514 research outputs found
Local quadratic convergence of polynomial-time interior-point methods for conic optimization problems
In this paper, we establish a local quadratic convergence of polynomial-time interior-point methods for general conic optimization problems. The main structural property used in our analysis is the logarithmic homogeneity of self-concordant barrier functions. We propose new path-following predictor-corrector schemes which work only in the dual space. They are based on an easily computable gradient proximity measure, which ensures an automatic transformation of the global linear rate of convergence to the local quadratic one under some mild assumptions. Our step-size procedure for the predictor step is related to the maximum step size (the one that takes us to the boundary). It appears that in order to obtain local superlinear convergence, we need to tighten the neighborhood of the central path proportionally to the current duality gapconic optimization problem, worst-case complexity analysis, self-concordant barriers, polynomial-time methods, predictor-corrector methods, local quadratic convergence
An interior-point method for the single-facility location problem with mixed norms using a conic formulation
We consider the single-facility location problem with mixed norms, i.e. the problem of minimizing the sum of the distances from a point to a set of fixed points in R, where each distance can be measured according to a different p-norm.We show how this problem can be expressed into a structured conic format by decomposing the nonlinear components of the objective into a series of constraints involving three-dimensional cones. Using the availability of a self-concordant barrier for these cones, we present a polynomial-time algorithm (a long-step path-following interior-point scheme) to solve the problem up to a given accuracy. Finally, we report computational results for this algorithm and compare with standard nonlinear optimization solvers applied to this problem.nonsymmetric conic optimization, conic reformulation, convex optimization, sum of norm minimization, single-facility location problems, interior-point methods
A predictor-corrector algorithm for semidefinite programming that uses the factor width cone
We propose an interior point method (IPM) for solving semidefinite programming problems (SDPs). The standard interior point algorithms used to solve SDPs work in the space of positive semidefinite matrices. Contrary to that the proposed algorithm works in the cone of matrices of constant factor width. We prove global convergence and provide a complexity analysis. Our work is inspired by a series of papers by Ahmadi, Dash, Majumdar and Hall, and builds upon a recent preprint by Roig-Solvas and Sznaier [arXiv:2202.12374, 2022]
Interior-point algorithms for convex optimization based on primal-dual metrics
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.Comment: 36 page
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