We develop explicit, piecewise-linear formulations of functions f (x) :
Rn → R, n ≤ 3, that are defined on an orthogonal grid of vertex points. If mixedinteger
linear optimization problems (MILPs) involving multidimensional piecewiselinear
functions can be easily and efficiently solved to global optimality, then nonanalytic
functions can be used as an objective or constraint function for large optimization
problems. Linear interpolation between fixed gridpoints can also be used to
approximate generic, nonlinear functions, allowing us to approximately solve problems
using mixed-integer linear optimization methods. Toward this end, we develop
two different explicit formulations of piecewise-linear functions and discuss the consequences
of integrating the formulations into an optimization problem.