339 research outputs found
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
Uncertainty principles for orthonormal sequences
The aim of this paper is to provide complementary quantitative extensions of
two results of H.S. Shapiro on the time-frequency concentration of orthonormal
sequences in . More precisely, Shapiro proved that if the elements of
an orthonormal sequence and their Fourier transforms are all pointwise bounded
by a fixed function in then the sequence is finite. In a related
result, Shapiro also proved that if the elements of an orthonormal sequence and
their Fourier transforms have uniformly bounded means and dispersions then the
sequence is finite. This paper gives quantitative bounds on the size of the
finite orthonormal sequences in Shapiro's uncertainty principles. The bounds
are obtained by using prolate sphero\"{i}dal wave functions and combinatorial
estimates on the number of elements in a spherical code. Extensions for Riesz
bases and different measures of time-frequency concentration are also given
Mirror-Descent Methods in Mixed-Integer Convex Optimization
In this paper, we address the problem of minimizing a convex function f over
a convex set, with the extra constraint that some variables must be integer.
This problem, even when f is a piecewise linear function, is NP-hard. We study
an algorithmic approach to this problem, postponing its hardness to the
realization of an oracle. If this oracle can be realized in polynomial time,
then the problem can be solved in polynomial time as well. For problems with
two integer variables, we show that the oracle can be implemented efficiently,
that is, in O(ln(B)) approximate minimizations of f over the continuous
variables, where B is a known bound on the absolute value of the integer
variables.Our algorithm can be adapted to find the second best point of a
purely integer convex optimization problem in two dimensions, and more
generally its k-th best point. This observation allows us to formulate a
finite-time algorithm for mixed-integer convex optimization
Nonlinear Integer Programming
Research efforts of the past fifty years have led to a development of linear
integer programming as a mature discipline of mathematical optimization. Such a
level of maturity has not been reached when one considers nonlinear systems
subject to integrality requirements for the variables. This chapter is
dedicated to this topic.
The primary goal is a study of a simple version of general nonlinear integer
problems, where all constraints are still linear. Our focus is on the
computational complexity of the problem, which varies significantly with the
type of nonlinear objective function in combination with the underlying
combinatorial structure. Numerous boundary cases of complexity emerge, which
sometimes surprisingly lead even to polynomial time algorithms.
We also cover recent successful approaches for more general classes of
problems. Though no positive theoretical efficiency results are available, nor
are they likely to ever be available, these seem to be the currently most
successful and interesting approaches for solving practical problems.
It is our belief that the study of algorithms motivated by theoretical
considerations and those motivated by our desire to solve practical instances
should and do inform one another. So it is with this viewpoint that we present
the subject, and it is in this direction that we hope to spark further
research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G.
Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50
Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art
Surveys, Springer-Verlag, 2009, ISBN 354068274
Abelian Chern-Simons Vortices and Holomorphic Burgers' Hierarchy
The Abelian Chern-Simons Gauge Field Theory in 2+1 dimensions and its
relation with holomorphic Burgers' Hierarchy is considered. It is shown that
the relation between complex potential and the complex gauge field as in
incompressible and irrotational hydrodynamics, has meaning of the analytic
Cole-Hopf transformation, linearizing the Burgers Hierarchy in terms of the
holomorphic Schr\"odinger Hierarchy. Then the motion of planar vortices in
Chern-Simons theory, appearing as pole singularities of the gauge field,
corresponds to motion of zeroes of the hierarchy. Using boost transformations
of the complex Galilean group of the hierarchy, a rich set of exact solutions,
describing integrable dynamics of planar vortices and vortex lattices in terms
of the generalized Kampe de Feriet and Hermite polynomials is constructed. The
results are applied to the holomorphic reduction of the Ishimori model and the
corresponding hierarchy, describing dynamics of magnetic vortices and
corresponding lattices in terms of complexified Calogero-Moser models.
