Modeling disjunctive constraints with a logarithmic number of binary variables and constraints

Many combinatorial constraints over continuous variables such as SOS1 and SOS2 constraints can be interpreted as disjunctive constraints that restrict the variables to lie in the union of a finite number of specially structured polyhedra. Known mixed integer binary formulations for these constraints have a number of binary variables and extra constraints linear in the number of polyhedra. We give sufficient conditions for constructing formulations for these constraints with a number of binary variables and extra constraints logarithmic in the number of polyhedra. Using these conditions we introduce mixed integer binary formulations for SOS1 and SOS2 constraints that have a number of binary variables and extra constraints logarithmic in the number of continuous variables. We also introduce the first mixed integer binary formulations for piecewise linear functions of one and two variables that use a number of binary variables and extra constraints logarithmic in the number of linear pieces of the functions. We prove that the new formulations for piecewise linear functions have favorable tightness properties and present computational results showing that they can significantly outperform other mixed integer binary formulations.