MIP Relaxations in Factorable Programming

TL;DR

This paper introduces new mixed-integer programming relaxations for factorable nonlinear functions, offering tighter bounds and fewer variables, which improve solution quality in polynomial optimization problems.

Contribution

The paper develops novel MIP relaxations based on convex hull formulations that are tighter and more efficient than existing relaxations for factorable programming.

Findings

Relaxations close 60-70% of the gap compared to McCormick relaxations.

Relaxations are often tighter and require fewer variables.

Significant improvements in solver performance on polynomial instances.

Abstract

In this paper, we develop new discrete relaxations for nonlinear expressions in factorable programming. We utilize specialized convexification results as well as composite relaxations to develop mixed-integer programming (MIP) relaxations. Our relaxations rely on ideal formulations of convex hulls of outer-functions over a combinatorial structure that captures local inner-function structure. The resulting relaxations often require fewer variables and are tighter than currently prevalent ones. Finally, we provide computational evidence to demonstrate that our relaxations close approximately 60-70% of the gap relative to McCormick relaxations and significantly improves the relaxations used in a state-of-the-art solver on various instances involving polynomial functions.