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We propose and analyse primal-dual interior-point algorithms for convexoptimization problems in conic form. The families of algorithms we analyse areso-called short-step algorithms and they match the current best iterationcomplexity bounds for primal-dual symmetric interior-point algorithm ofNesterov and Todd, for symmetric cone programming problems with givenself-scaled barriers. Our results apply to any self-concordant barrier for anyconvex cone. We also prove that certain specializations of our algorithms tohyperbolic cone programming problems (which lie strictly between symmetric coneprogramming and general convex optimization problems in terms of generality)can take advantage of the favourable special structure of hyperbolic barriers.We make new connections to Riemannian geometry, integrals over operator spaces,Gaussian quadrature, and strengthen the connection of our algorithms toquasi-Newton updates and hence first-order methods in general.

Interior-point algorithms for convex optimization based on primal-dual metrics