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.

• The paper compares six graph-based Douglas–Rachford algorithms on random subspace feasibility problems, using exact limit points to evaluate convergence through the governing sequence.
• Algorithms with identical graphs and subgraphs perform best at relaxation parameter θ = 1, Generalized Ryu improves toward θ = 1.9, and Malitsky–Tam’s optimum decreases toward 1 as the number of subspaces grows.
• Complete and Malitsky–Tam are generally fastest, Sequential is slowest, and iteration counts often track the product-space Friedrichs angle, although these numerical patterns remain theoretically unproven.

Overview and motivation

This paper presents a numerical investigation of six instances of the graph-based Douglas–Rachford (DR) splitting family introduced in [(2603.04231)'s framework, SIAM J. Optim., 34 (2024), pp. 1569–1594], applied to linear feasibility problems of the form

$\text{Find } x \in \bigcap_{i=1}^n U_i,$

where each $U_i \subseteq \mathbb{R}^p$ is a linear subspace. The study addresses two questions: which relaxation parameter $\theta \in \,]0,2[$ performs best for each algorithm, and how the algorithms compare in iteration count when each is equipped with its best parameter. The experiments are conducted on randomly generated subspaces in $\mathbb{R}^{50}$, with $n$ ranging from 3 to 12, and convergence is monitored on the governing sequence $\mathbf{v}^k$ rather than the shadow sequence, using the stopping criterion $\|\mathbf{v}^k - \mathbf{v}^*\| < 10^{-6}$.

A key methodological advantage enabling this design is the availability of a closed-form expression for the limit points of graph-based DR on subspaces [(2603.04231), Theorem 4.8 of "graph-linear"], which generalizes the classical characterization of the two-subspace DR limit. Specifically, with $\boldsymbol{\delta}$ the degree-balance vector of the graph $G$, $\boldsymbol{\alpha}$ solving $U_i \subseteq \mathbb{R}^p$0, and $U_i \subseteq \mathbb{R}^p$1 the subspace of $U_i \subseteq \mathbb{R}^p$2 defined by the orthogonality conditions $U_i \subseteq \mathbb{R}^p$3, the limits are

$U_i \subseteq \mathbb{R}^p$4

This permits exact computation of $U_i \subseteq \mathbb{R}^p$5 and hence an exact, instance-independent stopping test. The authors justify monitoring the governing sequence rather than the shadow sequence by the well-known spiraling behavior of DR iterates around the limit, which makes shadow-sequence distances non-monotone and can prematurely suggest convergence.

Algorithms considered

All six methods are instances of the graph-based DR framework, differing only in the pair of oriented graphs $U_i \subseteq \mathbb{R}^p$6, where $U_i \subseteq \mathbb{R}^p$7 is a connected order-preserving graph and $U_i \subseteq \mathbb{R}^p$8 is a connected spanning subgraph whose Laplacian factorization $U_i \subseteq \mathbb{R}^p$9 enters the iteration:

Algorithm	Graph $\theta \in \,]0,2[$0	Subgraph $\theta \in \,]0,2[$1
Sequential	path $\theta \in \,]0,2[$2	same as $\theta \in \,]0,2[$3
Complete	complete graph	same as $\theta \in \,]0,2[$4
Parallel down	star into node $\theta \in \,]0,2[$5	same as $\theta \in \,]0,2[$6
Parallel up	star out of node 1	same as $\theta \in \,]0,2[$7
Malitsky–Tam	ring (path plus edge $\theta \in \,]0,2[$8)	path
Generalized Ryu	complete graph	parallel-down star

The first four satisfy $\theta \in \,]0,2[$9; the last two do not. This structural distinction turns out to be predictive of the relaxation-parameter behavior observed empirically.

Best relaxation parameter

For each $\mathbb{R}^{50}$0, the authors generated 20 random problems, ran each algorithm from 10 common starting points, and evaluated the ratio $\mathbb{R}^{50}$1 over a grid of $\mathbb{R}^{50}$2 values in $\mathbb{R}^{50}$3. Three distinct regimes emerge.

Algorithms with $\mathbb{R}^{50}$4 (Sequential, Complete, Parallel down, Parallel up). The optimal parameter is uniformly $\mathbb{R}^{50}$5, independent of $\mathbb{R}^{50}$6. Moreover, the experiments exhibit an exact symmetry: the iteration counts for $\mathbb{R}^{50}$7 and $\mathbb{R}^{50}$8 coincide, so the curves for $\mathbb{R}^{50}$9 and $n$0 overlap completely, and likewise for all paired values. The four algorithms also produce numerically indistinguishable performance profiles, which the authors attribute to the shared property $n$1.

Generalized Ryu ($n$2, complete $n$3). Performance is monotonically improving in $n$4, with the best choice being the extreme value $n$5 for all $n$6. The $n$7/$n$8 symmetry disappears entirely. This is notable because $n$9 approaches the boundary of the guaranteed convergence interval, and the authors provide no theoretical explanation for why the empirical optimum lies at the grid boundary.

