972 research outputs found
From Steiner Formulas for Cones to Concentration of Intrinsic Volumes
The intrinsic volumes of a convex cone are geometric functionals that return
basic structural information about the cone. Recent research has demonstrated
that conic intrinsic volumes are valuable for understanding the behavior of
random convex optimization problems. This paper develops a systematic technique
for studying conic intrinsic volumes using methods from probability. At the
heart of this approach is a general Steiner formula for cones. This result
converts questions about the intrinsic volumes into questions about the
projection of a Gaussian random vector onto the cone, which can then be
resolved using tools from Gaussian analysis. The approach leads to new
identities and bounds for the intrinsic volumes of a cone, including a
near-optimal concentration inequality.Comment: This version corrects errors in Propositions 3.3 and 3.4 and in Lemma
8.3 that appear in the published versio
Gordon's inequality and condition numbers in conic optimization
The probabilistic analysis of condition numbers has traditionally been
approached from different angles; one is based on Smale's program in complexity
theory and features integral geometry, while the other is motivated by
geometric functional analysis and makes use of the theory of Gaussian
processes. In this note we explore connections between the two approaches in
the context of the biconic homogeneous feasiblity problem and the condition
numbers motivated by conic optimization theory. Key tools in the analysis are
Slepian's and Gordon's comparision inequalities for Gaussian processes,
interpreted as monotonicity properties of moment functionals, and their
interplay with ideas from conic integral geometry
Effective Condition Number Bounds for Convex Regularization
We derive bounds relating Renegar's condition number to quantities that
govern the statistical performance of convex regularization in settings that
include the -analysis setting. Using results from conic integral
geometry, we show that the bounds can be made to depend only on a random
projection, or restriction, of the analysis operator to a lower dimensional
space, and can still be effective if these operators are ill-conditioned. As an
application, we get new bounds for the undersampling phase transition of
composite convex regularizers. Key tools in the analysis are Slepian's
inequality and the kinematic formula from integral geometry.Comment: 17 pages, 4 figures . arXiv admin note: text overlap with
arXiv:1408.301
Intrinsic Volumes of Polyhedral Cones: A combinatorial perspective
The theory of intrinsic volumes of convex cones has recently found striking
applications in areas such as convex optimization and compressive sensing. This
article provides a self-contained account of the combinatorial theory of
intrinsic volumes for polyhedral cones. Direct derivations of the General
Steiner formula, the conic analogues of the Brianchon-Gram-Euler and the
Gauss-Bonnet relations, and the Principal Kinematic Formula are given. In
addition, a connection between the characteristic polynomial of a hyperplane
arrangement and the intrinsic volumes of the regions of the arrangement, due to
Klivans and Swartz, is generalized and some applications are presented.Comment: Survey, 23 page
Extended Formulations in Mixed-integer Convex Programming
We present a unifying framework for generating extended formulations for the
polyhedral outer approximations used in algorithms for mixed-integer convex
programming (MICP). Extended formulations lead to fewer iterations of outer
approximation algorithms and generally faster solution times. First, we observe
that all MICP instances from the MINLPLIB2 benchmark library are conic
representable with standard symmetric and nonsymmetric cones. Conic
reformulations are shown to be effective extended formulations themselves
because they encode separability structure. For mixed-integer
conic-representable problems, we provide the first outer approximation
algorithm with finite-time convergence guarantees, opening a path for the use
of conic solvers for continuous relaxations. We then connect the popular
modeling framework of disciplined convex programming (DCP) to the existence of
extended formulations independent of conic representability. We present
evidence that our approach can yield significant gains in practice, with the
solution of a number of open instances from the MINLPLIB2 benchmark library.Comment: To be presented at IPCO 201
The achievable performance of convex demixing
Demixing is the problem of identifying multiple structured signals from a
superimposed, undersampled, and noisy observation. This work analyzes a general
framework, based on convex optimization, for solving demixing problems. When
the constituent signals follow a generic incoherence model, this analysis leads
to precise recovery guarantees. These results admit an attractive
interpretation: each signal possesses an intrinsic degrees-of-freedom
parameter, and demixing can succeed if and only if the dimension of the
observation exceeds the total degrees of freedom present in the observation
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