Minimizing the Moreau Envelope of Nonsmooth Convex Functions over the Fixed Point Set of Certain Quasi-Nonexpansive Mappings

  • Isao Yamada
  • Masahiro Yukawa
  • Masao Yamagishi
Part of the Springer Optimization and Its Applications book series (SOIA, volume 49)


The first aim of this paper is to present a useful toolbox of quasi-nonexpansive mappings for convex optimization from the viewpoint of using their fixed point sets as constraints. Many convex optimization problems have been solved through elegant translations into fixed point problems. The underlying principle is to operate a certain quasi-nonexpansive mapping T iteratively and generate a convergent sequence to its fixed point. However, such a mapping often has infinitely many fixed points, meaning that a selection from the fixed point set Fix(T) should be of great importance. Nevertheless, most fixed point methods can only return an “unspecified” point from the fixed point set, which requires many iterations. Therefore, based on common sense, it seems unrealistic to wish for an “optimal” one from the fixed point set. Fortunately, considering the collection of quasi-nonexpansive mappings as a toolbox, we can accomplish this challenging mission simply by the hybrid steepest descent method, provided that the cost function is smooth and its derivative is Lipschitz continuous. A question arises: how can we deal with “nonsmooth” cost functions? The second aim is to propose a nontrivial integration of the ideas of the hybrid steepest descent method and the Moreau–Yosida regularization, yielding a useful approach to the challenging problem of nonsmooth convex optimization over Fix(T). The key is the use of smoothing of the original nonsmooth cost function by its Moreau–Yosida regularization whose the derivative is always Lipschitz continuous. The field of application of hybrid steepest descent method can be extended to the minimization of the ideal smooth approximation Fix(T). We present the mathematical ideas of the proposed approach together with its application to a combinatorial optimization problem: the minimal antenna-subset selection problem under a highly nonlinear capacity-constraint for efficient multiple input multiple output (MIMO) communication systems.


Nonsmooth convex optimization Moreau envelope Hybrid steepest descent method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The first author thank Heinz Bauschke, Patrick Combettes and Russell Luke for their kind encouragement and invitation of the first author to the dream meeting: The Interdisciplinary Workshop on Fixed-Point Algorithms for Inverse Problems in Science and Engineering in November 1–6, 2009 at the Banff International Research Station.


  1. 1.
    Apostol, T.M.: Mathematical Analysis, 2nd ed. Addison-Wesley (1974)Google Scholar
  2. 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)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Baillon, J.-B., Haddad, G.: Quelques propriétés des opérateurs angle-bornés et n-cycliquement monotones. Isr. J. Math. 26, 137–150 (1977)MathSciNetzbMATHGoogle Scholar
  4. 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)MathSciNetGoogle Scholar
  5. 5.
    Barbu, V., Precupanu, Th.: Convexity and Optimization in Banach Spaces, 3rd Ed. D. Reidel Publishing Company (1986)Google Scholar
  6. 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)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bauschke, H.H., Combettes, P.L.: A weak-to-strong convergence principle for Fejér monotone methods in Hilbert space. Math. Oper. Res. 26, 248–264 (2001)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bauschke, H.H., Combettes, P.L.: The Baillon-Haddad theorem revisited. J. Convex Anal. 17, 781–787 (2010)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer (2011)Google Scholar
  11. 11.
    Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sciences 2, 183–202 (2009)MathSciNetzbMATHGoogle Scholar
  12. 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)MathSciNetGoogle Scholar
  13. 13.
    Bertero, M., Boccacci, P.: Introduction to Inverse Problems in Imaging. IOP (1998)Google Scholar
  14. 14.
    Borwein, J.M., Fitzpatrick, S., Vanderwerff, J.: Examples of convex functions and classifications of normed spaces. J. Convex Anal. 1, 61–73 (1994)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Bougeard, M.L.: Connection between some statistical estimation criteria, lower-C 2 functions and Moreau-Yosida approximates. In: Bulletin International Statistical Institute 47th session 1, INSEE Paris Press, pp. 159–160 (1989)Google Scholar
  16. 16.
    Bougeard, M.L., Caquineau, C.D.: Parallel proximal decomposition algorithms for robust estimation. Ann. Oper. Res. 90, 247–270 (1999)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge Univ. Press (2004)zbMATHGoogle Scholar
  18. 18.
