31 research outputs found
Decomposition Methods for Nonlinear Optimization and Data Mining
We focus on two central themes in this dissertation. The first one is on
decomposing polytopes and polynomials in ways that allow us to perform
nonlinear optimization. We start off by explaining important results on
decomposing a polytope into special polyhedra. We use these decompositions and
develop methods for computing a special class of integrals exactly. Namely, we
are interested in computing the exact value of integrals of polynomial
functions over convex polyhedra. We present prior work and new extensions of
the integration algorithms. Every integration method we present requires that
the polynomial has a special form. We explore two special polynomial
decomposition algorithms that are useful for integrating polynomial functions.
Both polynomial decompositions have strengths and weaknesses, and we experiment
with how to practically use them.
After developing practical algorithms and efficient software tools for
integrating a polynomial over a polytope, we focus on the problem of maximizing
a polynomial function over the continuous domain of a polytope. This
maximization problem is NP-hard, but we develop approximation methods that run
in polynomial time when the dimension is fixed. Moreover, our algorithm for
approximating the maximum of a polynomial over a polytope is related to
integrating the polynomial over the polytope. We show how the integration
methods can be used for optimization.
The second central topic in this dissertation is on problems in data science.
We first consider a heuristic for mixed-integer linear optimization. We show
how many practical mixed-integer linear have a special substructure containing
set partition constraints. We then describe a nice data structure for finding
feasible zero-one integer solutions to systems of set partition constraints.
Finally, we end with an applied project using data science methods in medical
research.Comment: PHD Thesis of Brandon Dutr
Software for Exact Integration of Polynomials over Polyhedra
We are interested in the fast computation of the exact value of integrals of
polynomial functions over convex polyhedra. We present speed ups and extensions
of the algorithms presented in previous work. We present the new software
implementation and provide benchmark computations. The computation of integrals
of polynomials over polyhedral regions has many applications; here we
demonstrate our algorithmic tools solving a challenge from combinatorial voting
theory.Comment: Major updat
Coefficients of Sylvester's Denumerant
For a given sequence of positive integers, we consider
the combinatorial function that counts the nonnegative
integer solutions of the equation , where the right-hand side is a varying
nonnegative integer. It is well-known that is a
quasi-polynomial function in the variable of degree . In combinatorial
number theory this function is known as Sylvester's denumerant.
Our main result is a new algorithm that, for every fixed number , computes
in polynomial time the highest coefficients of the quasi-polynomial
as step polynomials of (a simpler and more explicit
representation). Our algorithm is a consequence of a nice poset structure on
the poles of the associated rational generating function for
and the geometric reinterpretation of some rational
generating functions in terms of lattice points in polyhedral cones. Our
algorithm also uses Barvinok's fundamental fast decomposition of a polyhedral
cone into unimodular cones. This paper also presents a simple algorithm to
predict the first non-constant coefficient and concludes with a report of
several computational experiments using an implementation of our algorithm in
LattE integrale. We compare it with various Maple programs for partial or full
computation of the denumerant.Comment: minor revision, 28 page
Top Coefficients of the Denumerant
International audienceFor a given sequence of positive integers, we consider the combinatorial function that counts the nonnegative integer solutions of the equation , where the right-hand side is a varying nonnegative integer. It is well-known that is a quasipolynomial function of of degree . In combinatorial number theory this function is known as the . Our main result is a new algorithm that, for every fixed number , computes in polynomial time the highest coefficients of the quasi-polynomial as step polynomials of . Our algorithm is a consequence of a nice poset structure on the poles of the associated rational generating function for and the geometric reinterpretation of some rational generating functions in terms of lattice points in polyhedral cones. Experiments using a implementation will be posted separately.Considérons une liste de entiers positifs. Le dénumérant est lafonction qui compte le nombre de solutions en entiers positifs ou nuls de l’équation , où varie dans les entiers positifs ou nuls. Il est bien connu que cette fonction est une fonction quasi-polynomiale de , de degré . Nous donnons un nouvel algorithme qui calcule, pour chaque entier fixé (mais n’est pas fixé, les plus hauts coefficients du quasi-polynôme en termes de fonctions en dents de scie. Notre algorithme utilise la structure d’ensemble partiellement ordonné des pôles de la fonction génératrice de . Les plus hauts coefficients se calculent à l’aide de fonctions génératrices de points entiers dans des cônes polyèdraux de dimension inférieure ou égale à
COVID-19 Severity and Cardiovascular Outcomes in SARS-CoV-2-Infected Patients With Cancer and Cardiovascular Disease
BACKGROUND: Data regarding outcomes among patients with cancer and co-morbid cardiovascular disease (CVD)/cardiovascular risk factors (CVRF) after SARS-CoV-2 infection are limited.
