959 research outputs found
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
Minimax and Adaptive Inference in Nonparametric Function Estimation
Since Stein's 1956 seminal paper, shrinkage has played a fundamental role in
both parametric and nonparametric inference. This article discusses minimaxity
and adaptive minimaxity in nonparametric function estimation. Three
interrelated problems, function estimation under global integrated squared
error, estimation under pointwise squared error, and nonparametric confidence
intervals, are considered. Shrinkage is pivotal in the development of both the
minimax theory and the adaptation theory. While the three problems are closely
connected and the minimax theories bear some similarities, the adaptation
theories are strikingly different. For example, in a sharp contrast to adaptive
point estimation, in many common settings there do not exist nonparametric
confidence intervals that adapt to the unknown smoothness of the underlying
function. A concise account of these theories is given. The connections as well
as differences among these problems are discussed and illustrated through
examples.Comment: Published in at http://dx.doi.org/10.1214/11-STS355 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Global optimisation in process design
This thesis concerns the development of rigorous global optimisation techniques and
their application to process engineering problems. Many Process Engineering optimisation
problems are nonlinear. Local optimisation approaches may not provide
global solutions to these problems if they are nonconvex.
The global optimisation approach utilised in this work is based on interval branch
and bound algorithms. The interval global optimisation approach is extended to take
advantage of information about the structure of the problem and facilitate efficient
solution of constrained NLPs using interval analysis. This is achieved by reformulating
the interval lower bounding procedure as a convex programming problem which
allows inclusion of convex constraints in the lower bounding problem. The approach
is applied to a number of standard constrained test problems indicating that this algorithm
retains the wide applicability of the interval methods while allowing efficient
solution of constrained problems.
A new approach to the construction of modular flowsheets is developed. This approach
allows construction of flowsheets from linked unit models which enable the
application of a number of global optimisation algorithms. The modular flowsheets
are constructed with 'generic' unit operations which provide interval bounds, linear
bounds, derivatives and derivative bounds using extended numerical types. The
genericity means that new 'extended types' can be devised and used without rewriting
the unit operations models.
The new interval global optimisation algorithm is applied to the generic modular
flowsheet. Using interval analysis and automatic differentiation as the arithmetic
types, lower bounding linear programs are constructed and used in a branch and
bound framework to globally optimise the modular flowsheet
Exploring first and second-order spatio-temporal structures of lightning strike impacts in the French Alps using subsampling
We model cloud-to-ground lightning strike impacts in the French Alps over the
period 2011-2021 (approximately 1.4 million of events) using spatio-temporal
point processes. We investigate first and higher-order structure for this point
pattern and address the questions of homogeneity of the intensity function,
first-order separability and dependence between events. The tuning of
nonparametric methods and the different tests we consider in this study make
the computational cost very expensive. We therefore suggest different
subsampling strategies to achieve these tasks
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