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An Acceleration Scheme for Solving Convex Feasibility Problems Using Incomplete Projection Algorithms

“…That method is particularly useful when the projections onto C; can be easily computed, as for instance in the particular case of hyperplanes or halfspaces. This last observation leads to generate for each iterate x", in more general cases, a suitable closed convex set S*, satisfying C C S*, that facilitates the computation of the exact projection onto it [6]. The choice of the sets S* may have a significant influence on the convergence rate of the algorithms.…”

“…The choice of the sets S* may have a significant influence on the convergence rate of the algorithms. Garcia-Palomares and Gonzdlez-Castano in [6] have proposed an incomplete projections algorithm (IPA) for obtaining an approximate projection of z* onto special convex sets S*, C C SF, i=1,2,...,q, arising from the separation of subsets of the violated constraints. The next iterate is defined by means of the projection of z* onto a hyperplane H* strictly separating C, generated by means of a combination of aggregate hyperplanes HF relative to the incomplete projections onto each set St .…”

“…This method is an extension of the projection methods for solving systems of linear equations given in [11] and [12]. The general scheme is similar to the IPA algorithm given in [6], and therefore is very convenient for parallel processing. The idea is, therefore, that at the current iterate the set of violated constraints is splitted into subsets or blocks St , In such a way that the required incomplete projection is obtained by combining exact projections onto simple convex sets.…”

“…In Section 2 the acceleration scheme for solving systems of linear inequalities is presented and applied to a version of the EPA algorithm that employs exact projections as in [6], and prove the main results leading to an improved rate of convergence. In subsection 2 we apply the acceleration scheme to the ACIPA algorithm, that uses incomplete projections onto blocks of inequalities.…”

Convergence analysis of a parallel projection algorithm for solving convex feasibility problems

“…A number of algorithms were presented based on projection method for solving the CFP [1,12,14,17]. According to the number of projections adopted at each step, the sequential projection algorithm, the parallel projection algorithm and the block-iterative projection algorithm had been intensively investigated.…”