Warm-IP: A Path-Following ADMM Warm Start for Interior-Point Quadratic Programming

Abstract

Interior-point (IP) methods are powerful second-order optimization algorithms but can make slow progress during early iterations. We propose Warm-IP, a hybrid framework that combines a path-following variant of the alternating direction method of multipliers (ADMM) with a primal--dual IP method. The ADMM phase generates a warm start that remains close to the central path, enabling the subsequent IP phase to rapidly refine the solution to high accuracy. We evaluate Warm-IP on a diverse collection of real-world and synthetic quadratic programming (QP) instances and compare it with leading commercial solvers (MOSEK and Gurobi) and open-source solvers (Clarabel, PIQP, and OSQP). We also curate and release a new benchmark of large-scale QPs from cancer radiotherapy treatment planning, built from real patient data and provided in canonical QP form, with instances containing hundreds of thousands of constraints. Warm-IP substantially outperforms existing solvers on large-scale problems. On the large-scale radiotherapy benchmark comprising 60 lung cancer cases, for example, Warm-IP solves an instance in 47 seconds on average with no failures, while the strongest competitors, MOSEK and Gurobi, average 55 seconds with failures on 23\% of the cases and roughly 240 seconds with no failures, respectively. To support reproducibility and future benchmarking, we publicly release the implementation and benchmark data through a \href{https://github.com/Radiotherapy-Optimization/Warm-IP}{GitHub repository}. These results demonstrate that appropriately warm-started IP methods can combine the rapid initial progress of first-order methods with the high-accuracy convergence of second-order methods.