    Bregman, L.M.: The method of successive projection for finding a common point of convex sets. Soviet Math. Dokl. 6, 688–692 (1965)zbMATHGoogle Scholar
  19. 19.
    Browder, F.E.: Convergence theorems for sequences of nonlinear operators in Banach spaces. Math. Z. 100, 201–225 (1967)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Byrne, C.L.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Byrne, C.L.: Applied Iterative Methods. A K Peters, Ltd., Wellesley, Massachusettes (2007)Google Scholar
  22. 22.
    Cai, J.F., Candés, E.J., Shen, Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20, 1956–1982 (2010)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Candés, E.J., Wakin, M.B.: An introduction to compressive sampling. IEEE Signal Process. Mag. 25, 21–30 (2008)Google Scholar
  24. 24.
    Capel, D., Zisserman, A.: Computer vision applied to super resolution. IEEE Signal Process. Mag. 20, 75–86 (2003)Google Scholar
  25. 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)MathSciNetGoogle Scholar
  26. 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)MathSciNetGoogle Scholar
  27. 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)MathSciNetGoogle Scholar
  28. 28.
    Censor, Y., Reich, S.: Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization. Optimization 37, 323–339 (1996)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Censor, Y., Zenios, S.A.: Parallel Optimization: Theory, Algorithm, and Optimization. Oxford University Press (1997)Google Scholar
  30. 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)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20, 89–97 (2004)MathSciNetGoogle Scholar
  32. 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)MathSciNetzbMATHGoogle Scholar
  33. 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)Google Scholar
  34. 34.
    Combettes, P.L.: Foundation of set theoretic estimation. Proc. IEEE. 81, 182–208 (1993)Google Scholar
  35. 35.
    Combettes, P.L.: Inconsistent signal feasibility problems: least squares solutions in a product space. IEEE Trans. Signal Process. 42, 2955–2966 (1994)Google Scholar
  36. 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)Google Scholar
  37. 37.
    Combettes, P.L.: Convex set theoretic image recovery by extrapolated iterations of parallel subgradient projections. IEEE Trans. Image Process. 6, 493–506 (1997)Google Scholar
  38. 38.
    Combettes, P.L.: Strong convergence of block-iterative outer approximation methods for convex optimization. SIAM J. Control Optim. 38, 538–565 (2000)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Combettes, P.L.: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 53, 475–504 (2004)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Combettes, P.L., Bondon, P.: Hard-constrained inconsistent signal feasibility problems. IEEE Trans. Signal Process. 47, 2460–2468 (1999)zbMATHGoogle Scholar
  41. 41.
    Combettes, P.L., Pesquet, J.-C.: Image restoration subject to a total variation constraint. IEEE Trans. Image Process. 13, 1213–1222 (2004)Google Scholar
  42. 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)Google Scholar
  43. 43.
    Combettes, P.L., Pesquet, J.-C.: A proximal decomposition method for solving convex variational inverse problems. Inverse Probl. 24 (2008)Google Scholar
  44. 44.
    Combettes, P.L., Pesquet, J.-C.: Split convex minimization algorithm for signal recovery. Proc. 2009 IEEE ICASSP (Taipei), 685–688 (2009)Google Scholar
  45. 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)Google Scholar
  46. 46.
    Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. SIAM Multiscale Model. Simul. 4, 1168–1200 (2005)MathSciNetzbMATHGoogle Scholar
  47. 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)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Deutsch, F., Best Approximation in Inner Product Spaces. Springer, New York (2001)zbMATHGoogle Scholar
  49. 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)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Dolidze, Z.O.: Solutions of variational inequalities associated with a class of monotone maps. Ekonomika i Matem. Metody 18, 925–927 (1982)MathSciNetGoogle Scholar
  51. 51.
    Donoho, D.L.: De-noising by soft-thresholding. IEEE Trans. Inf. Theory 41, 613–627 (1995)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theory 52, 1289–1306 (2006)MathSciNetGoogle Scholar
  53. 53.
    Donoho, D.L., Johnstone, I.M.: Ideal spatial adaptation via wavelet shrinkage. Biometrika 81, 425–455 (1994)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Dotson, Jr, W.G.: On the Mann iterative process. Trans. Amer. Math. Soc. 149, 65–73 (1970)MathSciNetzbMATHGoogle Scholar
  55. 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)Google Scholar
  56. 56.
