A comparative numerical study of graph-based splitting algorithms for linear subspaces

Abstract

In this note, we test the performance of six algorithms from the family of graph-based splitting methods [SIAM J. Optim., 34 (2024), pp. 1569-1594] specialized to normal cones of linear subspaces. To do this, we first implement some numerical experiments to determine the best relaxation parameter for each algorithm. Then, we compare the number of iterations each algorithm requires to reach a given stopping criterion, using the previously identified best relaxation parameter. The numerical results allow us to identify some relevant patterns and provide numerical evidence that may guide further theoretical analysis.

Figures (5)

• Figure 1: Douglas--Rachford algorithm for solving a feasibility problem involving two lines in $\mathbb{R}^2$
• Figure 2: Results of the numerical experiment comparing the performance of each algorithm along different values of the relaxation parameter $\theta$
• Figure 3: Best relaxation parameter $\theta$ with respect to the number of subspaces
• Figure 4: Results of the numerical experiment comparing the performance of the algorithms with respect to the Friedrichs angle of the subspaces under Pierra's product space reformulation
• Figure 5: Results of the numerical experiment comparing the performance of all the algorithms with respect to the number of subspaces