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Risk-averse decision strategies for influence diagrams using rooted junction trees

Olli Herrala, Topias Terho, Fabricio Oliveira

TL;DR

The paper extends a rooted junction tree (RJT)–based MIP formulation for influence diagrams to risk-averse decision-making by incorporating conditional value-at-risk ($CVaR$) and chance/budget constraints. It introduces two algorithms: a single-value-node transformation to expose joint consequence distributions and RJT modifications to include clusters capturing required joint distributions, enabling CVaR and constraint formulations within the RJT framework. The authors derive MILP formulations for CVaR and demonstrate how chance, logical, and budget constraints can be imposed on RJT clusters, preserving computational efficiency relative to alternative approaches like decision programming. Computational experiments on the pig farm problem show substantial speedups for the RJT-based risk-averse formulations, especially as the horizon grows, illustrating practical benefits in solving risk-aware IDs and highlighting avenues for decomposition and scalability improvements.

Abstract

This paper presents how a mixed-integer programming (MIP) formulation for influence diagrams, based on a gradual rooted junction tree representation of the diagram, can be generalized to incorporate risk considerations such as conditional value-at-risk and chance constraints. We present two algorithms on how targeted modifications can be made to the underlying influence diagram or to the gradual rooted junction tree representation to enable our reformulations. We present computational results comparing our reformulation with another MIP formulation for influence diagrams.

Risk-averse decision strategies for influence diagrams using rooted junction trees

TL;DR

The paper extends a rooted junction tree (RJT)–based MIP formulation for influence diagrams to risk-averse decision-making by incorporating conditional value-at-risk () and chance/budget constraints. It introduces two algorithms: a single-value-node transformation to expose joint consequence distributions and RJT modifications to include clusters capturing required joint distributions, enabling CVaR and constraint formulations within the RJT framework. The authors derive MILP formulations for CVaR and demonstrate how chance, logical, and budget constraints can be imposed on RJT clusters, preserving computational efficiency relative to alternative approaches like decision programming. Computational experiments on the pig farm problem show substantial speedups for the RJT-based risk-averse formulations, especially as the horizon grows, illustrating practical benefits in solving risk-aware IDs and highlighting avenues for decomposition and scalability improvements.

Abstract

This paper presents how a mixed-integer programming (MIP) formulation for influence diagrams, based on a gradual rooted junction tree representation of the diagram, can be generalized to incorporate risk considerations such as conditional value-at-risk and chance constraints. We present two algorithms on how targeted modifications can be made to the underlying influence diagram or to the gradual rooted junction tree representation to enable our reformulations. We present computational results comparing our reformulation with another MIP formulation for influence diagrams.
Paper Structure (15 sections, 2 theorems, 10 equations, 7 figures, 1 table, 2 algorithms)

This paper contains 15 sections, 2 theorems, 10 equations, 7 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Pig farm problem
  4. Influence Diagrams
  5. Rooted Junction Trees
  6. Our contributions
  7. Extracting the utility distribution
  8. Modifying the RJT
  9. Imposing chance, logical, and budget constraints
  10. Conditional Value-at-Risk
  11. Computational experiments
  12. Risk-averse pig farm problem
  13. Chance-constrained pig farm problem
  14. Conclusions
  15. Supplementary material

Key Result

Proposition 3.2

Let $G = (N,A)$ be an ID with $|N^{V}| > 1$. An ID $\overline{G} = (\overline{N},\overline{A})$ constructed using Algorithm alg:1 is equivalent (cf. Definition equivalent_diagrams) .

Figures (7)

  • Figure 1: ID of the pig farm problem lauritzen2001representing.
  • Figure 2: Gradual RJT of the pig farm problem.
  • Figure 3: The pig farm problem reformulated ID.
  • Figure 4: Gradual RJT of the reformulated pig farm problem.
  • Figure 5: Gradual RJT for pig farm problem with a cluster that contains nodes $H_1,...,H_4$.
  • ...and 2 more figures

Theorems & Definitions (6)

  • Definition 2.1
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof