Sequence of Polyhedral Relaxations for Nonlinear Univariate Functions

Given a nonlinear, univariate, bounded, and differentiable function f(x),this article develops a sequence of Mixed Integer Linear Programming (MILP) andLinear Programming (LP) relaxations that converge to the graph of f(x) andits convex hull, respectively. Theoretical convergence of the sequence ofrelaxations to the graph of the function and its convex hull is established.For nonlinear non-convex optimization problems, the relaxations presented inthis article can be used to construct tight MILP and LP relaxations. These MILPand the LP relaxations can also be used with MILP-based and spatialbranch-and-bound based global optimization algorithms, respectively.