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