A geometric way to build strong mixed-integer programming formulations.

Joey Huchette, Juan Pablo Vielma

Abstract: We give an explicit geometric way to build mixed-integer programming (MIP) formulations for unions of polyhedra. The construction is simply described in terms of spanning hyperplanes in an r -dimensional linear space. The resulting MIP formulation is ideal, and uses exactly r integer variables and 2 × ( # of spanning hyperplanes ) general inequality constraints. We use this result to derive novel logarithmic-sized ideal MIP formulations for discontinuous piecewise linear functions and structures appearing in robotics and power systems problems.