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On the Existence of Minimizers of Proximity Functions for Split Feasibility Problems

Author

Listed:
  • Xianfu Wang

    (University of British Columbia)

  • Xinmin Yang

    (Chongqing Normal University)

Abstract

Many inverse problems can be formulated as split feasibility problems. To find feasible solutions, one has to minimize proximity functions. We show that the existence of minimizers to the proximity function for Censor–Elfving’s split feasibility problem is equivalent to the existence of projections on appropriate convex sets and provide conditions under which such projections exist. These projections turn out to be the unique optimal solution of their Fenchel–Rockafellar duals and can be computed by the proximal point algorithm efficiently. Applications to linear equations and linear feasibility problems are given.

Suggested Citation

  • Xianfu Wang & Xinmin Yang, 2015. "On the Existence of Minimizers of Proximity Functions for Split Feasibility Problems," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 861-888, September.
    • Handle: RePEc:spr:joptap:v:166:y:2015:i:3:d:10.1007_s10957-015-0716-x
    DOI: 10.1007/s10957-015-0716-x
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    References listed on IDEAS

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    1. Unknown, 2005. "Forward," 2005 Conference: Slovenia in the EU - Challenges for Agriculture, Food Science and Rural Affairs, November 10-11, 2005, Moravske Toplice, Slovenia 183804, Slovenian Association of Agricultural Economists (DAES).
    2. Charles Byrne & Yair Censor, 2001. "Proximity Function Minimization Using Multiple Bregman Projections, with Applications to Split Feasibility and Kullback–Leibler Distance Minimization," Annals of Operations Research, Springer, vol. 105(1), pages 77-98, July.
    3. Yair Censor & Wei Chen & Patrick Combettes & Ran Davidi & Gabor Herman, 2012. "On the effectiveness of projection methods for convex feasibility problems with linear inequality constraints," Computational Optimization and Applications, Springer, vol. 51(3), pages 1065-1088, April.
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