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

Ordered Incidence geometry and the geometric foundations of convexity theory

An Ordered Incidence Geometry, that is a geometry with certain axioms of incidence and order, is proposed as a minimal setting for the fundamental convexity theorems, which usually appear in the context of a linear vector space, but require only incidence, order (and for separation, completeness), and none of the linear structure of a vector space.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42995/1/22_2005_Article_BF01227810.pd

Neuraminidase-Mediated, NKp46-Dependent Immune-Evasion Mechanism of Influenza Viruses

Natural killer (NK) cells play an essential role in the defense against influenza virus, one of the deadliest respiratory viruses known today. The NKp46 receptor, expressed by NK cells, is critical for controlling influenza infections, as influenza-virus-infected cells are eliminated through the recognition of the viral hemagglutinin (HA) protein by NKp46. Here, we describe an immune-evasion mechanism of influenza viruses that is mediated by the neuraminidase (NA) protein. By using various NA blockers, we show that NA removes sialic acid residues from NKp46 and that this leads to reduced recognition of HA. Furthermore, we provide in vivo and in vitro evidence for the existence of this NA-mediated, NKp46-dependent immune-evasion mechanism and demonstrate that NA inhibitors, which are commonly used for the treatment of influenza infections, are useful not only as blockers of virus budding but also as boosters of NKp46 recognition