Computational Optimization and Applications

, Volume 47, Issue 3, pp 455–476 | Cite as

Variable target value relaxed alternating projection method

  • A. CegielskiEmail author
  • R. Dylewski


In this paper we propose a modification of the von Neumann method of alternating projection x k+1=P A P B x k where A,B are closed and convex subsets of a real Hilbert space ℋ. If Fix P A P B then any sequence generated by the classical method converges weakly to a fixed point of the operator T=P A P B . If the distance δ=inf xA,yB xy is known then one can efficiently apply a modification of the von Neumann method, which has the form x k+1=P A (x k +λ k (P A P B x k x k )) for λ k >0 depending on x k (for details see: Cegielski and Suchocka, SIAM J. Optim. 19:1093–1106, 2008). Our paper contains a generalization of this modification, where we do not suppose that we know the value δ. Instead of δ we apply its approximation which is updated in each iteration.


Alternating projection method Relaxation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bauschke, H.H., Borwein, J.: Dykstra’s alternating projection algorithm for two sets. J. Approx. Theory 79, 418–443 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bauschke, H.H., Borwein, J.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bauschke, H.H., Combettes, P.L., Kruk, S.G.: Extrapolation algorithm for affine-convex feasibility problems. Numer. Algorithms 41, 239–274 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bauschke, H.H., Deutsch, F., Hundal, H., Park, S.-H.: Accelerating the convergence of the method of alternating projection. Trans. AMS 355, 3433–3461 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cegielski, A.: A method of projection onto an acute cone with level control in convex minimization. Math. Program. 85, 469–490 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cegielski, A., Dylewski, R.: Selection strategies in a projection method for convex minimization problems. Discuss. Math. Differ. Incl. Control Optim. 22, 97–123 (2002) zbMATHMathSciNetGoogle Scholar
  7. 7.
    Cegielski, A., Dylewski, R.: Residual selection in a projection method for convex minimization problems. Optimization 52, 211–220 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cegielski, A., Suchocka, A.: Relaxed alternating projection methods. SIAM J. Optim. 19, 1093–1106 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Combettes, P.: Inconsistent Signal Feasibility Problem: Lest Square Solutions in a Product Space. IEEE Trans. Signal Process. 42, 2955–2966 (1994) CrossRefGoogle Scholar
  10. 10.
    Gurin, L.G., Polyak, B.T., Raik, E.V.: The method of projection for finding the common point in convex sets. Z. Vychisl. Mat. Mat. Fiz. 7, 1211–1228 (1967) (in Russian). English translation in USSR Comput. Math. Phys. 7, 1–24 (1967) zbMATHGoogle Scholar
  11. 11.
    Kim, S., Ahn, H., Cho, S.-C.: Variable target value subgradient method. Math. Program. 49, 359–369 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kiwiel, K.C.: The efficiency of subgradient projection methods for convex optimization, part I: General level methods. SIAM J. Control Optim. 34, 660–676 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Polyak, B.T.: Minimization of unsmooth functionals. Z. Vychisl. Mat. Mat. Fiz. 9, 509–521 (1969) (in Russian). English translation in USSR Comput. Math. Phys. 9, 14–29 (1969) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.University of Zielona GóraZielona GóraPoland

Personalised recommendations