Minimizing the moreau envelope of nonsmooth convex functions over the fixed point set of certain quasi-nonexpansive mappings

Springer Optimization and Its Applications

Volumn 49, Issue , 2011, Pages 345-390

  Yamada, Isao ^a   Yukawa, Masahiro ^a   Yamagishi, Masao ^a  

^a TOKYO INSTITUTE OF TECHNOLOGY   (Japan)

Author keywords

Hybrid steepest descent method; Moreau envelope; Nonsmooth convex optimization

Indexed keywords

EID: 84976466459     PISSN: 19316828     EISSN: 19316836     Source Type: Book Series    
DOI: 10.1007/978-1-4419-9569-8_17     Document Type: Chapter

References (152)

1. Apostol, T.M.: Mathematical Analysis, 2nd ed. Addison-Wesley (1974)
2. Ascher, U.M., Haber, E., Huang, H.: On effective methods for implicit piecewise smooth surface recovery. SIAM J. Sci. Comput. 28, 339–358 (2006)
3. Baillon, J.-B., Haddad, G.: Quelques propriétés des opérateurs angle-bornés et n-cycliquement monotones. Isr. J. Math. 26, 137–150 (1977)
4. Baillon, J.-B., Bruck, R.E., Reich, S.: On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces. Houst. J. Math. 4, 1–9 (1978)
5. Barbu, V., Precupanu, Th.: Convexity and Optimization in Banach Spaces, 3rd Ed. D. Reidel Publishing Company (1986)
6. Bauschke, H.H.: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space. J. Math. Anal. Appl. 202, 150–159 (1996)
7. Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)
8. Bauschke, H.H., Combettes, P.L.: A weak-to-strong convergence principle for Fejérmonotone methods in Hilbert space. Math. Oper. Res. 26, 248–264 (2001)
9. Bauschke, H.H., Combettes, P.L.: The Baillon-Haddad theorem revisited. J. Convex Anal. 17, 781–787 (2010)
10. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer (2011)
11. Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sciences 2, 183–202 (2009)
12. Beck, A., Teboulle, M.: Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. IEEE Trans. Image Process. 18, 2419–2434 (2009)
13. Bertero, M., Boccacci, P.: Introduction to Inverse Problems in Imaging. IOP (1998)
14. Borwein, J.M., Fitzpatrick, S., Vanderwerff, J.: Examples of convex functions and classifications of normed spaces. J. Convex Anal. 1, 61–73 (1994)
15. Bougeard, M.L.: Connection between some statistical estimation criteria, lower-C2 functions and Moreau-Yosida approximates. In: Bulletin International Statistical Institute 47th session 1, INSEE Paris Press, pp. 159–160 (1989)
16. Bougeard, M.L., Caquineau, C.D.: Parallel proximal decomposition algorithms for robust estimation. Ann. Oper. Res. 90, 247–270 (1999)
17. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge Univ. Press (2004)
18. Bregman, L.M.: The method of successive projection for finding a common point of convex sets. Soviet Math. Dokl. 6, 688–692 (1965)
19. Browder, F.E.: Convergence theorems for sequences of nonlinear operators in Banach spaces. Math. Z. 100, 201–225 (1967)
20. Byrne, C.L.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)
21. Byrne, C.L.: Applied Iterative Methods. A K Peters, Ltd., Wellesley, Massachusettes (2007)
22. Cai, J.F., Candés, E.J., Shen, Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20, 1956–1982 (2010)
23. Candés, E.J., Wakin, M.B.: An introduction to compressive sampling. IEEE Signal Process. Mag. 25, 21–30 (2008)
24. Capel, D., Zisserman, A.: Computer vision applied to super resolution. IEEE Signal Process. Mag. 20, 75–86 (2003)
25. Cavalcante, R., Yamada, I.: Multiaccess interference suppression in orthogonal space-time block coded MIMO systems by adaptive projected subgradient method. IEEE Trans. Signal Process. 56, 1028–1042 (2008)
