1,594 research outputs found
Faster Convex Optimization: Simulated Annealing with an Efficient Universal Barrier
This paper explores a surprising equivalence between two seemingly-distinct
convex optimization methods. We show that simulated annealing, a well-studied
random walk algorithms, is directly equivalent, in a certain sense, to the
central path interior point algorithm for the the entropic universal barrier
function. This connection exhibits several benefits. First, we are able improve
the state of the art time complexity for convex optimization under the
membership oracle model. We improve the analysis of the randomized algorithm of
Kalai and Vempala by utilizing tools developed by Nesterov and Nemirovskii that
underly the central path following interior point algorithm. We are able to
tighten the temperature schedule for simulated annealing which gives an
improved running time, reducing by square root of the dimension in certain
instances. Second, we get an efficient randomized interior point method with an
efficiently computable universal barrier for any convex set described by a
membership oracle. Previously, efficiently computable barriers were known only
for particular convex sets
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
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
A lower bound on the barrier parameter of barriers for convex cones
International audienceLet K⊂Rn be a regular convex cone, let e1,...,en∈∂K be linearly independent points on the boundary of a compact affine section of the cone, and let x∗∈K0 be a point in the relative interior of this section. For k = 1, . . . , n, let l k be the line through the points e k and x *, let y k be the intersection point of l k with ∂K opposite to e k , and let z k be the intersection point of l k with the linear subspace spanned by all points e l , l = 1, . . . , n except e k . We give a lower bound on the barrier parameter ν of logarithmically homogeneous self-concordant barriers F:K0→R on K in terms of the projective cross-ratios qk=(ek,x∗;yk,zk) . Previously known lower bounds by Nesterov and Nemirovski can be obtained from our result as a special case. As an application, we construct an optimal barrier for the epigraph of the ||⋅||∞ -norm in Rn and compute lower bounds on the barrier parameter for the power cone and the epigraph of the ||⋅||p -norm in R2
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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