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On the relationship between MESP and 0/1 D-Opt and their upper bounds

Gabriel Ponte, Marcia Fampa, Jon Lee

TL;DR

Strong connections are established between two fundamental nonlinear 0/1 optimization problems coming from the area of experimental design, namely maximum entropy sampling and 0/1 D-Optimality, and basic upper-bounding methods are transported between these two problems.

Abstract

We establish strong connections between two fundamental nonlinear 0/1 optimization problems coming from the area of experimental design, namely maximum entropy sampling and 0/1 D-Optimality. The connections are based on maps between instances, and we analyze the behavior of these maps. Using these maps, we transport basic upper-bounding methods between these two problems, and we are able to establish new domination results and other inequalities relating various basic upper bounds. Further, we establish results relating how different branch-and-bound schemes based on these maps compare. Additionally, we observe some surprising numerical results, where bounding methods that did not seem promising in their direct application to real-data MESP instances, are now useful for MESP instances that come from 0/1 D-Optimality.

On the relationship between MESP and 0/1 D-Opt and their upper bounds

TL;DR

Strong connections are established between two fundamental nonlinear 0/1 optimization problems coming from the area of experimental design, namely maximum entropy sampling and 0/1 D-Optimality, and basic upper-bounding methods are transported between these two problems.

Abstract

We establish strong connections between two fundamental nonlinear 0/1 optimization problems coming from the area of experimental design, namely maximum entropy sampling and 0/1 D-Optimality. The connections are based on maps between instances, and we analyze the behavior of these maps. Using these maps, we transport basic upper-bounding methods between these two problems, and we are able to establish new domination results and other inequalities relating various basic upper bounds. Further, we establish results relating how different branch-and-bound schemes based on these maps compare. Additionally, we observe some surprising numerical results, where bounding methods that did not seem promising in their direct application to real-data MESP instances, are now useful for MESP instances that come from 0/1 D-Optimality.

Paper Structure

This paper contains 26 sections, 46 theorems, 91 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction.
  2. On the equivalence of MESP and 0/1 D-Opt.
  3. Transferring and comparing upper bounds for MESP and 0/1 D-Opt.
  4. Basic MESP upper bounds.
  5. Basic 0/1 D-Opt upper bounds.
  6. Transferring upper bounds
  7. Spectral bounds.
  8. Diagonal and Hadamard bounds.
  9. NLP-Id and Natural bounds.
  10. NLP-Id and NLP-Tr.
  11. Comparison between upper bounds.
  12. Previous results on upper bound domination.
  13. Comparisons with the NLP-Id bound.
  14. Comparisons with the spectral bound for MESP.
  15. Comparisons with the diagonal bound.
  16. ...and 11 more sections

Key Result

Proposition 2.2

$I_n-A(A^ { \raisebox{\depth}{$\m@th\mathsf{T}$}} A + B^ { \raisebox{\depth}{$\m@th\mathsf{T}$}} B)^{-1}A^ { \raisebox{\depth}{$\m@th\mathsf{T}$}}$ is positive semidefinite and has maximum eigenvalue less than or equal to one.

Figures (8)

  • Figure 2.1: From 0/1 D-Opt
  • Figure 2.2: From MESP
  • Figure 5.1: $\mathcal{M}$-induced bounds for pure 0/1 $\hbox{\rm D-Opt}$
  • Figure 5.2: $\hbox{\rm MESP}$ bounds for some particular $\hbox{\rm MESP}$ instances
  • Figure 5.3: Computing the equivalent NLP-Id bound for $\hbox{\rm MESP}$ and $\mathcal{D}$-induced natural bound ($n=2000$)
  • ...and 3 more figures

Theorems & Definitions (96)

  • Definition 2.1
  • Proposition 2.2
  • Proof 1
  • Lemma 2.3
  • Proof 2
  • Lemma 2.4
  • Proof 3
  • Theorem 2.5: li2022d
  • Proof 4
  • Theorem 2.6: li2022d
  • ...and 86 more