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

Improved Measurement of the dˉ/uˉ\bar d / \bar u Asymmetry in the Nucleon Sea

Measurements of the ratio of Drell-Yan yields from an 800 \rm{GeV/c} proton beam incident on liquid hydrogen and deuterium targets are reported. Approximately 360,000 Drell-Yan muon pairs remained after all cuts on the data. From these data, the ratio of anti-down ($\bar{d}$) to anti-up ($\bar{u}$) quark distributions in the proton sea is determined over a wide range in Bjorken-$x$. These results confirm previous measurements by E866 and extend them to lower $x$. From these data, $(\bar{d}-\bar{u})$ and $\int(\bar{d}-\bar{u})dx$ are evaluated for $0.015<x<0.35$. These results are compared with parameterizations of various parton distribution functions, models and experimental results from NA51, NMC, and HERMES.Comment: 17 pages, 15 figure