A Fast Temporal Decomposition Procedure for Long-horizon Nonlinear Dynamic Programming

The paper proves that FOTD (an SQP method whose Newton system is approximately solved by OTD) achieves global convergence and a uniform local linear rate, under Assumptions 4.1–4.3 and a compactness assumption, with explicit parameter choices for η, b, and µ. Key steps are: (i) a uniform bound γG on GG^T from controllability and boundedness (Lemma 4.1), (ii) an exponentially accurate OTD step with error δ = Cρ^b (Theorem 4.2), (iii) descent of the exact augmented Lagrangian along the (inexact) direction given suitable η and δ (Theorem 5.1), yielding global convergence (Theorem 5.2), and (iv) local acceptance of α = 1 and uniform linear contraction Ψ^{τ+1} ≤ (Cρ^b)Ψ^τ (Theorems 6.1, 6.3, 6.4). The candidate solution reproduces these ingredients in essentially the same order and with the same parameter relations (η1 ≥ 17/(η2γG), η2 ≤ γRH/(12Υ^2), µ ≥ 32Υ^{4t+1}/γC) and the same message on the role of the augmented Lagrangian merit. Minor differences are mostly bookkeeping of constants (use of Υupper vs. the paper’s Υ), and presenting an equivalent residual bound for the OTD-assembled step. Overall, the logic and conclusions match the paper’s argument and the same core lemmas are invoked, hence both are correct with substantially the same proof structure (e.g., see Theorem 5.1, Theorem 5.2, Lemma 4.1, Lemma 4.2, and Theorems 6.1–6.4 in the paper).