Malitsky–Tam (ring over path). The best parameter depends on $\mathbf{v}^k$0: it is large for small $\mathbf{v}^k$1 (close to the Generalized Ryu optimum) and decreases toward $\mathbf{v}^k$2 as $\mathbf{v}^k$3 grows. This interpolation between the two other regimes is the most distinctive empirical finding of the relaxation study, and the authors explicitly state that a theoretical explanation is left for future work.

Algorithmic performance comparison

With each algorithm set to its best parameter, the authors compare mean iteration counts across 100 random problems per $\mathbf{v}^k$4, relating performance to the Friedrichs angle of the subspaces under Pierra's product-space reformulation $\mathbf{v}^k$5 — equivalently, to the Friedrichs number introduced in prior work. The authors acknowledge an important assumption here: there is no theoretical guarantee that this product-space Friedrichs angle captures the geometry of the original subspaces, although the numerical results support its usefulness as a predictor.

The principal observations are:

• Iteration counts correlate strongly with the Friedrichs angle, most clearly for the Complete algorithm, where iterations follow a well-defined curve as a function of the angle. For Sequential, this relationship degrades as $\mathbf{v}^k$6 grows, with iterates spreading away from any single curve.
• Complete and Malitsky–Tam are the fastest methods and are nearly indistinguishable. For small $\mathbf{v}^k$7 and small angles, Malitsky–Tam is slightly faster; for larger angles it exhibits a wider spread. For $\mathbf{v}^k$8, Complete is the best performer on nearly all instances, while Malitsky–Tam's iteration counts gradually increase with $\mathbf{v}^k$9.
• Parallel up and Parallel down are almost indistinguishable, which the authors explain by the fact that their graphs are topologically isomorphic up to edge orientation — a structural observation consistent with the relaxation study.
• Generalized Ryu behaves anomalously at large angles: for $\|\mathbf{v}^k - \mathbf{v}^*\| < 10^{-6}$0 and angles above 0.5, iteration counts range from fewer than 10 to over 100, with no clear curve. The curve sharpens as $\|\mathbf{v}^k - \mathbf{v}^*\| < 10^{-6}$1 increases but flattens at a higher level, so the algorithm underperforms the leaders for larger $\|\mathbf{v}^k - \mathbf{v}^*\| < 10^{-6}$2.
• Sequential is the slowest overall, with performance deteriorating as $\|\mathbf{v}^k - \mathbf{v}^*\| < 10^{-6}$3 increases.

The overall ranking by mean iterations is therefore: Complete and Malitsky–Tam (best, with Complete dominating for $\|\mathbf{v}^k - \mathbf{v}^*\| < 10^{-6}$4), followed by Generalized Ryu, then the two parallel variants (essentially tied), and Sequential (worst).

Limitations and open questions

The study is purely numerical; none of the observed patterns is backed by proof. The authors identify three open questions, which they intend to address in future work:

1. Is $\|\mathbf{v}^k - \mathbf{v}^*\| < 10^{-6}$5 provably optimal when $\|\mathbf{v}^k - \mathbf{v}^*\| < 10^{-6}$6, and can the observed symmetry between $\|\mathbf{v}^k - \mathbf{v}^*\| < 10^{-6}$7 and $\|\mathbf{v}^k - \mathbf{v}^*\| < 10^{-6}$8 be established theoretically?
2. Does a general expression exist for the optimal relaxation parameter when $\|\mathbf{v}^k - \mathbf{v}^*\| < 10^{-6}$9? The Generalized Ryu result — optimum at the grid boundary $\boldsymbol{\delta}$0 — suggests the answer, if affirmative, differs qualitatively from the $\boldsymbol{\delta}$1 case, and it remains possible that values closer to 2 would perform even better.
3. Can a convergence rate be derived in terms of the Friedrichs angle under the product-space reformulation (or the Friedrichs number)? The authors note that no theoretical guarantee currently links this quantity to the geometry of the original subspaces, so the observed correlations rest on empirical evidence alone.

Additional implicit limitations include the restriction to linear subspaces in $\boldsymbol{\delta}$2 with randomly generated instances, and the reliance on the closed-form limit characterization, which is available only in the linear-subspace setting; the findings do not automatically transfer to general convex sets or to non-Euclidean geometries.

Conclusion

The paper provides a systematic empirical calibration of six graph-based DR splitting methods for subspace feasibility problems. Its main contributions are the identification of three distinct relaxation-parameter regimes tied to the relationship between $\boldsymbol{\delta}$3 and $\boldsymbol{\delta}$4 (constant $\boldsymbol{\delta}$5 with $\boldsymbol{\delta}$6/$\boldsymbol{\delta}$7 symmetry when $\boldsymbol{\delta}$8; monotone improvement toward $\boldsymbol{\delta}$9 for Generalized Ryu; and an $G$0-dependent transition for Malitsky–Tam), and a performance ranking in which the Complete and Malitsky–Tam methods dominate, with iteration counts strongly correlated with the product-space Friedrichs angle. The numerical evidence is consistent across problem sizes but remains unexplained by theory; the paper's value lies partly in posing precise questions — optimality of $G$1, the symmetry phenomenon, and a Friedrichs-angle-based rate theory — that a subsequent theoretical analysis could resolve.