2365. [25] Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms 8 (1994), 221–239. [26] Y. Censor, T. Elfving, N. Kopf, T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems 21 (2005), 2071–2084. [27] Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numerical Algorithms 59 (2012), 301–323. [28] Y. Censor, G. T. Herman, On some optimization techniques in image reconstruction from projections, Appl. Numer. Math., 3 (1987), 365–391. [29] Y. Censor, A. Segal, The split common fixed point problem for directed operators, J. Convex Anal. 16 (2009), 587–600. [30] A. Chambolle, Ch. Dossal, On the convergence of the iterates of the “Fast Iterative Shrinkage Thresholding Algorithm,” J. Optim. Theory Appl., 166 (2015), 968–982. [31] C. E. Chidume, S¸ t. Marus¸ter, Iterative methods for the computation of fixed points of demicontractive mappings. J. Comput. Appl. Math. 234 (2010), no. 3, 861–882. [32] P. L. Combettes, V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul. 4 (2005), 1168–1200. [33] J. B. Conway, A Course in Functional Analysis, Springer-Verlag New York, 2007. [34] Y. Dang, , J. Sun, and H. Xu Inertial accelerated algorithms for solving a split feasibility problem, J. Ind. Manag. Optim. 13 (2017), no. 3, 1383–1394. [35] F, Deutsch, The method of alternating orthogonal projections Approximation Theory, Spline Functions and Applications ed S P Singh (Dordrecht: Kluwer). (1992) 105–21. [36] W. G. Dotson Jr., Fixed points of quasi-nonexpansive mappings, J. Austral. Math. Soc. 13 (1972), 167–170. [37] I. Ekeland, R. Temam, Convex Analysis and Variational Problems, Amsterdam, North-Holland, 1976. [38] K. Goebel, W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28, Cambridge University Press, Cambridge, UK, 1990 [39] O. Guler, On the convergence of the proximal point algorithm for convex minimization, SIAM Journal on Control and Optimization, 20 (1991), 403–419. [40] S. He, C. Yang Solving the variational inequality problem defined on intersection of finite level sets. Abstr. Appl. Anal. (2013) [41] G. T. Herman, Image Reconstructionfrom Projections, Academic Press, New York, 1980. [42] G. T. Herman, L.B. Meyer, Algebraic reconstruction techniques can be made computationally efficient, IEEE Trans. Medical Imaging 12 (1993) 600–609. [43] T. L. Hicks, J.D. Kubicek, On the Mann iteration process in a Hilbert spaces, J. Math. Anal. Appl. 59 (1977), 498–504. [44] T. H. Kim, H. K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005), 51–60 [45] A. Latif, D. R. Sahu, Q. H. Ansari, Variable KM-like algorithms for fixed point problems and split feasibility problems,Fixed Point Theory Appl. 211 (2014), 20. [46] J. Liang, J. Fadili, and G. Peyre, Local linear convergence of forward-backward under partial smoothness, in Advances in Neural Information Processing Systems, 2014, 1970–1978. [47] D. Lorenz, T. Pock: An inertial forward-backward algorithm for monotone inclusions. J. Math. Imaging Vis. 51 (2015), 311–325. [48] P. E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal. 16 (2008), 899–912. [49] P. E. Mainge, Convergence theorem for inertial KM-type algorithms, Journal of Computational and Applied Mathematics, 219 (2008), 223–236. [50] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506–510. [51] G. Marino, H.K. Xu, Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces, J. Math. Anal. Appl. 329, (2007), 336–346. [52] T.S. Motzkin, I.I. Schoenberg, The relaxation method for linear inequalities, Canad. J. Math., 6 (1954), 393–404. [53] A. Moudafi, Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241 (2000), 46–55. [54] A. Moudafi, The split common fixed point problem for demicontractive mappings, Inverse Problems, 26, (2010), 055007, 6. [55] A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl. 150 (2011), 275–283. [56] A. Moudafi, A note on the split common fixed-point problem for quasi-nonexpansive operators, Nonlinear Anal. 74 (2011), 4083–4087. [57] A. Moudafi, Viscosity-type algorithms for the split common fixed-point problem, Adv. Nonlinear Var. Inequal. 