  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.
