446 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
A Schwarz lemma for K\"ahler affine metrics and the canonical potential of a proper convex cone
This is an account of some aspects of the geometry of K\"ahler affine metrics
based on considering them as smooth metric measure spaces and applying the
comparison geometry of Bakry-Emery Ricci tensors. Such techniques yield a
version for K\"ahler affine metrics of Yau's Schwarz lemma for volume forms. By
a theorem of Cheng and Yau there is a canonical K\"ahler affine Einstein metric
on a proper convex domain, and the Schwarz lemma gives a direct proof of its
uniqueness up to homothety. The potential for this metric is a function
canonically associated to the cone, characterized by the property that its
level sets are hyperbolic affine spheres foliating the cone. It is shown that
for an -dimensional cone a rescaling of the canonical potential is an
-normal barrier function in the sense of interior point methods for conic
programming. It is explained also how to construct from the canonical potential
Monge-Amp\`ere metrics of both Riemannian and Lorentzian signatures, and a mean
curvature zero conical Lagrangian submanifold of the flat para-K\"ahler space.Comment: Minor corrections. References adde
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
Interior-point methods on manifolds: theory and applications
Interior-point methods offer a highly versatile framework for convex
optimization that is effective in theory and practice. A key notion in their
theory is that of a self-concordant barrier. We give a suitable generalization
of self-concordance to Riemannian manifolds and show that it gives the same
structural results and guarantees as in the Euclidean setting, in particular
local quadratic convergence of Newton's method. We analyze a path-following
method for optimizing compatible objectives over a convex domain for which one
has a self-concordant barrier, and obtain the standard complexity guarantees as
in the Euclidean setting. We provide general constructions of barriers, and
show that on the space of positive-definite matrices and other symmetric
spaces, the squared distance to a point is self-concordant. To demonstrate the
versatility of our framework, we give algorithms with state-of-the-art
complexity guarantees for the general class of scaling and non-commutative
optimization problems, which have been of much recent interest, and we provide
the first algorithms for efficiently finding high-precision solutions for
computing minimal enclosing balls and geometric medians in nonpositive
curvature.Comment: 85 pages. v2: Merged with independent work arXiv:2212.10981 by
Hiroshi Hira
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