Sequence of Polyhedral Relaxations for Nonlinear Univariate Functions

TL;DR

This paper introduces a sequence of MILP and LP relaxations for nonlinear univariate functions that converge to the function's graph and convex hull, aiding in global optimization.

Contribution

It develops a theoretically convergent sequence of relaxations for univariate functions, enhancing the tightness of bounds in nonlinear optimization.

Findings

Convergence of relaxations to the function's graph and convex hull is proven.

Relaxations can be integrated into global optimization algorithms.

Provides a method for constructing tight relaxations for non-convex problems.

Abstract

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