Learned Preconditioning for a Primal-Dual Interior-Point Method

2026-09-28

AI Summary

This study addresses the high computational cost and sensitivity to boundary perturbations inherent in interior-point methods for solving linear systems. To this end, it proposes the pdLIP framework, which integrates learned preconditioning with projected search to enable GPU-parallel optimization. Methodologically, a self-supervised shared network predicts preconditioners, thereby eliminating the need for explicit Hessian computation, while a coordinate recurrent network is combined with a primal-dual projected search strategy for efficient problem solving. Experimental results demonstrate that the proposed warm-start mechanism reduces iteration counts by 63%–67%, effectively accelerating the solution of high-dimensional non-convex problems.

Abstract

Interior-point methods (IPMs) are among the most widely used algorithms for constrained optimization, yet their Newton-based search directions require costly second-order information and large linear-system solves. Learning to optimize offers cheaper updates learned from data, but the singular behavior of logarithmic barriers near constraint boundaries makes IPMs highly sensitive to perturbations, complicating both warm starting and learning reliable updates. We introduce pdLIP, an IPM for smooth nonlinear programs that integrates learned preconditioning with pdProj, an all-shifted primal-dual projected-search IPM. A shared coordinate-wise recurrent network predicts a positive diagonal preconditioner that scales the right-hand side of the reduced Newton system for the primal step, and the remaining slack and multiplier directions are recovered analytically. The learned iterations avoid Hessian evaluations and Newton-system solves, using only first-order and coordinate-wise operations amenable to GPU parallelization. Training is self-supervised, with a loss based on a penalty-barrier merit function and the residual of perturbed optimality conditions, requiring neither target directions nor precomputed solutions. Primal and dual shifts mitigate the barrier's sensitivity to perturbations near constraint boundaries, enabling effective warm starting. Across four classes of 200-dimensional convex and nonconvex constrained problems, pdLIP warm starts reduce pdProj refinement iterations by 63-67% compared with cold starts at the same KKT residual tolerance of $10^{-8}$, with negligible warm-start generation cost relative to the subsequent pdProj solve. Improvements persist on box-constrained QPs with 1000 variables and extend to applications including portfolio optimization, support vector machines, and a nonlinear control example.

Problem

• Interior-point methods
• Constrained optimization
• Learned preconditioning
• Warm starting
• Logarithmic barrier

Innovation

• Learned Preconditioning
• Primal-Dual Interior-Point Method
• Self-Supervised Training
• Warm Starting
• Recurrent Neural Network