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 of the variables of a standard primal-dual IPM. With elementary arguments, we establish polynomial-time convergence matching the standard \scrO(\surdn) bound. Finally, we prove the global convergence of a longstep variant and provide an implementation that supports all symmetric cones. For linear programming, our algorithms reduce to central-path tracking in the log-domain.