Neural Compression of Long ADMM Trajectory for Multiparametric Quadratic Program

The Alternating Direction Method of Multipliers (ADMM) is a widely used method for structured convex optimization, and its practical performance depends strongly on the choice of penalty and relaxation parameters. Motivated by settings such as Model Predictive Control (MPC), where one repeatedly solves related optimization problems with fixed structure and changing parameter values, we propose learning online updates of the relaxation parameter to improve average performance on problem classes of interest, while guaranteeing that asymptotic convergence is not compromised for the worst-case realization of such problems. This choice is computationally attractive in the Operator Splitting Quadratic Program (OSQP)-like architectures, since adapting relaxation does not trigger the matrix refactorizations associated with penalty updates. We establish convergence guarantees for ADMM with time-varying penalty and relaxation parameters under mild assumptions, and show on benchmark quadratic programs that the resulting learned policies improve both iteration count and wall-clock time on average over baseline OSQP.

This paper introduces numerical optimizations for maximizing throughput on GPU when solving large batches (10,000 to over 100,000) of sequential quadratic programming (SQP) iterations, where all problems have the same structure. The optimizations are implemented in a toolbox WarpMPC for model-predictive control (MPC) in JAX and Warp. Based on the insight that all MPC problem instances in a batch share the same sparsity in time, cost, and constraints, we propose unrolling sparse linear factorizations and solves, which dominate alternating direction method of multipliers (ADMM) solver runtime. We avoid memory access bottlenecks and wasting computations via optimized memory layout, padding-reducing segmentation of the unrolled factorization, and dependency level scheduled backsolves, additionally accelerating sensitivity computation. We achieve throughputs of 8,000 to 250,000 SQP iterations per second on nonlinear cartpole, quadrotor, and humanoid robot benchmarks, outperforming baselines by 3$\times$ to 25$\times$. We illustrate practical usefulness by synthesizing a dataset and training a neural network approximation of an MPC in under 4 minutes that stabilizes a nano quadrotor in hardware experiments.

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.