Extracting structure from functional expressions for continuous and discrete relaxations of MINLP

TL;DR

This paper introduces novel continuous and discrete relaxations for nonlinear expressions in MINLP, leveraging inner-function structure to improve tightness and efficiency over existing methods.

Contribution

It develops new relaxations that utilize inner-function structure and discretization, providing tighter bounds and reducing variable count for supermodular functions.

Findings

Tighter relaxations for nonlinear expressions in MINLP.

Reduced number of variables for supermodular functions.

Enhanced discretization-based relaxations outperform existing methods.

Abstract

In this paper, we develop new continuous and discrete relaxations for nonlinear expressions in an MINLP. In contrast to factorable programming, our techniques utilize the inner-function structure by encapsulating it in a polyhedral set, using a technique first proposed in [12]. We tighten the relaxations derived in [33,13] and obtain new relaxations for functions that could not be treated using prior techniques. We develop new discretization-based mixed-integer programming relaxations that yield tighter relaxations than similar relaxations in the literature. These relaxations utilize the simplotope that captures inner-function structure to generalize the incremental formulation of [8] to multivariate functions. In particular, when the outer-function is supermodular, our formulations require exponentially fewer continuous variables than any previously known formulation.