Advertisement

Iterative Averaging of Entropic Projections for Solving Stochastic Convex Feasibility Problems

  • Dan Butnariu
  • Yair Censor
  • Simeon Reich
Article

Abstract

The problem considered in this paper is that of finding a point which iscommon to almost all the members of a measurable family of closed convexsubsets of R++ n , provided that such a point exists.The main results show that this problem can be solved by an iterative methodessentially based on averaging at each step the Bregman projections withrespect to f(x)=∑i=1nxi· ln xi ofthe current iterate onto the given sets.

stochastic convex feasibility problem Bregman projection entropic projection modulus of local convexity very convex function 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Y. Alber and D. Butnariu, "Convergence of Bregman-projection methods for solving consistent convex feasibility problems in reflexive Banach spaces," J. Optim. Theory and Appl., vol. 92, pp. 33–61, 1997.Google Scholar
  2. 2.
    J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser: Boston, 1990.Google Scholar
  3. 3.
    H.H. Bauschke and J.M. Borwein, "On projection algorithms for solving convex feasibility problems," SIAM Review, vol. 38, pp. 367–426, 1996.Google Scholar
  4. 4.
    H.H. Bauschke and J.M. Borwein, "Legendre functions and the method of random Bregman projections," Convex Analysis (to appear).Google Scholar
  5. 5.
    L.M. Bregman, "The relaxation method for finding the common point of convex sets and its application to the solution of convex programming," USSR Comp. Math. and Math. Phys., vol. 7, pp. 200–217, 1967.Google Scholar
  6. 6.
    D. Butnariu, "The expected-projection method: Its behavior and applications to linear operator equations and convex optimization," J. Applied Analysis, vol. 1, pp. 95–108, 1995.Google Scholar
  7. 7.
    D. Butnariu and Y. Censor, "Strong convergence of almost simultaneous projection methods in Hilbert spaces," J. Comput. Appl. Math., vol. 53, pp. 33–42, 1994.Google Scholar
  8. 8.
    D. Butnariu and S.D. Flåm, "Strong convergence of expected-projection methods in Hilbert spaces," Numer. Funct. Anal. Optim., vol. 15, pp. 601–636, 1995.Google Scholar
  9. 9.
    Y. Censor and A. Lent, "An iterative row-action method for interval convex programming," J. Optim. Theory and Appl., vol. 34, no. 3, pp. 321–353, 1981.Google Scholar
  10. 10.
    Y. Censor and T. Elfving, "A multiprojection algorithm using Bregman projections in a product space," Numerical Algorithms, vol. 8, pp. 221–239, 1994.Google Scholar
  11. 11.
    Y. Censor and S. Reich, "Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization," Optimization, vol. 37, pp. 323–339, 1996.Google Scholar
  12. 12.
    F.H. Clarke, Optimization and Nonsmooth Analysis, John Willey & Sons: New York, 1983.Google Scholar
  13. 13.
    P.L. Combettes, "The foundations of set theoretic estimation," Proc. IEEE, vol. 81, pp. 182–208, 1993.Google Scholar
  14. 14.
    A.R. De Pierro and A.N. Iusem, "A relaxed version of Bregman's method for convex programming," J. Optim. Theory Appl., vol. 51, pp. 421–440, 1986.Google Scholar
  15. 15.
    P.R. Halmos, Measure Theory, Springer-Verlag: New York, 1974.Google Scholar
  16. 16.
    A.N. Iusem and A.R. De Pierro, "Convergence results for an accelerated Cimmino algorithm," Numer. Math., vol. 49, pp. 347–368, 1986.Google Scholar
  17. 17.
    R.R. Phelps, Convex Functions, Monotone Operators and Differentiability, Springer-Verlag: Berlin, 1993.Google Scholar
  18. 18.
    R.T. Rockafellar and R.J.-B. Wets, "Scenarios and policy aggregation in optimization under uncertaunty," Mathematics of Operation Research, vol. 16, pp. 119–147, 1991.Google Scholar
  19. 19.
    A.A. Vladimirov, Y.E. Nesterov, and Y.N. Cekanov, "Uniformly convex functions," Vestnik Moskovskaya Universiteta, Series Matematika i Kybernetika, vol. 3, pp. 12–23, 1978.Google Scholar
  20. 20.
    R.J.-B. Wets, "Stochastic Programming," in Handbook of Operation Research and Management Sciences, G.L. Nemhauser, A.H.G. Rinnooy Kan and M.J. Todd (Eds.), vol. 1: Optimization, North-Holland, Amsterdam, pp. 573–629, 1989.Google Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • Dan Butnariu
    • 1
  • Yair Censor
    • 1
  • Simeon Reich
    • 2
  1. 1.Department of Mathematics and Computer ScienceUniversity of HaifaHaifaIsrael
  2. 2.Department of MathematicsThe Technion–Israel Institute of TechnologyHaifaIsrael

Personalised recommendations