Treffer: Interior-Point Algorithms with Full Newton Steps for Nonsymmetric Convex Conic Optimization.

We design and analyze primal-dual, feasible interior-point algorithms (IPAs) employing full Newton steps to solve convex optimization problems in standard conic form. Unlike most nonsymmetric cone programming methods, the algorithms presented in this paper require only a logarithmically homogeneous self-concordant barrier (LHSCB) of the primal cone but compute feasible and \(\varepsilon\) -optimal solutions to both primal and dual problems in \(\mathcal{O}(\sqrt \nu \log (1/\varepsilon))\) iterations, where \(\nu\) is the barrier parameter of the LHSCB; this matches the best currently known theoretical iteration complexity of IPAs for both symmetric and nonsymmetric cone programming. The definition of the neighborhood of the central path and feasible starts ensures that the computed solutions are compatible with the dual certificates framework introduced by Lee, Papp, and Varga [Dual Certificates of Primal Cone Membership, 2025]. Several initialization strategies are discussed, including two-phase methods that can be applied if a strictly feasible primal solution is available and one based on a homogeneous self-dual embedding that allows the rigorous detection of feasible or optimal solutions of large norm. In a detailed study of a classic, notoriously difficult polynomial optimization problem, we demonstrate that the methods are efficient and numerically reliable. Although the standard approach using semidefinite programming fails for this problem with the solvers we tried, the new IPAs compute highly accurate near-optimal solutions that can be certified to be near optimal in exact arithmetic. [ABSTRACT FROM AUTHOR]

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