Interior-point algorithms with full Newton steps for nonsymmetric convex conic optimization
Dávid Papp, Anita Varga
TL;DR
This work develops feasible, full-step primal-dual interior-point algorithms for convex conic optimization on general nonsymmetric cones using only a primal logarithmically homogeneous self-concordant barrier with parameter $ν$. The authors establish a path-following framework around a central path with neighborhood $\mathcal{N}(η,τ)$, solve a Newton system, and prove $O(\sqrt{ν}\log(1/ε))$ iteration complexity to obtain $ε$-optimal primal-dual pairs, without relying on a dual barrier. They present multiple initialization strategies, including two-phase schemes and a homogeneous self-dual embedding (HSD), to handle unknown feasibility while preserving polynomial-time guarantees and dual certificates. A key application to sums-of-squares cones demonstrates that the proposed nonsymmetric IPAs attain high-quality, certifiable lower bounds far beyond what SDP can reliably achieve in large instances, confirming practical efficiency and numerical reliability. Overall, the results broaden the reach of interior-point methods to general cones with LHSCBs and have implications for SOS optimization, quantum information, and risk-aware modeling.
Abstract
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 the primal and dual problems in $O(\sqrtν\log(1/\varepsilon))$ iterations, where $ν$ is the barrier parameter of the LHSCB; this matches the best 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 ensure that the computed solutions are compatible with the dual certificates framework of (Davis and Papp, 2022). 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 large feasible or optimal solutions. 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.