Corrections on two vortex dynamics from the Moyal space-time non-commutativity
in terms of Airy functions are found.Comment: 15 pages, talk presented in Workshop `Nonlinear Physics IV: Theory
and Experiment`, 22-30 June 2006, Gallipoli, Ital
High-dimensional analysis reveals distinct endotypes in patients with idiopathic inflammatory myopathies
The idiopathic inflammatory myopathies (IIM) are a rare clinically heterogeneous group of conditions affecting the skin, muscle, joint, and lung in various combinations. While myositis specific autoantibodies are well described, we postulate that broader immune endotypes exist in IIM spanning B cell, T cell, and monocyte compartments. This study aims to identify immune endotypes through detailed immunophenotyping of peripheral blood mononuclear cells (PBMCs) in IIM patients compared to healthy controls. We collected PBMCs from 17 patients with a clinical diagnosis of inflammatory myositis and characterized the B, T, and myeloid cell subsets using mass cytometry by time of flight (CyTOF). Data were analyzed using a combination of the dimensionality reduction algorithm t-distributed stochastic neighbor embedding (t-SNE), cluster identification, characterization, and regression (CITRUS), and marker enrichment modeling (MEM); supervised biaxial gating validated populations identified by these methods to be differentially abundant between groups. Using these approaches, we identified shared immunologic features across all IIM patients, despite different clinical features, as well as two distinct immune endotypes. All IIM patients had decreased surface expression of RP105/CD180 on B cells and a reduction in circulating CD3+CXCR3+ subsets relative to healthy controls. One IIM endotype featured CXCR4 upregulation across all cellular compartments. The second endotype was hallmarked by an increased frequency of CD19+CD21loCD11c+ and CD3+CD4+PD1+ subsets. The experimental and analytical methods we describe here are broadly applicable to studying other immune-mediated diseases (e.g., autoimmunity, immunodeficiency) or protective immune responses (e.g., infection, vaccination)
Hypercontractive measures, Talagrand's inequality, and influences
We survey several Talagrand type inequalities and their application to
influences with the tool of hypercontractivity for both discrete and
continuous, and product and non-product models. The approach covers similarly
by a simple interpolation the framework of geometric influences recently
developed by N. Keller, E. Mossel and A. Sen. Geometric Brascamp-Lieb
decompositions are also considered in this context
Local Hardy Spaces of Musielak-Orlicz Type and Their Applications
Let \phi: \mathbb{R}^n\times[0,\fz)\rightarrow[0,\fz) be a function such
that is an Orlicz function and (the class of local weights
introduced by V. S. Rychkov). In this paper, the authors introduce a local
Hardy space of Musielak-Orlicz type by the local grand
maximal function, and a local -type space
which is further proved to be the
dual space of . As an application, the authors prove
that the class of pointwise multipliers for the local
-type space ,
characterized by E. Nakai and K. Yabuta, is just the dual of
L^1(\rn)+h_{\Phi_0}(\mathbb{R}^n), where is an increasing function on
satisfying some additional growth conditions and a
Musielak-Orlicz function induced by . Characterizations of
, including the atoms, the local vertical and the local
nontangential maximal functions, are presented. Using the atomic
characterization, the authors prove the existence of finite atomic
decompositions achieving the norm in some dense subspaces of
, from which, the authors further deduce some
criterions for the boundedness on of some sublinear
operators. Finally, the authors show that the local Riesz transforms and some
pseudo-differential operators are bounded on .Comment: Sci. China Math. (to appear
CD19 + CD21lo/neg cells are increased in systemic sclerosis-associated interstitial lung disease
Interstitial lung disease (ILD) represents a significant cause of morbidity and mortality in systemic sclerosis (SSc). The purpose of this study was to examine recirculating lymphocytes from SSc patients for potential biomarkers of interstitial lung disease (ILD). Peripheral blood mononuclear cells (PBMCs) were isolated from patients with SSc and healthy controls enrolled in the Vanderbilt University Myositis and Scleroderma Treatment Initiative Center cohort between 9/2017-6/2019. Clinical phenotyping was performed by chart abstraction. Immunophenotyping was performed using both mass cytometry and fluorescence cytometry combined with t-distributed stochastic neighbor embedding analysis and traditional biaxial gating. This study included 34 patients with SSc-ILD, 14 patients without SSc-ILD, and 25 healthy controls. CD2
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