2014/11/08 by Tor G. J. Myklebust, Tor Myklebust, Myklebust, Tor +3 · 1 citation
Computer Science · Engineering · Mathematics · #49M37 #52A41 #65Y20 #90C05 #90C22 #90C25 #90C30 #90C51 #90C60 #Advanced Optimization Algorithms Research #FOS: Mathematics #Optimization and Control (math.OC) #Optimization and Variational Analysis #Sparse and Compressive Sensing Techniques #math.OC #msc:49M37 #msc:52A41 #msc:65Y20 #msc:90C05 #msc:90C22 #msc:90C25 #msc:90C30 #msc:90C51 #msc:90C60
paper · pdf · doi:10.48550/arxiv.1411.2129
36 pages
arxiv created 2014/11/08 · openalex publication_date 2014/11/08 · arxiv updated 2014/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose and analyse primal-dual interior-point algorithms for convex optimization problems in conic form. The families of algorithms we analyse are so-called short-step algorithms and they match the current best iteration complexity bounds for primal-dual symmetric interior-point algorithm of Nesterov and Todd, for symmetric cone programming problems with given self-scaled barriers. Our results apply to any self-concordant barrier for any convex cone. We also prove that certain specializations of our algorithms to hyperbolic cone programming problems (which lie strictly between symmetric cone programming 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 to quasi-Newton updates and hence first-order methods in general.