Block-iterative projection methods for parallel computation of solutions to convex feasibility problems

AbstractAn iterative method is proposed for solving convex feasibility problems. Each iteration is a convex combination of projections onto the given convex sets where the weights of the combination may vary from step to step. It is shown that any sequence of iterations generated by the algorithm converges if the intersection of the given family of convex sets is nonempty and that the limit point of the sequence belongs to this intersection under mild conditions on the sequence of weight functions. Special cases are block-iterative processes where in each iterative step a certain subfamily of the given family of convex sets is used. In particular, a block-iterative version of the Agmon-Motzkin-Schoenberg relaxation method for solving systems of linear inequalities is derived. Such processes lend themselves to parallel implementation and will be useful in various areas of applications, including image reconstruction from projections, image restoration, and other fully discretized inversion problems