Abstract:Augmented Lagrangian and optimistic primal--dual methods stabilize equality-constrained optimization through seemingly different mechanisms: the former adds constraint-dependent primal curvature, while the latter adds dual memory. Recent work has shown that these mechanisms are equivalent for scalar parameters. We extend this equivalence to matrix-valued correction. We prove an additivity principle: for symmetric matrix parameters, the ideal primal trajectory depends only on the summed correction matrix, not on how it is split between augmented and optimistic channels. This exposes a design freedom: algebraically equivalent decompositions can have different finite-step feasibility because augmented correction affects primal curvature, whereas optimistic correction affects the scale of the dual memory correction. We formulate the resulting step-size-limited design problem and derive a closed-form hybrid rule that selects a matrix correction, splits it between the two channels, and chooses primal and dual steps using local spectral weights. Experiments on nonlinear equality-constrained problems with controlled constraint-Jacobian conditioning show that the hybrid design improves over pure augmented and pure optimistic endpoints, closely tracks a grid-search hybrid oracle, and is competitive with first-order primal--dual baselines under mild-to-moderate ill-conditioning. The experiments also identify the expected limitation: exact cancellation requires increasingly large matrix corrections as the constraint Jacobian becomes ill-conditioned.
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2605.06141 [cs.LG] |
| (or arXiv:2605.06141v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.06141 arXiv-issued DOI via DataCite (pending registration) |
From: Jiayi Zhao [view email]
[v1]
Thu, 7 May 2026 12:39:22 UTC (33 KB)
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