A new numerical solver showed its clearest speed advantage on larger tested systems of monotone absolute-value equations, according to an arXiv preprint. The proposed coordinate proximal predictor-corrector method, or CPPC, was consistently fastest when the system dimension reached 1,000 or more. For dimensions of 500 or below, the results were mixed and the SGP spectral method was often faster.
The result is a benchmark finding, not a claim that CPPC is uniformly faster. The comparisons used constructed dense matrix instances, deterministic single-threaded runs and one run per instance, with the methods given the same inputs. The study did not report uncertainty estimates, so the measured rankings describe the tested cases rather than a general performance guarantee.
What the method changes
CPPC combines an exact scalar proximal predictor that changes one coordinate with a correction based on the full equation residual. The correction uses a separating hyperplane, a geometric boundary defined by the residual, to update the candidate solution.
The coordinate step is intended to reduce the cost of updating the residual. With cached quantities, changing one coordinate requires a single column update, followed by one new full matrix-vector product per iteration. That accounting lowers the amount of global work per step, but it does not by itself show that total runtime will be shorter, because the number of iterations still matters.
The correction also avoids a line search, the trial-and-error process often used to choose a step length. The paper gives a positive algebraic alignment certificate for the full-residual correction, and under the method's stated conditions uses that certificate to remove line search or backtracking without changing the correction geometry.
Guarantees depend on the matrix and the solution set
The analysis gives an exact condition for the equation's residual mapping to be monotone: the symmetric part of the matrix must satisfy sym(A) greater than or equal to the identity in the paper's matrix ordering. This allows nonsymmetric matrices and places no restriction on their skew-symmetric part.
Under the stated monotonicity condition, a nonempty solution set and the required parameter conditions, the algorithm is well defined. It either stops at a solution or produces a sequence that converges to one, while both the coordinate proximal residual and the original equation residual tend to zero. The paper also proves Fejér monotonicity, meaning the iterates do not move farther from the solution set in the relevant distance measure, along with an order-of-K-to-the-minus-one-half bound for the best residual seen through iteration K.
A stronger condition gives a stronger conclusion. If the symmetric part of the matrix exceeds the identity by a positive margin m, the analysis states that every right-hand side has a unique solution and that the method converges globally with R-linear convergence of the AVE residual. These are mathematical results under the specified assumptions; they do not remove the need for a solution set in the more general convergence statement.
The proximal residual is also more than a bookkeeping device: it vanishes exactly when the original absolute-value-equation residual vanishes. That equivalence makes a zero coordinate-proximal residual a valid stopping certificate for the equation itself.
Performance shifted with the problem's shape
The strongest runtime results appeared when the nonzero residual was localized. CPPC was fastest in the reported tests when the support fraction—the share of coordinates carrying nonzero residual—was 0.01 or 0.05, and it was also fastest for contiguous support. As the support became more diffuse, SGP overtook it. CPPC reached the iteration cap for dispersed support and for a tested bandwidth of 100, while it remained competitive through bandwidth 5.
Matrix geometry produced another split. CPPC was fastest at skew-strength values of 0 and 0.5 and at monotonicity margins of 0.02 and 0.05. Larger skew-symmetric parts favored SGP, and spectral methods were faster at tested monotonicity margins of 0.5 or more.
A benchmark that needs checking
The paper flags a discrepancy in the SGP comparison. It reports different iteration outcomes for two formula variants: one variant matches the published counts, while the formula stated in the cited work gives substantially fewer iterations. That issue matters because the baseline implementation affects how the runtime comparison should be interpreted.
The numerical instances were built with a prescribed exact solution and an initial point of zero. The study lists seven dimension levels and describes additional sweeps over residual localization, coupling width and matrix geometry, but it does not report a total number of instances for those structural sweeps. Code and data supporting the numerical results are stated to be publicly available.
Taken together, the evidence supports a conditional algorithmic result and a regime-specific benchmark advantage. It does not establish that CPPC will be faster on arbitrary absolute-value equations, and the iteration-cap cases and baseline discrepancy leave room for independent checks. The numerical conclusions remain limited to the constructed dense instances and tested parameter ranges.
Paper data and sources
Original title: A Line-Search-Free Coordinate Proximal Predictor-Corrector Method for Monotone Absolute Value Equations
Authors: Haotian Wang, Yong Xia
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text