Convergence of String-Averaging Projection Schemes for Inconsistent Convex Feasibility Problems

We study iterative projection algorithms for the convex feasibility problem of finding a point in the intersection of finitely many nonempty, closed and convex subsets in the Euclidean space. We propose (without proof) an algorithmic scheme which genenralizes both the string-averaging algorithm and the block-iterative projections (BIP) method with fixed blocks and prove convergence of the string-averaging method in the inconsistent case by translating it into a fully sequential algorithm in the product space.

[1]  Heinz H. Bauschke,et al.  The method of cyclic projections for closed convex sets in Hilbert space , 1997 .

[2]  Felix E. Browder,et al.  Nonlinear Operators and Nonlinear Equations of Evolution , 1976 .

[3]  R. Aharoni,et al.  Successive Projections on Hyperplanes , 1984 .

[4]  Y. Censor,et al.  Parallel Optimization: Theory, Algorithms, and Applications , 1997 .

[5]  S. Reich,et al.  Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings , 1984 .

[6]  Yair Censor,et al.  Averaging Strings of Sequential Iterations for Convex Feasibility Problems , 2001 .

[7]  Claude Brezinski,et al.  Projection methods for systems of equations , 1997 .

[8]  Heinz H. Bauschke,et al.  On Projection Algorithms for Solving Convex Feasibility Problems , 1996, SIAM Rev..

[9]  Boris Polyak,et al.  The method of projections for finding the common point of convex sets , 1967 .

[10]  F. Browder.  Nonlinear operators and nonlinear equations of evolution in Banach spaces , 1976 .

[11]  Simeon Reich,et al.  A limit theorem for projections , 1983 .

[12]  W. Cheney,et al.  Proximity maps for convex sets , 1959 .

[13]  Gilbert Crombez,et al.  Finding common fixed points of strict paracontractions by averaging strings of sequential iterations. , 2002 .

[14]  F. Browder,et al.  The solution by iteration of nonlinear functional equations in Banach spaces , 1966 .

[15]  Patrick L. Combettes,et al.  Inconsistent signal feasibility problems: least-squares solutions in a product space , 1994, IEEE Trans. Signal Process..

[16]  Heinz H. Bauschke,et al.  Projection and proximal point methods: convergence results and counterexamples , 2004 .

[17]  Felix E. Browder,et al.  Nonlinear equations of evolution and nonlinear accretive operators in Banach spaces , 1967 .

[18]  Y. Censor,et al.  Block-iterative projection methods for parallel computation of solutions to convex feasibility problems , 1989 .

[19]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[20]  F. Browder,et al.  NONEXPANSIVE NONLINEAR OPERATORS IN A BANACH SPACE. , 1965, Proceedings of the National Academy of Sciences of the United States of America.

[21]  R. D. Murphy,et al.  Iterative solution of nonlinear equations , 1994 .

[22]  Yongyi Yang,et al.  Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics , 1998 .

[23]  Hyang-Joo Rhee.  AN APPLICATION OF THE STRING AVERAGING METHOD TO ONE-SIDED BEST SIMULTANEOUS APPROXIMATION , 2003 .

[24]  Roy Meshulam.  On products of projections , 1996, Discret. Math..

[25]  Guy Pierra,et al.  Decomposition through formalization in a product space , 1984, Math. Program..