    Dunn, J.C.: Convexity, monotonicity, and gradient processes. J. Math. Anal. Appl. 53, 145–158 (1976)MathSciNetzbMATHGoogle Scholar
  57. 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)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Eicke, B.: Iteration methods for convexly constrained ill-posed problems in Hilbert space. Numer. Funct. Anal. Optim. 13, 413–429 (1992)MathSciNetzbMATHGoogle Scholar
  59. 59.
    Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. Classics in Applied Mathematics 28, SIAM (1999)Google Scholar
  60. 60.
    Elsner, L., Koltracht, L., Neumann, M.: Convergence of sequential and asynchronous nonlinear paracontractions. Numer. Math. 62, 305–319 (1992)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Engle, H.W., Leit\(\tilde{\mbox{ a}}\)o, A.: A Mann iterative regularization method for elliptic Cauchy problems. Numer. Funct. Anal. Optim. 22, 861–884 (2001)Google Scholar
  62. 62.
    Fadili, M.J., Starck, J.-L.: Monotone operator splitting for optimization problems in sparse recovery. Proc. 2009 IEEE ICIP, Cailo (2009)Google Scholar
  63. 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)Google Scholar
  64. 64.
    Fukushima, M.: A relaxed projection method for variational inequalities. Math. Program. 35, 58–70 (1986)MathSciNetzbMATHGoogle Scholar
  65. 65.
    Fukushima, M., Qi, L.: A globally and superlinearly convergent algorithm for nonsmooth convex minimization. SIAM J. Optim. 6, 1106–1120 (1996)MathSciNetzbMATHGoogle Scholar
  66. 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)Google Scholar
  67. 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)Google Scholar
  68. 68.
    Gandy, S., Recht, B., Yamada, I.: Tensor completion and low-n-rank tensor recovery via convex optimization. Inverse Probl. 27(2), 025010 (2011)MathSciNetGoogle Scholar
  69. 69.
    Gharavi-Alkhansari, M., Gershman, A.B.: Fast antenna subset selection in MIMO systems. IEEE Trans. Signal Process. 52, 339–347 (2004)MathSciNetGoogle Scholar
  70. 70.
    Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge Univ. Press. (1990)Google Scholar
  71. 71.
    Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. New York and Basel Dekker (1984)Google Scholar
  72. 72.
    Goldstein, A.A.: Convex programming in Hilbert space. Bull. Amer. Math. Soc. 70, 709–710 (1964)MathSciNetzbMATHGoogle Scholar
  73. 73.
    Golshtein, E.G., Tretyakov, N.V.: Modified Lagrangians and Monotone Maps in Optimization. Wiley (1996)zbMATHGoogle Scholar
  74. 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)MathSciNetGoogle Scholar
  75. 75.
    Groetsch, C.W.: A note on segmenting Mann iterates. J. Math. Anal. Appl. 40, 369–372 (1972)MathSciNetzbMATHGoogle Scholar
  76. 76.
    Groetsch, C.W.: Inverse Problems in Mathematical Sciences. Wiesbaden-Vieweg (1993)Google Scholar
  77. 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)Google Scholar
  78. 78.
    Halpern, B.: Fixed points of nonexpanding maps. Bull. Amer. Math. Soc. 73, 957–961 (1967)MathSciNetzbMATHGoogle Scholar
  79. 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)Google Scholar
  80. 80.
    Haugazeau, Y.: Sur les Inéquations variationnelles et la Minimisation de Fonctionnelles Convexes. Thèse, Universite de Paris (1968)Google Scholar
  81. 81.
    Haykin, S.: Adaptive Filter Theory, 4th edn. Prentice Hall (2002)Google Scholar
  82. 82.
    Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. Springer (1993)Google Scholar
  83. 83.
    Huber, P.J.: Robust estimation of a location parameter. Ann. Math. Statist. 35, 73–101 (1964)MathSciNetzbMATHGoogle Scholar
  84. 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)Google Scholar
  85. 85.
    Kiwiel, K.C.: Block-iterative surrogate projection methods for convex feasibility problems. Linear Alg. Appl. 215, 225–259 (1995)MathSciNetzbMATHGoogle Scholar
  86. 86.
    Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley Classics Library, Wiley, New York (1989)zbMATHGoogle Scholar
  87. 87.
