Parallelizable Interior-Point ADMM for Block-Structured Nonconvex Programs

Physics is increasingly integrated into AI systems for learning, control, and predictive world modeling. This makes scalable numerical simulation an important computational primitive. Local–global optimization recasts physics simulation as a parallel nonconvex optimization problem by exploiting the splitting structure of physical models. However, existing convergence analyses of splitting algorithms do not match the empirical success of the local–global scheme. We close this gap by analyzing two variants of the ADMM algorithm, which we call Interior-Point ADMM (IP-ADMM), for block-structured composite programs with smooth nonlinear inequality and affine equality constraints. Our key idea is a one-sided finite-activation wall function that keeps the solution inside the feasible domain. As a result, the constraint blocks have closed-form update rules, each objective block requires one call to the original convex proximal oracle, and one equality-constrained quadratic solve per iteration performs the consensus update after the independent local updates. Ordinary IP-ADMM obtains a CQ-free best-iterate weak-KKT bound of $O(K^{-1/3})$, or a normalized banded Fritz–John bound of $O(K^{-1/4})$. Under Uniform Band-MFCQ, a sufficiently accurate banded Fritz–John certificate converts to KKT at a rate of $O(K^{-1/4})$. We further propose a checkpoint variant of IP-ADMM, called IP-ADMM-Ckpt, which attains a latest-checkpoint KKT bound at a rate of $O(K^{-1/3})$. Finally, we show how these upper bounds depend on the objective-block count $N$, the constraint-block count $M$, and the coupling norm $\Vert A{\Vert }_2$, which reveals how the cost grows as we solve larger physical systems.