26. Cavalcante, R., Yamada, I.: A flexible peak-to-average power ratio reduction scheme for OFDM systems by the adaptive projected subgradient method. IEEE Trans. Signal Process. 57, 1456–1468 (2009)
27. Cavalcante, R., Yamada, I., Mulgrew, B.: An adaptive projected subgradient approach to learning in diffusion networks. IEEE Trans. Signal Process. 57, 2762–2774 (2009)
28. Censor, Y., Reich, S.: Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization. Optimization 37, 323–339 (1996)
29. Censor, Y., Zenios, S.A.: Parallel Optimization: Theory, Algorithm, and Optimization. Oxford University Press (1997)
30. Censor, Y., Iusem, A.N., Zenios, S.A.: An interior point method with Bregman functions for the variational inequality problem with paramonotone operators. Math. Program. 81, 373–400 (1998)
31. Chambolle, A.: An algorithm for total variation minimization and applications. J.Math. Imaging Vis. 20, 89–97 (2004)
32. Chambolle, A., DeVore, R.A., Lee, N.Y., Lucier, B.J.: Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage. IEEE Trans. Image Process. 7, 319–335 (1998)
33. Chidume, C.: Geometric Properties of Banach Spaces and Nonlinear Iterations (Chapter 7: Hybrid steepest descent method for variational inequalities). Lecture Notes in Mathematics 1965, Springer (2009)
34. Combettes, P.L.: Foundation of set theoretic estimation. Proc. IEEE. 81, 182–208 (1993)
35. Combettes, P.L.: Inconsistent signal feasibility problems: least squares solutions in a product space. IEEE Trans. Signal Process. 42, 2955–2966 (1994)
36. Combettes, P.L.: Construction d’un point fixe commun à une famille de contractions fermes. C.R. Acad. Sci.Paris Sèr. I Math. 320, 1385–1390 (1995)
37. Combettes, P.L.: Convex set theoretic image recovery by extrapolated iterations of parallel subgradient projections. IEEE Trans. Image Process. 6, 493–506 (1997)
38. Combettes, P.L.: Strong convergence of block-iterative outer approximation methods for convex optimization. SIAM J. Control Optim. 38, 538–565 (2000)
39. Combettes, P.L.: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 53, 475–504 (2004)
40. Combettes, P.L., Bondon, P.: Hard-constrained inconsistent signal feasibility problems. IEEE Trans. Signal Process. 47, 2460–2468 (1999)
41. Combettes, P.L., Pesquet, J.-C.: Image restoration subject to a total variation constraint. IEEE Trans. Image Process. 13, 1213–1222 (2004)
42. Combettes, P.L., Pesquet, J.-C.: A Douglas-Rachford splitting approach to nonsmooth convex variational signal recovery. IEEE J. Sel. Top. Signal Process. 1, 564–574 (2007)
43. Combettes, P.L., Pesquet, J.-C.: A proximal decomposition method for solving convex variational inverse problems. Inverse Probl. 24 (2008)
44. Combettes, P.L., Pesquet, J.-C.: Split convex minimization algorithm for signal recovery. Proc. 2009 IEEE ICASSP (Taipei), 685–688 (2009)
45. Combettes, P.L., Pesquet, J.-C.: Proximal splitting methods in signal processing. In: H. H. Bauschke, R. Burachik, P. L. Combettes, V. Elser, D. R. Luke, H. Wolkowicz (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer (2010)
46. Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. SIAM Multiscale Model. Simul. 4, 1168–1200 (2005)
47. Daubechies, I., Defrise, M., Mol, C.D.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm. Pure Appl. Math. 57, 1413–1457 (2004)
48. Deutsch, F., Best Approximation in Inner Product Spaces. Springer, New York (2001)
49. Deutsch, F., Yamada, I.: Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numer. Funct. Anal. Optim. 19, 33–56 (1998)