    Levitin, E.S., Polyak, B.T.: Constrained minimization method. USSR Comput. Maths. Phys. 6, 1–50 (1966)Google Scholar
  88. 88.
    Li, W., Swetits, J.J.: The linear 1 estimator and the Huber M-estimator. SIAM J. Optim. 8, 457–475 (1998)MathSciNetzbMATHGoogle Scholar
  89. 89.
    Lions, P.L.: Approximation de points fixes de contractions. C. R. Acad. Sci. Paris Sèrie A-B 284, 1357–1359 (1977)zbMATHGoogle Scholar
  90. 90.
    Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)MathSciNetzbMATHGoogle Scholar
  91. 91.
    Liu, F., Nashed, M.Z.: Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set-Valued Anal. 6, 313–344 (1998)MathSciNetzbMATHGoogle Scholar
  92. 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)MathSciNetzbMATHGoogle Scholar
  93. 93.
    Mangasarian, O.L., Muscicant, D.R.: Robust linear and support vector regression. IEEE Trans. Pattern Anal. Mach. Intell. 22, 950–955 (2000)Google Scholar
  94. 94.
    Mann, W.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4, 506–510 (1953)MathSciNetzbMATHGoogle Scholar
  95. 95.
    Mehta, N.B., Molisch, A.F.: MIMO System Technology for Wireless Communications, chapter 6, CRC Press (2006)Google Scholar
  96. 96.
    Michelot, C., Bougeard, M.L.: Duality results and proximal solutions of the Huber M-estimator problem. Appl. Math. Optim. 30, 203–221 (1994)MathSciNetzbMATHGoogle Scholar
  97. 97.
    Molisch, A.F., Win, M.Z.: MIMO systems with antenna selection. IEEE Microw. Mag. 5, 46–56 (2004)Google Scholar
  98. 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)MathSciNetzbMATHGoogle Scholar
  99. 99.
    Moreau, J.J.: Proximité et dualité dans un espace hilbertien. Bull. Soc. Math. France 93, 273–299 (1965)MathSciNetzbMATHGoogle Scholar
  100. 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)MathSciNetzbMATHGoogle Scholar
  101. 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)MathSciNetzbMATHGoogle Scholar
  102. 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)MathSciNetzbMATHGoogle Scholar
  103. 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)MathSciNetzbMATHGoogle Scholar
  104. 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)MathSciNetzbMATHGoogle Scholar
  105. 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)Google Scholar
  106. 106.
    Pierra, G.: Decomposition through formalization in a product space. Math. Program. 28, 96–115 (1984)MathSciNetzbMATHGoogle Scholar
  107. 107.
    Polyak, B.T.: Minimization of unsmooth functionals. USSR Comput. Maths. Phys. 9, 14–29 (1969)Google Scholar
  108. 108.
    Rockafellar, R.T.: Monotone operators and proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)MathSciNetzbMATHGoogle Scholar
  109. 109.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis, 1st edn. Springer (1998)Google Scholar
  110. 110.
    Sabharwal, A., Potter, L.C.: Convexly constrained linear inverse problems: Iterative least-squares and regularization. IEEE Trans. Signal Process. 46, 2345–2352 (1998)MathSciNetzbMATHGoogle Scholar
  111. 111.
    Sanayei, S., Nosratinia, A.: Antenna selection in MIMO systems. IEEE Commun. Mag. 42, 68–73 (2004)Google Scholar
  112. 112.
    Sayed, A.H.: Fundamentals of Adaptive Filtering. Wiley-IEEE Press (2003)Google Scholar
  113. 113.
    Slavakis, K., Yamada, I.: Robust wideband beamforming by the hybrid steepest descent method. IEEE Trans. Signal Process. 55, 4511–4522 (2007)MathSciNetGoogle Scholar
  114. 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)MathSciNetzbMATHGoogle Scholar
  115. 115.
    Slavakis, K., Theodoridis, S., Yamada, I.: Online kernel-based classification using adaptive projection algorithms. IEEE Trans. Signal Process. 56, 2781–2796 (2008)MathSciNetGoogle Scholar
  116. 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)MathSciNetGoogle Scholar
  117. 117.
    Starck, J.-L., Murtagh, F.: Astronomical Image and Data Analysis, 2nd.edn. Springer (2006)Google Scholar
  118. 118.
