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

Santanu S Dey, Dahye Han, Yang Wang

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.

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

Abstract

In this paper, we study the strength of convex relaxations obtained by convexification of aggregation of constraints for a set 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 can be described by the intersection of convex hulls of a finite number of aggregations, examples when can only be obtained as the intersection of the convex hull of an infinite number of aggregations, and examples when 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 . 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.

Paper Structure

This paper contains 28 sections, 3 theorems, 51 equations, 29 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Aggregations
  3. Contributions of this paper
  4. Notation and organization of the paper
  5. Theoretical evaluation of aggregation
  6. Computational evaluation of aggregation
  7. Finding aggregation weights
  8. Convex relaxation
  9. Evaluation metric
  10. Environment and software
  11. Randomized experiment
  12. Instance generation
  13. Evaluation of results on random instances
  14. Finite Element Model updating problem
  15. Selection of rows for aggregation
  16. ...and 13 more sections

Key Result

Theorem 1

Consider the set $S$ described in (eq:bb_set) with $n_1 = n_2 = 1$. Then there exists $T \subseteq \mathbb{R}^2$ where $|T| \leq 3$ such that:

Figures (29)

  • Figure 1: Convex hull of each constraint
  • Figure 2: Zoom-in of Figure \ref{['figure:n1=n2=1_a']}
  • Figure 3: $\textup{conv}(S)$ achieved
  • Figure 5: Average relative improvement (%) against the gap achieved from the McCormick relaxation for different choices of $(n_1,n_2)$.
  • Figure 6: Average relative improvement (%) against the gap achieved from the one-row relaxation for different choices of $(n_1,n_2)$ for instances excluding $\rho_{1row} < 0.01\%$.
  • ...and 24 more figures

Theorems & Definitions (9)

  • Theorem 1
  • Example 1
  • Theorem 2
  • Theorem 3
  • Claim 1
  • Claim 2
  • proof
  • Claim 3
  • proof