Aggregation of Bilinear Bipartite Equality Constraints and its Application to Structural Model Updating Problem

TL;DR

This paper investigates how aggregating bilinear bipartite equality constraints can produce tighter convex relaxations, with theoretical insights and practical applications to structural model updating, improving solution bounds.

Contribution

It provides theoretical conditions for convex hull descriptions via aggregation and demonstrates improved relaxations in structural model updating problems.

Findings

Aggregation can produce tighter convex relaxations.

Including aggregated convex hulls improves bounds in structural model updating.

Exact convex hulls may not always be achievable through aggregation.

Abstract

In this paper, we study the strength of convex relaxations obtained by convexification of aggregation of constraints for a set $S$ described by two bilinear bipartite equalities. Aggregation is the process of rescaling the original constraints by scalar weights and adding the scaled constraints together. It is natural to study the aggregation technique as it yields a single bilinear bipartite equality whose convex hull is already understood from previous literature. On the theoretical side, we present sufficient conditions when $\text{conv}(S)$ can be described by the intersection of convex hulls of a finite number of aggregations, examples when $\text{conv}(S)$ can only be obtained as the intersection of the convex hull of an infinite number of aggregations, and examples when $\text{conv}(S)$ cannot be achieved exactly from the process of aggregation. Computationally, we explore different methods to derive aggregation weights in order to obtain tight convex relaxations. We show that even if an exact convex hull may not be achieved using aggregations, including the convex hull of an aggregation often significantly tightens the outer approximation of $\text{conv}(S)$. Finally, we apply the aggregation method to obtain convex relaxation for the structural model updating problem and show that this yields better bounds within a branch-and-bound tree as compared to not using aggregations.