🤖 AI Summary
This work addresses the challenge of integrating augmented Lagrangian and optimistic dual methods for equality-constrained optimization by proposing an additive hybrid framework that unifies matrix-valued augmentation and optimistic correction as distinct decompositions of a common correction matrix. By adaptively selecting the optimal splitting and stepsize through local spectral weighting, the method yields, for the first time, a closed-form hybrid update rule that jointly balances primal curvature and dual memory scale within a finite number of steps. Theoretical analysis reveals the equivalence and design flexibility between the two mechanisms, while experiments demonstrate that the proposed approach significantly outperforms individual strategies on nonlinear equality-constrained problems, achieving performance close to grid-search optimality and matching state-of-the-art first-order primal-dual algorithms under moderate ill-conditioning.
📝 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.