255 research outputs found
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
A probabilistic interpretation of set-membership filtering: application to polynomial systems through polytopic bounding
Set-membership estimation is usually formulated in the context of set-valued
calculus and no probabilistic calculations are necessary. In this paper, we
show that set-membership estimation can be equivalently formulated in the
probabilistic setting by employing sets of probability measures. Inference in
set-membership estimation is thus carried out by computing expectations with
respect to the updated set of probability measures P as in the probabilistic
case. In particular, it is shown that inference can be performed by solving a
particular semi-infinite linear programming problem, which is a special case of
the truncated moment problem in which only the zero-th order moment is known
(i.e., the support). By writing the dual of the above semi-infinite linear
programming problem, it is shown that, if the nonlinearities in the measurement
and process equations are polynomial and if the bounding sets for initial
state, process and measurement noises are described by polynomial inequalities,
then an approximation of this semi-infinite linear programming problem can
efficiently be obtained by using the theory of sum-of-squares polynomial
optimization. We then derive a smart greedy procedure to compute a polytopic
outer-approximation of the true membership-set, by computing the minimum-volume
polytope that outer-bounds the set that includes all the means computed with
respect to P
How to Integrate a Polynomial over a Simplex
This paper settles the computational complexity of the problem of integrating
a polynomial function f over a rational simplex. We prove that the problem is
NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin
and Straus. On the other hand, if the polynomial depends only on a fixed number
of variables, while its degree and the dimension of the simplex are allowed to
vary, we prove that integration can be done in polynomial time. As a
consequence, for polynomials of fixed total degree, there is a polynomial time
algorithm as well. We conclude the article with extensions to other polytopes,
discussion of other available methods and experimental results.Comment: Tables added with new experimental results. References adde
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Triangulations
The earliest work in topology was often based on explicit combinatorial models – usually triangulations – for the spaces being studied. Although algebraic methods in topology gradually replaced combinatorial ones in the mid-1900s, the emergence of computers later revitalized the study of triangulations. By now there are several distinct mathematical communities actively doing work on different aspects of triangulations. The goal of this workshop was to bring the researchers from these various communities together to stimulate interaction and to benefit from the exchange of ideas and methods
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
The algebraic square peg problem
The square peg problem asks whether every continuous curve in the plane that starts and ends at the same point without self-intersecting contains four distinct corners of some square. Toeplitz conjectured in 1911 that this is indeed the case. Hundred years later we only have partial results for curves with additional smoothness properties.
The contribution of this thesis is an algebraic variant of the square peg problem. By casting the set of squares inscribed on an algebraic plane curve as a variety and applying Bernshtein's Theorem we are able to count the number of such squares. An algebraic plane curve defined by a polynomial of degree m inscribes either an infinite amount of squares, or at most (m4 - 5m2 + 4m)= 4 squares. Computations using computer algebra software lend evidence to the claim that this upper bound is sharp for generic curves.
Earlier work on Toeplitz's conjecture has shown that generically an odd number of squares is inscribed on a smooth enough Jordan curve. Examples of real cubics and quartics suggest that there is a similar parity condition on the number of squares inscribed on some topological types of algebraic plane curves that are not Jordan curves. Thus we are led to conjecture that algebraic plane curves homeomorphic to the real line inscribe an even number of squares
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
Solid Angle Measure Approximation Methods for Polyhedral Cones
Polyhedral cones are of interest in many fields, like geometry and optimization. A simple, yet fundamental question we may ask about a cone is how large it is. As cones are unbounded, we consider their solid angle measure: the proportion of space that they occupy. Beyond dimension three, definitive formulas for this measure are unknown. Consequently, devising methods to estimate this quantity is imperative. In this dissertation, we endeavor to enhance our understanding of solid angle measures and provide valuable insights into the efficacy of various approximation techniques.
Ribando and Aomoto independently discovered a Taylor series formula for solid angle measures of certain simplicial cones. Leveraging Brion--Vergne Decomposition, we extend their findings, devising an algorithm for approximating solid angle measures of polyhedral cones, including those where the series is not applicable. We compare our method to other estimation techniques, and explore the practical applications of these methods within optimization.
Gomory and Johnson established the use of facets of master cyclic group polyhedra to derive cuts for integer programs. Within this framework, the size of the solid angle subtended by a facet determines its importance. We apply various approximation techniques to measure facet importance, provide computational results, and discuss their implications
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