OBJECTIVES: To compare Coronavirus disease 2019 (COVID-19) related complications among cancer patients with and without co-morbid CVD/CVRF.
METHODS: Retrospective cohort study of patients with cancer and laboratory-confirmed SARS-CoV-2, reported to the COVID-19 and Cancer Consortium (CCC19) registry from 03/17/2020 to 12/31/2021. CVD/CVRF was defined as established CVD
RESULTS: Among 10,876 SARS-CoV-2 infected patients with cancer (median age 65 [IQR 54-74] years, 53% female, 52% White), 6253 patients (57%) had co-morbid CVD/CVRF. Co-morbid CVD/CVRF was associated with higher COVID-19 severity (adjusted OR: 1.25 [95% CI 1.11-1.40]). Adverse CV events were significantly higher in patients with CVD/CVRF (all
CONCLUSIONS: Co-morbid CVD/CVRF is associated with higher COVID-19 severity among patients with cancer, particularly those not receiving active cancer therapy. While infrequent, COVID-19 related CV complications were higher in patients with comorbid CVD/CVRF. (COVID-19 and Cancer Consortium Registry [CCC19]; NCT04354701)
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Decomposition Methods for Nonlinear Optimization and Data Mining
We focus on two central themes in this dissertation. The first one is on
decomposing polytopes and polynomials in ways that allow us to perform nonlinear
optimization. We start off by explaining important results on decomposing a polytope into
special polyhedra. We use these decompositions and develop methods for computing a special
class of integrals exactly. Namely, we are interested in computing the exact value of
integrals of polynomial functions over convex polyhedra. We present prior work and new
extensions of the integration algorithms. Every integration method we present requires that
the polynomial has a special form. We explore two special polynomial decomposition
algorithms that are useful for integrating polynomial functions. Both polynomial
decompositions have strengths and weaknesses, and we experiment with how to practically use
them. After developing practical algorithms and efficient software tools for integrating a
polynomial over a polytope, we focus on the problem of maximizing a polynomial function
over the continuous domain of a polytope. This maximization problem is NP-hard, but we
develop approximation methods that run in polynomial time when the dimension is fixed.
Moreover, our algorithm for approximating the maximum of a polynomial over a polytope is
related to integrating the polynomial over the polytope. We show how the integration
methods can be used for optimization. The second central topic in this dissertation is on
problems in data science. We first consider a heuristic for mixed-integer linear
optimization. We show how many practical mixed-integer linear have a special substructure
containing set partition constraints. We then describe a nice data structure for finding
feasible zero-one integer solutions to systems of set partition constraints. Finally, we
end with an applied project using data science methods in medical research
Recommended from our members
Approximating the maximum of a polynomial over a polytope: Handelman decomposition and continuous generating functions
We investigate a way to approximate the maximum of a polynomial over a polytopal
region by using Handelman's polynomial decomposition and continuous multivariate generating
functions. The maximization problem is NP-hard, but our approximation methods will run in
polynomial time when the dimension is fixed
Approximating the maximum of a polynomial over a polytope: Handelman decomposition and continuous generating functions
We investigate a way to approximate the maximum of a polynomial over a polytopal
region by using Handelman's polynomial decomposition and continuous multivariate generating
functions. The maximization problem is NP-hard, but our approximation methods will run in
polynomial time when the dimension is fixed
A Modular Geometrical Framework for Modelling the Force-Contraction Profile of Vacuum-Powered Soft Actuators
In this paper, we present a generalized modeling tool for predicting the output force profile of vacuum-powered soft actuators using a simplified geometrical approach and the principle of virtual work. Previous work has derived analytical formulas to model the force-contraction profile of specific actuators. To enhance the versatility and the efficiency of the modelling process we propose a generalized numerical algorithm based purely on geometrical inputs, which can be tailored to the desired actuator, to estimate its force-contraction profile quickly and for any combination of varying geometrical parameters. We identify a class of linearly contracting vacuum actuators that consists of a polymeric skin guided by a rigid skeleton and apply our model to two such actuators-vacuum bellows and Fluid-driven Origami-inspired Artificial Muscles-to demonstrate the versatility of our model. We perform experiments to validate that our model can predict the force profile of the actuators using its geometric principles, modularly combined with design-specific external adjustment factors. Our framework can be used as a versatile design tool that allows users to perform parametric studies and rapidly and efficiently tune actuator dimensions to produce a force-contraction profile to meet their needs, and as a pre-screening tool to obviate the need for multiple rounds of time-intensive actuator fabrication and testing.National Science Foundation (Award 1847541)Muscular Dystrophy Association Research (Grant 577961