1,554 research outputs found
Self-scaled barrier functions on symmetric cones and their classification
Self-scaled barrier functions on self-scaled cones were introduced through a
set of axioms in 1994 by Y.E. Nesterov and M.J. Todd as a tool for the
construction of long-step interior point algorithms. This paper provides firm
foundation for these objects by exhibiting their symmetry properties, their
intimate ties with the symmetry groups of their domains of definition, and
subsequently their decomposition into irreducible parts and algebraic
classification theory. In a first part we recall the characterisation of the
family of self-scaled cones as the set of symmetric cones and develop a
primal-dual symmetric viewpoint on self-scaled barriers, results that were
first discovered by the second author. We then show in a short, simple proof
that any pointed, convex cone decomposes into a direct sum of irreducible
components in a unique way, a result which can also be of independent interest.
We then show that any self-scaled barrier function decomposes in an essentially
unique way into a direct sum of self-scaled barriers defined on the irreducible
components of the underlying symmetric cone. Finally, we present a complete
algebraic classification of self-scaled barrier functions using the
correspondence between symmetric cones and Euclidean Jordan algebras.Comment: 17 page
Self-scaled barriers for irreducible symmetric cones
Self-scaled barrier functions are fundamental objects in the theory of
interior-point methods for linear optimization over symmetric cones, of which
linear and semidefinite programming are special cases. We are classifying all
self-scaled barriers over irreducible symmetric cones and show that these
functions are merely homothetic transformations of the universal barrier
function. Together with a decomposition theorem for self-scaled barriers this
concludes the algebraic classification theory of these functions. After
introducing the reader to the concepts relevant to the problem and tracing the
history of the subject, we start by deriving our result from first principles
in the important special case of semidefinite programming. We then generalise
these arguments to irreducible symmetric cones by invoking results from the
theory of Euclidean Jordan algebras.Comment: 12 page
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
Inner approximation of convex cones via primal-dual ellipsoidal norms
We study ellipsoids from the point of view of approximating convex sets. Our focus is
on finding largest volume ellipsoids with specified centers which are contained in certain
convex cones. After reviewing the related literature and establishing some fundamental
mathematical techniques that will be useful, we derive such maximum volume ellipsoids
for second order cones and the cones of symmetric positive semidefinite matrices. Then we
move to the more challenging problem of finding a largest pair (in the sense of geometric
mean of their radii) of primal-dual ellipsoids (in the sense of dual norms) with specified
centers that are contained in their respective primal-dual pair of convex cones
Linear optimization over homogeneous matrix cones
A convex cone is homogeneous if its automorphism group acts transitively on
the interior of the cone, i.e., for every pair of points in the interior of the
cone, there exists a cone automorphism that maps one point to the other. Cones
that are homogeneous and self-dual are called symmetric. The symmetric cones
include the positive semidefinite matrix cone and the second order cone as
important practical examples. In this paper, we consider the less well-studied
conic optimization problems over cones that are homogeneous but not necessarily
self-dual. We start with cones of positive semidefinite symmetric matrices with
a given sparsity pattern. Homogeneous cones in this class are characterized by
nested block-arrow sparsity patterns, a subset of the chordal sparsity
patterns. We describe transitive subsets of the automorphism groups of the
cones and their duals, and important properties of the composition of log-det
barrier functions with the automorphisms in this set. Next, we consider
extensions to linear slices of the positive semidefinite cone, i.e.,
intersection of the positive semidefinite cone with a linear subspace, and
review conditions that make the cone homogeneous. In the third part of the
paper we give a high-level overview of the classical algebraic theory of
homogeneous cones due to Vinberg and Rothaus. A fundamental consequence of this
theory is that every homogeneous cone admits a spectrahedral (linear matrix
inequality) representation. We conclude by discussing the role of homogeneous
cone structure in primal-dual symmetric interior-point methods.Comment: 59 pages, 10 figures, to appear in Acta Numeric
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