Interior-point algorithms with full Newton steps for nonsymmetric convex conic optimization

TL;DR

This paper introduces primal-dual interior-point algorithms with full Newton steps for nonsymmetric convex conic optimization, achieving optimal iteration complexity using only a logarithmically homogeneous self-concordant barrier.

Contribution

The paper presents a novel interior-point algorithm that works with nonsymmetric cones using minimal barrier assumptions and achieves optimal iteration complexity.

Findings

Algorithms compute feasible, near-optimal solutions in $O( oot u extstyle ext{log}(1/ extvarepsilon))$ iterations.

Method successfully solves a difficult polynomial optimization problem where standard SDP approaches fail.

Numerical experiments demonstrate efficiency and reliability of the proposed algorithms.

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 $O(\sqrt\nu\log(1/\varepsilon))$ iterations, where $\nu$ 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.