The problem with integer programming

Integer programming (IP), also known as discrete optimization, is a way of modelling a very wide range of problems involving indivisibilities (e.g. yes/no investment decisions) and non-convexities (e.g. economies of scale and fixed cost allocation). Such problems arise in many areas; these are mentioned in the paper. However, IP demands ingenuity in both building models and in their solution. Much is still not properly understood. This paper investigates the question: ‘Is IP like Linear Programming (LP)? ’ The mathematical and economic properties of IP will be contrasted with LP. It will be suggested that the mathematics and economics of IP are still not properly understood. Many of the results which apply to LP do not apply to IP. It will be asserted that this lack of understanding reveals inadequacies in both the mathematics and economics. Published online October 5th, 2010.