41,448 research outputs found
Integer Polynomial Optimization in Fixed Dimension
We classify, according to their computational complexity, integer
optimization problems whose constraints and objective functions are polynomials
with integer coefficients and the number of variables is fixed. For the
optimization of an integer polynomial over the lattice points of a convex
polytope, we show an algorithm to compute lower and upper bounds for the
optimal value. For polynomials that are non-negative over the polytope, these
sequences of bounds lead to a fully polynomial-time approximation scheme for
the optimization problem.Comment: In this revised version we include a stronger complexity bound on our
algorithm. Our algorithm is in fact an FPTAS (fully polynomial-time
approximation scheme) to maximize a non-negative integer polynomial over the
lattice points of a polytop
On the complexity of nonlinear mixed-integer optimization
This is a survey on the computational complexity of nonlinear mixed-integer
optimization. It highlights a selection of important topics, ranging from
incomputability results that arise from number theory and logic, to recently
obtained fully polynomial time approximation schemes in fixed dimension, and to
strongly polynomial-time algorithms for special cases.Comment: 26 pages, 5 figures; to appear in: Mixed-Integer Nonlinear
Optimization, IMA Volumes, Springer-Verla
Integer programming, lattices, and results in fixed dimension
We review and describe several results regarding integer programming problems in fixed dimension. First, we describe various lattice basis reduction algorithms that are used as auxiliary algorithms when solving integer feasibility and optimization problems. Next, we review three algorithms for solving the integer feasibility problem. These algorithms are based on the idea of branching on lattice hyperplanes, and their running time is polynomial in fixed dimension. We also briefly describe an algorithm, based on a different principle, to count integer points in an integer polytope. We then turn the attention to integer optimization. Again, we describe three algorithms: binary search, a linear algorithm for a fixed number of constraints, and a randomized algorithm for a varying number of constraints. The topic of the next part of our chapter is how to use lattice basis reduction in problem reformulation. Finally, we review cutting plane results when the dimension is fixe
Integer convex minimization by mixed integer linear optimization
Minimizing a convex function over the integral points of a bounded convex set is polynomial in fixed dimension (Grötschel et al., 1988). We provide an alternative, short, and geometrically motivated proof of this result. In particular, we present an oracle-polynomial algorithm based on a mixed integer linear optimization oracle
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
FPTAS for optimizing polynomials over the mixed-integer points of polytopes in fixed dimension
We show the existence of a fully polynomial-time approximation scheme (FPTAS)
for the problem of maximizing a non-negative polynomial over mixed-integer sets
in convex polytopes, when the number of variables is fixed. Moreover, using a
weaker notion of approximation, we show the existence of a fully
polynomial-time approximation scheme for the problem of maximizing or
minimizing an arbitrary polynomial over mixed-integer sets in convex polytopes,
when the number of variables is fixed.Comment: 16 pages, 4 figures; to appear in Mathematical Programmin
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
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