A Scalable Learned Iterative Solver for Nonlinear MPC via Temporal Decomposition

Nonlinear Model Predictive Control requires solving a constrained nonlinear program (NLP) in real-time at every sampling instant, a computational bottleneck that limits deployment on resource-constrained hardware or at high sampling rates. We address this challenge for the broad class of input-affine nonlinear systems to show that the optimal control move can be approximated by a state-dependent quadratic program (QP) whose cost parameters depend on the current state and reference. We propose a single-network residual-corrector architecture: a state-dependent analytic baseline provides initial QP parameters, and the network learns only the corrections needed to match the full NLP solution; the QP is solved by a differentiable interior-point layer, guaranteeing constraint satisfaction for the first control action. The network is trained offline on data generated by an NLP solver using a hybrid loss that combines supervised imitation and KKT-residual penalties. We validate the approach on a three-link planar robotic arm with Cartesian end-effector tracking, demonstrating orders-of-magnitude speedup over the NLP solver while maintaining comparable tracking performance.

The practical deployment of nonlinear model predictive control (NMPC) is often limited by online computation: solving a nonlinear program at high control rates can be expensive on embedded hardware, especially when models are complex or horizons are long. Learning-based NMPC approximations shift this computation offline but typically demand large expert datasets and costly training. We propose Sequential-AMPC, a sequential neural policy that generates MPC candidate control sequences by sharing parameters across the prediction horizon. For deployment, we wrap the policy in a safety-augmented online evaluation and fallback mechanism, yielding Safe Sequential-AMPC. Compared to a naive feedforward policy baseline across several benchmarks, Sequential-AMPC requires substantially fewer expert MPC rollouts and yields candidate sequences with higher feasibility rates and improved closed-loop safety. On high-dimensional systems, it also exhibits better learning dynamics and performance in fewer epochs while maintaining stable validation improvement where the feedforward baseline can stagnate.

Nonlinear Parametric Optimization Network (NLPOpt-Net) is an unsupervised learning architecture to solve constrained nonlinear programs (NLP). Given the structure of an NLP, it learns the parametric solution maps with guaranteed constraint satisfaction. The architecture consists of a backbone neural network (NN) followed by a multilayer ($k$-layered) projection. While the NN drives toward optimality through a loss function consisting of a modified Lagrangian augmented with a consistency loss, the projection ensures feasibility by projecting the NN predictions in the original constraint manifold. Instead of typical distance minimization, our projection exploits local quadratic approximations of the original NLP. Under certain conditions (such as convexity), the projection has a descent property, which improves the NN predictions further. NLPOpt-Net deploys an inversion-free, modified Chambolle-Pock algorithm to solve the constrained quadratic projections during the forward pass and uses the implicit function theorem for efficient backpropagation. The fixed structure of the projection further allows decoupling of the NN and the projection once the training is complete. NLPOpt-Net solves large-scale convex QP, QCQP, NLP, and nonconvex problems with near zero optimality gap and constraint violations reduced to machine precision. Additionally, it provides near accurate prediction of the active sets and corresponding dual variables, thereby enabling a scalable approach for multiparametric programming. Compiling the projection in C provides order of magnitude improvement in inference time compared to JAX. We provide the codes and NLPOpt-Net as a ready to use package that includes GPU support.