An acceleration scheme for Dykstra’s algorithm

Abstract

Dykstra's algorithm is an iterative alternating projection procedure for solving the best approximation problem: find the closest point, to a given one, in the intersection of a finite number of closed and convex sets. The main drawback of Dykstra's algorithm is its frequent slow convergence. In this work we develop an acceleration scheme with a strong geometrical flavor, which guarantees termination at the solution in two cycles of projections in the case of two closed subspaces. The proposed scheme can also be applied to any other alternating projection algorithm that solves the best approximation problem.

Anderson acceleration of the alternating projections method for computing the nearest correlation matrix

“…As a final experiment we use the four matrices from Experiment 1 to compare Anderson acceleration with the acceleration scheme from [29]. Table 10 shows the number of iterations, it 2, for that scheme, in which we set its safeguard parameter ε to 10 −14 and use the same convergence tolerance as in all our experiments.…”

“…The number of iterations for the acceleration scheme is the same as for the unaccelerated method in each case except for the matrix with n = 6, and in that case we see a reduction in the number of iterations by a factor 1.1 versus 3.8 for Anderson acceleration. In all test cases, after a few initial iterations the mixing parameter α k needed for the scheme [29] could not be computed because the safeguard was triggered. We conclude that the acceleration scheme of [29] is not To summarize, in these experiments we have found that Anderson acceleration of the alternating projections method for the nearest correlation matrix, with an appropriate choice of m ∈ [1,6], results in a reduction in the number of iterations by a factor of at least 2 for the standard algorithm and a factor at least 3 when additional constraints are included.…”

“…In all test cases, after a few initial iterations the mixing parameter α k needed for the scheme [29] could not be computed because the safeguard was triggered. We conclude that the acceleration scheme of [29] is not To summarize, in these experiments we have found that Anderson acceleration of the alternating projections method for the nearest correlation matrix, with an appropriate choice of m ∈ [1,6], results in a reduction in the number of iterations by a factor of at least 2 for the standard algorithm and a factor at least 3 when additional constraints are included. The factors can be much larger than these worst-cases, especially in the experiments with additional constraints, where we saw a reduction in the number of iterations by a factor 21.8 in Table 7.…”

“…We also thank Professor Marcos Raydan for a helpful discussion about the acceleration scheme in [29].…”

“…Recently, López and Raydan [29] have proposed a geometrically-based acceleration scheme for the alternating projections method that builds a new sequence from the original one by taking linear combinations of successive pairs of iterates. The new sequence is tested for convergence and the original iteration remains unchanged.…”