9,982 research outputs found
Geometry of Online Packing Linear Programs
We consider packing LP's with rows where all constraint coefficients are
normalized to be in the unit interval. The n columns arrive in random order and
the goal is to set the corresponding decision variables irrevocably when they
arrive so as to obtain a feasible solution maximizing the expected reward.
Previous (1 - \epsilon)-competitive algorithms require the right-hand side of
the LP to be Omega((m/\epsilon^2) log (n/\epsilon)), a bound that worsens with
the number of columns and rows. However, the dependence on the number of
columns is not required in the single-row case and known lower bounds for the
general case are also independent of n.
Our goal is to understand whether the dependence on n is required in the
multi-row case, making it fundamentally harder than the single-row version. We
refute this by exhibiting an algorithm which is (1 - \epsilon)-competitive as
long as the right-hand sides are Omega((m^2/\epsilon^2) log (m/\epsilon)). Our
techniques refine previous PAC-learning based approaches which interpret the
online decisions as linear classifications of the columns based on sampled dual
prices. The key ingredient of our improvement comes from a non-standard
covering argument together with the realization that only when the columns of
the LP belong to few 1-d subspaces we can obtain small such covers; bounding
the size of the cover constructed also relies on the geometry of linear
classifiers. General packing LP's are handled by perturbing the input columns,
which can be seen as making the learning problem more robust
Hypoconstrained Jammed Packings of Nonspherical Hard Particles: Ellipses and Ellipsoids
Continuing on recent computational and experimental work on jammed packings
of hard ellipsoids [Donev et al., Science, vol. 303, 990-993] we consider
jamming in packings of smooth strictly convex nonspherical hard particles. We
explain why the isocounting conjecture, which states that for large disordered
jammed packings the average contact number per particle is twice the number of
degrees of freedom per particle (\bar{Z}=2d_{f}), does not apply to
nonspherical particles. We develop first- and second-order conditions for
jamming, and demonstrate that packings of nonspherical particles can be jammed
even though they are hypoconstrained (\bar{Z}<2d_{f}). We apply an algorithm
using these conditions to computer-generated hypoconstrained ellipsoid and
ellipse packings and demonstrate that our algorithm does produce jammed
packings, even close to the sphere point. We also consider packings that are
nearly jammed and draw connections to packings of deformable (but stiff)
particles. Finally, we consider the jamming conditions for nearly spherical
particles and explain quantitatively the behavior we observe in the vicinity of
the sphere point.Comment: 33 pages, third revisio
A Parallelizable Acceleration Framework for Packing Linear Programs
This paper presents an acceleration framework for packing linear programming
problems where the amount of data available is limited, i.e., where the number
of constraints m is small compared to the variable dimension n. The framework
can be used as a black box to speed up linear programming solvers dramatically,
by two orders of magnitude in our experiments. We present worst-case guarantees
on the quality of the solution and the speedup provided by the algorithm,
showing that the framework provides an approximately optimal solution while
running the original solver on a much smaller problem. The framework can be
used to accelerate exact solvers, approximate solvers, and parallel/distributed
solvers. Further, it can be used for both linear programs and integer linear
programs
A Class of Semidefinite Programs with rank-one solutions
We show that a class of semidefinite programs (SDP) admits a solution that is
a positive semidefinite matrix of rank at most , where is the rank of
the matrix involved in the objective function of the SDP. The optimization
problems of this class are semidefinite packing problems, which are the SDP
analogs to vector packing problems. Of particular interest is the case in which
our result guarantees the existence of a solution of rank one: we show that the
computation of this solution actually reduces to a Second Order Cone Program
(SOCP). We point out an application in statistics, in the optimal design of
experiments.Comment: 16 page
MGOS: A library for molecular geometry and its operating system
The geometry of atomic arrangement underpins the structural understanding of molecules in many fields. However, no general framework of mathematical/computational theory for the geometry of atomic arrangement exists. Here we present "Molecular Geometry (MG)'' as a theoretical framework accompanied by "MG Operating System (MGOS)'' which consists of callable functions implementing the MG theory. MG allows researchers to model complicated molecular structure problems in terms of elementary yet standard notions of volume, area, etc. and MGOS frees them from the hard and tedious task of developing/implementing geometric algorithms so that they can focus more on their primary research issues. MG facilitates simpler modeling of molecular structure problems; MGOS functions can be conveniently embedded in application programs for the efficient and accurate solution of geometric queries involving atomic arrangements. The use of MGOS in problems involving spherical entities is akin to the use of math libraries in general purpose programming languages in science and engineering. (C) 2019 The Author(s). Published by Elsevier B.V
Lecture notes: Semidefinite programs and harmonic analysis
Lecture notes for the tutorial at the workshop HPOPT 2008 - 10th
International Workshop on High Performance Optimization Techniques (Algebraic
Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg
University, The Netherlands.Comment: 31 page
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