# A geodesic interior-point method for linear optimization over symmetric cones

arXiv: https://arxiv.org/abs/2008.08047 · HTML: https://mlsanity.com/paper/2008.08047/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Aug 2020.

## Built on by (most cited first, 1 of 1)

- [Differential Privacy for Euclidean Jordan Algebra with Applications to Private Symmetric Cone Programming](https://mlsanity.com/paper/2509.16915/) (2025)

## Abstract (arXiv)

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.