50. Dolidze, Z.O.: Solutions of variational inequalities associated with a class of monotone maps. Ekonomika i Matem. Metody 18, 925–927 (1982)
51. Donoho, D.L.: De-noising by soft-thresholding. IEEE Trans. Inf. Theory 41, 613–627 (1995)
52. Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theory 52, 1289–1306 (2006)
53. Donoho, D.L., Johnstone, I.M.: Ideal spatial adaptation via wavelet shrinkage. Biometrika 81, 425–455 (1994)
54. Dotson, Jr, W.G.: On the Mann iterative process. Trans. Amer. Math. Soc. 149, 65–73 (1970)
55. Dua, A., Medepalli, K., Paulraj, A.J.: Receive antenna selection in MIMO systems using convex optimization. IEEE Trans. Wirel. Commun. 5, 2353–2357 (2006)
56. Dunn, J.C.: Convexity, monotonicity, and gradient processes. J. Math. Anal. Appl. 53, 145–158 (1976)
57. Eckstein, J., Bertsekas, D.P.: On the Douglas-Rachfold splitting method and proximal point algorithm for maximal monotone operators. Math. Program. 55, 293–318 (1992)
58. Eicke, B.: Iteration methods for convexly constrained ill-posed problems in Hilbert space. Numer. Funct. Anal. Optim. 13, 413–429 (1992)
59. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. Classics in Applied Mathematics28, SIAM (1999)
60. Elsner, L., Koltracht, L., Neumann, M.: Convergence of sequential and asynchronous nonlinear paracontractions. Numer. Math. 62, 305–319 (1992)
61. Engle, H.W., Leitão, A.: A Mann iterative regularization method for elliptic Cauchy problems. Numer. Funct. Anal. Optim. 22, 861–884 (2001)
62. Fadili, M.J., Starck, J.-L.: Monotone operator splitting for optimization problems in sparse recovery. Proc. 2009 IEEE ICIP, Cailo (2009)
63. Foschini, G.J., Gans, M.J.: On limits of wireless communications in a fading environment when using multiple antennas. Wirel. Pers. Commun. 6, 311–335 (1998)
64. Fukushima, M.: A relaxed projection method for variational inequalities. Math. Program. 35, 58–70 (1986)
65. Fukushima, M., Qi, L.: A globally and superlinearly convergent algorithm for nonsmooth convex minimization. SIAM J. Optim. 6, 1106–1120 (1996)
66. Gabay, D.: Applications of the method of multipliers to variational inequalities. In: M. Fortin and R. Glowinski (eds.) Augmented Lagrangian Methods: Applications to the solution of boundary value problems, North-Holland, Amsterdam (1983)
67. Gandy, S., Yamada, I.: Convex optimization techniques for the efficient recovery of a sparsely corrupted low-rank matrix. Journal of Math-for-Industry 2(2010B-5), 147–156 (2010)
68. Gandy, S., Recht, B., Yamada, I.: Tensor completion and low-n-rank tensor recovery via convex optimization. Inverse Probl. 27(2), 025010 (2011)
69. Gharavi-Alkhansari, M., Gershman, A.B.: Fast antenna subset selection in MIMO systems. IEEE Trans. Signal Process. 52, 339–347 (2004)
70. Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge Univ. Press. (1990)
71. Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. New York and Basel Dekker (1984)
72. Goldstein, A.A.: Convex programming in Hilbert space. Bull. Amer. Math. Soc. 70, 709–710 (1964)
73. Golshtein, E.G., Tretyakov, N.V.: Modified Lagrangians and Monotone Maps in Optimization. Wiley (1996)
74. Gorokhov, A., Gore, D.A., Paulraj, A.J.: Receive antenna selection for MIMO spatial multiplexing: theory and algorithms. IEEE Trans. Signal Process. 51, 2796–2807 (2003)
75. Groetsch, C.W.: A note on segmenting Mann iterates. J. Math. Anal. Appl. 40, 369–372 (1972)
76. Groetsch, C.W.: Inverse Problems in Mathematical Sciences. Wiesbaden-Vieweg (1993)
77. Gubin, L.G., Polyak, B.T., Raik, E.V.: The method of projections for finding the common point of convex sets. USSR Comput. Maths. Phys. 7, 1–24 (1967)
