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

TL;DR

This paper introduces a novel geodesic interior-point method for symmetric-cone optimization, offering a more efficient, scale-invariant algorithm with proven polynomial-time convergence and broad applicability to various cone types.

Contribution

It presents a new geodesic-based interior-point method that improves efficiency and symmetry over classical approaches for symmetric-cone optimization problems.

Findings

Algorithm uses half the variables of standard IPMs.

Achieves polynomial-time convergence with square-root-n bound.

Supports all symmetric cones with proven global convergence.

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.