16 (2013), 61–68. [58] A. Moudafi, M. Oliny, Convergence of a splitting inertial proximal method for monotone operators. J. Comput. Appl. Math. 155 (2003), 447–454. [59] Yu. E. Nesterov, A method for solving the convex programming problem with convergence rate O(1=k2), Dokl. Akad. Nauk SSSR 269 (1983), no. 3, 543–547. [60] Z. Opial, Nonexpansive and Monotone Mappings in Banach Spaces. Lecture Notes 67-1, Center for Dynamical Systems, Brown University, Providence, RI, 1967. [61] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73 (1967), 591–597. [62] N. Parikh, S.P. Boyd, Proximal algorithms. Foundations and Trends in Optimization, 1(3), (2014), 123–231. [63] B. T. Polyak, Gradient methods for minimizing functionals, USSR Comput. Math. Math. Phys., 3 (1963), 864–878. [64] B. T. Polyak, Some methods of speeding up the convergence of iterative methods. Zh. Vychisl. Mat. Mat. Fiz. 4 (1964), 1–17. [65] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67 (1979), 274–276. [66] S. Reich, A limit theorem for projections, Linear and Multilinear Algebra 13 (1983), 281–290. [67] R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math. Volume 33, Number 1 (1970), 209-216. [68] D. R. Sahu, Fixed points of demicontinuous nearly Lipschitzian mappings in Banach spaces, Comment. Math. Univ. Carolin. 46 (2005), 653–666. [69] D. R. Sahu, Applications of the S -iteration process to constrained minimization problems and split feasibility problems, Fixed Point Theory, 12 (2011), 187–204. [70] D.R. Sahu, Q.H. Ansari, J.C. Yao, Convergence of inexact mann iterations generated by nearly nonexpansive sequences and applications, Numer. Funct. Anal. Optim. 37 (2016), 1312–1338. [71] D.R. Sahu, N.C. Wong, J.C. Yao, A unified hybrid iterative method for solving variational inequalities involving generalized pseudo-contractive mappings, SIAM J. Control Optim. 50 (2012), 2335-2354. [72] D.R. Sahu, N.C. Wong, J.C. Yao, A generalized hybrid steepest-descent method for variational inequalities in Banach spaces, Fixed Point Theory Appl. 2011 :Article ID 754702. [73] W. Takahashi, N. C. Wong, J.C. Yao, Attractive points and Halpern-type strong convergence theorems in Hilbert spaces. J. Fixed Point Theory Appl. 17 (2015), no. 2, 301–311. [74] Y. C. Tang, J. G. Peng, L.W. Liu, A cyclic algorithm for the split common fixed point problem of demicontractive mappings in Hilbert spaces, Math. Model. Anal. 17 (2012), 457–466. [75] Y. C. Tang, J. G. Peng, L.W. Liu, A cyclic and simultaneous iterative algorithm for the multiple split common fixed point problem of demicontractive mappings, Bull. Korean Math. Soc. 51 (2014), 1527–1538. [76] Y. C. Tang, L. W. Liu, Several iterative algorithms for solving the split common fixed point problem of directed operators with applications, Optimization 65 (2014), 53–65. [77] D. V. Thong, Viscosity approximation methods for solving fixed-point problems and split common fixed-point problems, J. Fixed Point Theory Appl. 19 (2017), 1481–1499. [78] V. V. Vasin, Ill-posed problems and iterative approximation of fixed points of pseudocontractive mappings, Ill posed Problem in Natural Sciences (1992), 214-223. [79] Y. Wang, T.-H. Kim, X. Fang, H. He, The split common fixed-point problem for demicontractive mappings and quasi-nonexpansive mappings, J. Nonlinear Sci. Appl. 10 (2017), 2976–2985. [80] F. Wang, H. K. Xu, Cyclic algorithms for split feasibility problems in Hilbert spaces, Nonlinear Analysis 74 (2011), 4105–4111. [81] H. K. Xu, T. H. Kim, Convergence of Hybrid Steepest-Descent Methods for Variational Inequalities, Journal of Optimization Theory and Applications, 119 (2003), 185–201. [82] H. K. Xu, Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66 (2002), 240–256. [83] H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004), 279–291. [84] H. K. Xu, A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems 22 (2006), 021–2034. [85] H. K. Xu, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Problems 26 (2010), 105018, 17. [86] I. Yamada, N. Ogura, Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings. Numer. Funct. Anal. Optim. 25 (2004), no. 7-8, 619–655. [87] Y. Yao, R. Chen, J. C. Yao, Strong convergence and certain control conditions of modified Mann iteration. Nonlinear Anal. 68 (2008), 1687–1693. [88] D.C. Youla, H. Webb, Image Restoration by the Method of Convex Projections: Part 1-Theory, IEEE Trans. Medical Imaging, (1982), 81-94. [89] J. Zhao and S. He, Strong convergence of the viscosity approximation process for the split common fixed-point problem of quasi-nonexpansive mappings, J. Appl. Math. 2012 (2012), 12 pages. |
電子全文 Fulltext |
本電子全文僅授權使用者為學術研究之目的,進行個人非營利性質之檢索、閱讀、列印。請遵守中華民國著作權法之相關規定,切勿任意重製、散佈、改作、轉貼、播送,以免觸法。 論文使用權限 Thesis access permission:自定論文開放時間 user define 開放時間 Available: 校內 Campus: 已公開 available 校外 Off-campus: 已公開 available |
紙本論文 Printed copies |
紙本論文的公開資訊在102學年度以後相對較為完整。如果需要查詢101學年度以前的紙本論文公開資訊,請聯繫圖資處紙本論文服務櫃台。如有不便之處敬請見諒。 開放時間 available 已公開 available |
QR Code |