A warmstarting technique for general conic optimization in interior point methods
Yuwen Chen, Paul Goulart, Colin Jones
TL;DR
<3-5 sentence high-level summary>The paper presents a smoothing-operator-based warmstarting strategy for primal-dual interior-point methods applied to general conic optimization. By generating a starting point on the central path from the previous optimum, the method preserves proximity to the central path and achieves comparable residuals to the prior solution, while enabling efficient, parallelizable computations across cones. The authors provide theoretical results showing that the smoothed start lies on the central path and that residuals remain controlled for nonnegative, SOC, and PSD cones, with extensions to general cones via damped Newton methods. Empirically, the approach reduces iteration counts and solve times across diverse problems, including SVM hyperparameter tuning and portfolio optimization, underscoring its practical impact for parametric and reoptimization workflows.
Abstract
We propose a novel warmstarting method for primal-dual interior point methods based on a smoothing operator that generates a starting point on the central path from the previous optimum. Compared to traditional approaches that prioritize minimizing infeasibility residuals, our method focuses on maintaining proximity to the central path. Computation of a smoothing operator is efficient and can be parallelized for conic constraints. We also prove that the residual of the smoothed starting point remains comparable to the one before the smoothing step. The numerical tests show that the proposed warmstarting strategy can reduce iteration numbers and computational time effectively across test problems.