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This chapter surveys a selection of results from the interplay of integer programming and the geometry of numbers. Apart from being a survey, the text is also intended as an entry point into the field. I therefore added exercises at the end of each section to invite the reader to delve deeper into the presented material.
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...
The first 50 years of Integer and Mixed-Integer Programming have taken us to a very stable paradigm for solving problems in a reliable and effective way. We run over these 50 exciting years by showing some crucial milestones and we highlight the building blocks that are making nowadays solvers effective from both a performance and an application viewpoint. Finally, we show that a lot of work must...
An integer linear program (ILP) is symmetric if its variables can be permuted without changing the structure of the problem. Areas where symmetric ILPs arise range from applied settings (scheduling on identical machines), to combinatorics (code construction), and to statistics (statistical designs construction). Relatively small symmetric ILPs are extremely difficult to solve using branch-and-cut...
We survey some recent developments in the area of semidefinite optimization applied to integer programming. After recalling some generic modeling techniques to obtain semidefinite relaxations for NP-hard problems, we look at the theoretical power of semidefinite optimization in the context of the Max-Cut and the Coloring Problem. In the second part, we consider algorithmic questions related to semidefinite...
In this chapter, we provide an overview of the mathematical foundations and recent theoretical and computational advances in the study of the grouptheoretic approach in mixed integer programming. We motivate the definition of group relaxation geometrically and present methods to optimize linear functions over this set. We then discuss fundamental results about the structure of group relaxations. We...
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