A geodesic interior-point method for linear optimization over symmetric cones
Abstract
We develop a new interior-point method (IPM) for symmetric-cone optimization, a common generalization of linear, second-order-cone, and semidefinite programming. In contrast to classical IPMs, we update iterates with a geodesic of the cone instead of the kernel of the linear constraints. This approach yields a primal-dual-symmetric, scale-invariant, and line-search-free algorithm that uses just half the variables of a standard primal-dual IPM. With elementary arguments, we establish polynomial-time convergence matching the standard square-root-n bound. Finally, we prove global convergence of a long-step variant and provide an implementation that supports all symmetric cones. For linear programming, our algorithms reduce to central-path tracking in the log domain.
Cite
@article{arxiv.2008.08047,
title = {A geodesic interior-point method for linear optimization over symmetric cones},
author = {Frank Permenter},
journal= {arXiv preprint arXiv:2008.08047},
year = {2023}
}
Comments
The first revision adds new results, including a manifold optimization interpretation of the presented algorithms, a simplified step-size rule for Newton's method, an energy interpretation of divergence, connections with self-scaled and self-concordant barriers, square-root-free evaluation of the Riemannian exponential map, and expanded computational results. This revision corrects typos