78. Halpern, B.: Fixed points of nonexpanding maps. Bull. Amer. Math. Soc. 73, 957–961 (1967)
79. Hasegawa, H., Ohtsuka, T., Yamada, I., Sakaniwa, K.: An edge-preserving super-precision for simultaneous enhancement of spacial and grayscale resolutions. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E91-A, 673–681 (2008)
80. Haugazeau, Y.: Sur les Inéquations variationnelles et la Minimisation de Fonctionnelles Convexes. Thèse, Universite de Paris (1968)
81. Haykin, S.: Adaptive Filter Theory, 4th edn. Prentice Hall (2002)
82. Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. Springer (1993)
83. Huber, P.J.: Robust estimation of a location parameter. Ann. Math. Statist. 35, 73–101 (1964)
84. Iemoto, S., Takahashi, W.: Strong convergence theorems by a hybrid steepest descent method for countable nonexpansive mappings in Hilbert spaces. Sci.Math. Jpn. 69 (online: 2008-49), 227–240 (2009)
85. Kiwiel, K.C.: Block-iterative surrogate projection methods for convex feasibility problems. Linear Alg. Appl. 215, 225–259 (1995)
86. Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley Classics Library, Wiley, New York (1989)
87. Levitin, E.S., Polyak, B.T.: Constrained minimization method. USSR Comput. Maths. Phys. 6, 1–50 (1966)
88. Li, W., Swetits, J.J.: The linear _1 estimator and the Huber M-estimator. SIAM J. Optim. 8, 457–475 (1998)
89. Lions, P.L.: Approximation de points fixes de contractions. C. R. Acad. Sci. Paris Sèrie A-B 284, 1357–1359 (1977)
90. Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)
91. Liu, F., Nashed, M.Z.: Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set-Valued Anal. 6, 313–344 (1998)
92. Mainge, P.E.: Extension of the hybrid steepest descent method to a class of variational inequalities and fixed point problems with nonself-mappings. Numer. Funct. Anal. Optim. 29, 820–834 (2008)
93. Mangasarian, O.L., Muscicant, D.R.: Robust linear and support vector regression. IEEE Trans. Pattern Anal. Mach. Intell. 22, 950–955 (2000)
94. Mann, W.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4, 506–510 (1953)
95. Mehta, N.B., Molisch, A.F.:MIMO System Technology forWireless Communications, chapter 6, CRC Press (2006)
96. Michelot, C., Bougeard, M.L.: Duality results and proximal solutions of the Huber M-estimator problem. Appl. Math. Optim. 30, 203–221 (1994)
97. Molisch, A.F., Win, M.Z.: MIMO systems with antenna selection. IEEE Microw. Mag. 5, 46–56 (2004)
98. Moreau, J.J.: Fonctions convexes duales et points proximaux dans un espace hilbertien. C. R. Acad. Sci. Paris Ser. A Math. 255, 2897–2899 (1962)
99. Moreau, J.J.: Proximité et dualité dans un espace hilbertien. Bull. Soc. Math. France 93, 273–299 (1965)
100. Nikolova, M.: Minimizing of cost functions involving nonsmooth data-fidelity terms – Application to the processing of outliers. SIAM J. Numer. Anal. 40, 965–994 (2002)
101. Ogura, N., Yamada, I.: Non-strictly convex minimization over the fixed point set of the asymptotically shrinking nonexpansive mapping. Numer. Funct. Anal. Optim. 23, 113–137 (2002)
102. Ogura, N., Yamada, I.: Non-strictly convex minimization over the bounded fixed point set of nonexpansive mapping. Numer. Funct. Anal. Optim. 24, 129–135 (2003)
103. Ogura, N., Yamada, I.: A deep outer approximating half space of the level set of certain Quadratic Functions. J. Nonlinear Convex Anal. 6, 187–201 (2005)
104. Passty, G.B.: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J. Math. Anal. Appl. 72, 383–390 (1979)
