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We develop a new interior-point method (IPM) for symmetric-cone optimization,a common generalization of linear, second-order-cone, and semidefiniteprogramming. In contrast to classical IPMs, we update iterates with a geodesicof the cone instead of the kernel of the linear constraints. This approachyields a primal-dual-symmetric, scale-invariant, and line-search-free algorithmthat uses just half the variables of a standard primal-dual IPM. Withelementary arguments, we establish polynomial-time convergence matching thestandard square-root-n bound. Finally, we prove global convergence of along-step variant and provide an implementation that supports all symmetriccones. For linear programming, our algorithms reduce to central-path trackingin the log domain.

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