    Starck, J.-L., Murtagh, F., Fadili, J.M.: Sparse Image and Signal Processing – Wavelets, Curvelets, Morphological Diversity. Cambridge Univ. Press (2010)zbMATHGoogle Scholar
  119. 119.
    Stark, H., Yang, Y.: Vector Space Projections – A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics. Wiley (1998)Google Scholar
  120. 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)MathSciNetzbMATHGoogle Scholar
  121. 121.
    Takahashi, W.: Nonlinear Functional Analysis – Fixed Point Theory and its Applications. Yokohama Publishers (2000)Google Scholar
  122. 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)MathSciNetzbMATHGoogle Scholar
  123. 123.
    Takahashi, N., Yamada, I.: Steady-state mean-square performance analysis of a relaxed set-membership NLMS algorithm by the energy conservation argument. IEEE Trans. Signal Process. 57, 3361–3372 (2009)MathSciNetGoogle Scholar
  124. 124.
    Telatar, I.E.: Capacity of multi-antenna Gaussian channels. Eur. Trans. Telecomm. 10, 585–595 (1999)Google Scholar
  125. 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)Google Scholar
  126. 126.
    Tseng, P.: Applications of a splitting algorithm to decomposition in convex programming and variational inequalities. SIAM J. Control Optim. 29, 119–138 (1991)MathSciNetzbMATHGoogle Scholar
  127. 127.
    Vasin, V.V., Ageev, A.L.: Ill-Posed Problems with A Priori Information. VSP (1995)Google Scholar
  128. 128.
    Widrow, B., Stearns, S.D.: Adaptive Signal Processing. Prentice Hall (1985)Google Scholar
  129. 129.
    Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. Math. 58, 486–491 (1992)MathSciNetzbMATHGoogle Scholar
  130. 130.
    Xu, H.K., Kim, T.H.: Convergence of hybrid steepest descent methods for variational inequalities. J. Optim. Theory Appl. 119, 185–201 (2003)MathSciNetzbMATHGoogle Scholar
  131. 131.
    Yamada, I.: Approximation of convexly constrained pseudoinverse by Hybrid Steepest Descent Method. Proc. 1999 IEEE ISCAS, Florida (1999)Google Scholar
  132. 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)Google Scholar
  133. 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)Google Scholar
  134. 134.
    Yamada, I.: Kougaku no Tameno Kansu Kaiseki (Functional Analysis for Engineering), Suurikougaku-Sha/Saiensu-Sha (2009)Google Scholar
  135. 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)MathSciNetzbMATHGoogle Scholar
  136. 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)MathSciNetzbMATHGoogle Scholar
  137. 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)Google Scholar
  138. 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)MathSciNetzbMATHGoogle Scholar
  139. 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)Google Scholar
  140. 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)Google Scholar
  141. 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)Google Scholar
  142. 142.
    Yosida, K.: Functional Analysis, 4th edn. Springer (1974)Google Scholar
  143. 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)Google Scholar
  144. 144.
    Yukawa, M., Yamada, I.: Pairwise optimal weight realization – acceleration technique for set-theoretic adaptive parallel subgradient projection algorithm. IEEE Trans. Signal Process. 54, 4557–4571 (2006)Google Scholar
  145. 145.
    Yukawa, M., Yamada, I.: Minimal antenna-subset selection under capacity constraint for power-efficient MIMO systems: a relaxed 1-minimization approach. Proc. 2010 IEEE ICASSP, Dallas (2010)Google Scholar
  146. 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)Google Scholar
  147. 147.
    Yukawa, M., Slavakis, K., Yamada, I.: Adaptive parallel quadratic-metric projection algorithms. IEEE Trans. Audio Speech Lang. Process. 15, 1665–1680 (2007)Google Scholar
  148. 148.
    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific (2002)Google Scholar
  149. 149.
    Zeidler, E.: Nonlinear Functional Analysis and its Applications, III – Variational Methods and Optimization. Springer (1985)Google Scholar
  150. 150.
    Zeidler, E.: Nonlinear Functional Analysis and its Applications, II/B – Nonlinear Monotone Operators. Springer (1990)Google Scholar
  151. 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)MathSciNetzbMATHGoogle Scholar
  152. 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)MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Communications and Integrated SystemsTokyo Institute of TechnologyTokyoJapan

Personalised recommendations