On the Convergence of Fenchel Cutting Planes in Mixed-Integer Programming

Fenchel cutting planes are based on the dual relationship between separation and optimization and can be applied in many instances where alternative cutting planes cannot. They are deep in the sense of providing the maximum separation between a point $\hat x$ and a polyhedron P as measured by an arbitrary norm which is specified in the process of generating a Fenchel cut. This paper demonstrates a number of fundamental convergence properties of Fenchel cuts and addresses the question of which norms lead to the most desirable Fenchel cuts. The strengths and weaknesses of the related class of 1-polar cuts are also examined.