Learned Preconditioning for a Primal-Dual Interior-Point Method

Abhinav Madabhushi, Jialin Liu, Minxin Zhang

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…