105. Pierra, G.: Eclatement de contraintes en parallèle pour la minimisation d’une forme quadratique. Lecture Notes in Computer Science 41, 200–218, Springer (1976)
106. Pierra, G.: Decomposition through formalization in a product space. Math. Program. 28, 96–115 (1984)
107. Polyak, B.T.: Minimization of unsmooth functionals. USSR Comput. Maths. Phys. 9, 14–29 (1969)
108. Rockafellar, R.T.:Monotone operators and proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)
109. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis, 1st edn. Springer (1998)
110. Sabharwal, A., Potter, L.C.: Convexly constrained linear inverse problems: Iterative leastsquares and regularization. IEEE Trans. Signal Process. 46, 2345–2352 (1998)
111. Sanayei, S., Nosratinia, A.: Antenna selection in MIMO systems. IEEE Commun. Mag. 42, 68–73 (2004)
112. Sayed, A.H.: Fundamentals of Adaptive Filtering. Wiley-IEEE Press (2003)
113. Slavakis, K., Yamada, I.: Robust wideband beamforming by the hybrid steepest descent method. IEEE Trans. Signal Process. 55, 4511–4522 (2007)
114. Slavakis, K., Yamada, I., Ogura, N.: The adaptive projected subgradient method over the fixed point set of strongly attracting nonexpansive mappings. Numer. Funct. Anal. Optim. 27, 905–930 (2006)
115. Slavakis, K., Theodoridis, S., Yamada, I.: Online kernel-based classification using adaptive projection algorithms. IEEE Trans. Signal Process. 56, 2781–2796 (2008)
116. Slavakis, K., Theodoridis, S., Yamada, I.: Adaptive constrained filtering in reproducing kernel Hilbert spaces: the beamforming case. IEEE Trans. Signal Process. 57, 4744–4764 (2009)
117. Starck, J.-L., Murtagh, F.: Astronomical Image and Data Analysis, 2nd.edn. Springer (2006)
118. Starck, J.-L., Murtagh, F., Fadili, J.M.: Sparse Image and Signal Processing – Wavelets, Curvelets, Morphological Diversity. Cambridge Univ. Press (2010)
119. Stark, H., Yang, Y.: Vector Space Projections – A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics. Wiley (1998)
120. Suzuki, T.: A sufficient and necessary condition for Halpern-type strong convergence to fixed points of nonexpansive mappings. Proc. Amer. Math. Soc. 135, 99–106 (2007)
121. Takahashi, W.: Nonlinear Functional Analysis – Fixed Point Theory and its Applications. Yokohama Publishers (2000)
122. Takahashi, N., Yamada, I.: Parallel algorithms for variational inequalities over the Cartesian product of the intersections of the fixed point sets of nonexpansive mappings. J. Approx. Theory 153, 139–160 (2008)
123. Takahashi, N., Yamada, I.: Steady-state mean-square performance analysis of a relaxed setmembership NLMS algorithm by the energy conservation argument. IEEE Trans. Signal Process. 57, 3361–3372 (2009)
124. Telatar, I.E.: Capacity of multi-antenna Gaussian channels. Eur. Trans. Telecomm. 10, 585–595 (1999)
125. Theodoridis, S., Slavakis, K., Yamada, I.: Adaptive learning in a world of projections – A unifying framework for linear and nonlinear classification and regression tasks. IEEE Signal Processing Mag. 28, 97–123 (2011)
126. Tseng, P.: Applications of a splitting algorithm to decomposition in convex programming and variational inequalities. SIAM J. Control Optim. 29, 119–138 (1991)
127. Vasin, V.V., Ageev, A.L.: Ill-Posed Problems with A Priori Information. VSP (1995)
128. Widrow, B., Stearns, S.D.: Adaptive Signal Processing. Prentice Hall (1985)
129. Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. Math. 58, 486–491 (1992)
130. Xu, H.K., Kim, T.H.: Convergence of hybrid steepest descent methods for variational inequalities. J. Optim. Theory Appl. 119, 185–201 (2003)
