# Convexification of bilinear terms over network polytopes

## TL;DR

This paper develops methods to precisely convexify bilinear terms over network polytopes, improving bounds in network optimization problems by explicitly characterizing the convex hull using network structures.

## Contribution

It introduces systematic procedures to obtain the convex hull of bilinear sets over network polytopes, explicitly deriving facet inequalities using tree and forest structures.

## Key findings

- The proposed convexification methods improve dual bounds in network optimization.
- Explicit facet inequalities are derived for bilinear sets over network polytopes.
- Computational experiments demonstrate the effectiveness of the new convexification techniques.

## Abstract

It is well-known that the McCormick relaxation for the bilinear constraint $z=xy$ gives the convex hull over the box domains for $x$ and $y$. In network applications where the domain of bilinear variables is described by a network polytope, the McCormick relaxation, also referred to as linearization, fails to provide the convex hull and often leads to poor dual bounds. We study the convex hull of the set containing bilinear constraints $z_{i,j}=x_iy_j$ where $x_i$ represents the arc-flow variable in a network polytope, and $y_j$ is in a simplex. For the case where the simplex contains a single $y$ variable, we introduce a systematic procedure to obtain the convex hull of the above set in the original space of variables, and show that all facet-defining inequalities of the convex hull can be obtained explicitly through identifying a special tree structure in the underlying network. For the generalization where the simplex contains multiple $y$ variables, we design a constructive procedure to obtain an important class of facet-defining inequalities for the convex hull of the underlying bilinear set that is characterized by a special forest structure in the underlying network. Computational experiments are presented to evaluate the effectiveness of the proposed methods.