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.
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References
Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221–239 (1994)
Combettes, P.L.: Inconsistent signal feasibility problems: least-squares solutions in a product space. IEEE Trans. Signal Process. 42, 2955–2966 (1994)
Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18, 441–453 (2002)
Censor, Y., Chen, W., Combettes, P.L., Davidi, R., Herman, G.T.: On the effectiveness of projection methods for convex feasibility problems with linear inequality constraints. Comput. Optim. Appl. 51, 1065–1088 (2012)
Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4, 1168–1200 (2005)
Combettes, P.L., Dung, Dinh, Vu, B.C.: Dualization of signal recovery problems. Set-Valued Var. Anal. 18, 373–404 (2010)
Bauschke, H.H., Borwein, J.M.: Dykstra’s alternating projection algorithm for two sets. J. Approx. Theory 79, 418–443 (1994)
Bauschke, H.H., Borwein, J.M.: On the convergence of von Neumann’s alternating projection algorithm for two sets. Set-Valued Anal. 1, 185–212 (1993)
Combettes, P.L., Dung, Dinh, Vu, B.C.: Proximity for sums of composite functions. J. Math. Anal. Appl. 380, 680–688 (2011)
De Pierro, A.R., Iusem, A.N.: On the asymptotic behavior of some alternate smoothing series expansion iterative methods. Linear Algebr. Appl. 130, 3–24 (1990)
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Berlin (2011)
Cegielski, A.: Iterative Methods for Fixed Point Problems in Hilbert Spaces. In: Lecture Notes in Mathematics 2057. Springer, Heidelberg (2012)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Byrne, C.: Iterative Algorithms in Inverse Problems. http://faculty.uml.edu/cbyrne/ITER2.pdf (2006)
Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)
Bot, R.I., Csetnek, E.R.: On the convergence rate of a forward–backward type primal-dual splitting algorithm for convex optimization problems. Optimization 64, 5–23 (2015)
Byrne, C.: An elementary proof of convergence for the forward–backward splitting algorithm. J. Nonlinear Convex Anal. 15, 681–691 (2014)
Attouch, H., Peypouquet, J., Redont, P.: A dynamical approach to an inertial forward–backward algorithm for convex minimization. SIAM J. Optim. 24, 232–256 (2014)
Bauschke, H.H., Phan, H.M., Wang, X.: The method of alternating relaxed projections for two nonconvex sets. Vietnam J. Math. 42, 421–450 (2014)
Cegielski, A., Suchocka, A.: Relaxed alternating projection methods. SIAM J. Optim. 19, 1093–1106 (2008)
Bauschke, H.H., Luke, D.R., Phan, H.M., Wang, X.: Restricted normal cones and the method of alternating projections: applications. Set-Valued Var. Anal. 21, 475–501 (2013)
Byrne, C., Censor, Y.: Proximity function minimization using multiple Bregman projections, with applications to split feasibility and Kullback-Leibler distance minimization. Geometric programming. Ann. Oper. Res. 105(2001), 77–98 (2002)
Xu, H.: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 26, 105018 (2010). 17 pp
López, G., Martin-Márquez, V., Wang, F., Xu, H.: Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Probl. 28, 085004 (2012). 18 pp
Yang, Q.: The relaxed CQ algorithm solving the split feasibility problem. Inverse Probl. 20, 1261–1266 (2004)
Borwein, J.M., Lewis, A.S.: Convex Analysis and Nonlinear Optimization: Theory and Examples, 2nd edn. Springer, New York (2006)
Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing, Singapore (2002)
Adly, S., Ernst, E., Théra, M.: On the closedness of the algebraic difference of closed convex sets. J. Math. Pures Appl. 82, 1219–1249 (2003)
Bauschke, H.H., Wang, X., Wylie, C.J.S.: Fixed points of averages of resolvents: geometry and algorithms. SIAM J. Optim. 22, 24–40 (2012)
Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)
Burachik, R.S., Jeyakumar, V.: A dual condition for the convex subdifferential sum formula with applications. J. Convex Anal. 12, 279–290 (2005)
Acknowledgments
The authors wish to thank two anonymous referees and Dr. Franco Giannessi for their suggestions which have improved the presentation of the paper. X. Wang was partially supported by a Discovery Grant of NSERC. X. Yang was partially supported by a Grant of NSFC (No. 11271391).
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Wang, X., Yang, X. On the Existence of Minimizers of Proximity Functions for Split Feasibility Problems. J Optim Theory Appl 166, 861–888 (2015). https://doi.org/10.1007/s10957-015-0716-x
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DOI: https://doi.org/10.1007/s10957-015-0716-x
Keywords
- \(CQ\)-algorithm
- Fenchel–Rockafellar’s duality
- Proximal point method
- Proximity function
- Split feasibility problem