131. Yamada, I.: Approximation of convexly constrained pseudoinverse by Hybrid Steepest Descent Method. Proc. 1999 IEEE ISCAS, Florida (1999)
132. Yamada, I.: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In: D. Butnariu, Y. Censor, S. Reich (eds.) Inherently Parallel Algorithm for Feasibility and Optimization and Their Applications, Elsevier, 473–504 (2001)
133. Yamada, I.: Adaptive projected subgradient method: A unified view for projection based adaptive algorithms. The Journal of IEICE 86, 654–658 (2003)(in Japanese)
134. Yamada, I.: Kougaku no Tameno Kansu Kaiseki (Functional Analysis for Engineering), Suurikougaku-Sha/Saiensu-Sha (2009)
135. Yamada, I., Ogura, N.: Adaptive projected subgradient method for asymptotic minimization of sequence of nonnegative convex functions. Numer. Funct. Anal. Optim. 25, 593–617 (2004)
136. Yamada, I., Ogura, N.: Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings. Numer. Funct. Anal. Optim. 25, 619–655 (2004)
137. Yamada, I., Ogura, N., Yamashita, Y., Sakaniwa, K.: An extension of optimal fixed point theorem for nonexpansive operator and its application to set theoretic signal estimation. Technical Report of IEICE DSP96-106, 63–70 (1996)
138. Yamada, I., Ogura, N., Yamashita, Y., Sakaniwa, K.: Quadratic optimization of fixed points of nonexpansive mappings in Hilbert space. Numer. Funct. Anal. Optim. 19, 165–190 (1998)
139. Yamada, I., Ogura, N., Shirakawa, N.: A numerically robust hybrid steepest descent method for the convexly constrained generalized inverse problems. In: Z. Nashed, O. Scherzer (eds.) Inverse Problems, Image Analysis, and Medical Imaging, Contemporary Mathematics 313, 269–305 (2002)
140. Yamada, I., Slavakis, K., Yamada, K.: An efficient robust adaptive filtering algorithm based on parallel subgradient projection techniques. IEEE Trans. Signal Process. 50, 1091–1101 (2002)
141. Yamagishi, M., Yamada, I.: A deep monotone approximation operator based on the best quadratic lower bound of convex functions. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E91-A, 1858–1866 (2008)
142. Yosida, K.: Functional Analysis, 4th edn. Springer (1974)
143. Youla, D.C., Webb, H.: Image restoration by the method of convex projections: Part 1 – Theory. IEEE Trans. Med. Imaging 1, 81–94 (1982)
144. Yukawa, M., Yamada, I.: Pairwise optimal weight realization – acceleration technique for settheoretic adaptive parallel subgradient projection algorithm. IEEE Trans. Signal Process. 54, 4557–4571 (2006)
145. 1-minimization approach. Proc. 2010 IEEE ICASSP, Dallas (2010)
146. Yukawa, M., Cavalcante, R., Yamada, I.: Efficient blind MAI suppression in DS/CDMA by embedded constraint parallel projection techniques. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E88-A, 2427–2435 (2005)
147. Yukawa, M., Slavakis, K., Yamada, I.: Adaptive parallel quadratic-metric projection algorithms. IEEE Trans. Audio Speech Lang. Process. 15, 1665–1680 (2007)
148. Zălinescu, C: Convex Analysis in General Vector Spaces. World Scientific (2002)
149. Zeidler, E.: Nonlinear Functional Analysis and its Applications, III – Variational Methods and Optimization. Springer (1985)
150. Zeidler, E.: Nonlinear Functional Analysis and its Applications, II/B – Nonlinear Monotone Operators. Springer (1990)
151. Zeng, L.C., Schaible, S., Yao, J.C.: Hybrid steepest descent methods for zeros of nonlinear operators with applications to variational inequalities. J. Optim. Theory Appl. 141, 75–91 (2009)
152. Zhang, B., Fadili, J.M., Starck, J.-L.: Wavelet, ridgelet, and curvelets for Poisson noise removal. IEEE Trans. Image Process. 17, 1093–1108 (2008)