# A decomposition strategy for decision problems with endogenous uncertainty using mixed-integer programming

**Authors:** Olli Herrala, Tommi Ekholm, Fabricio Oliveira  
**arXiv:** [2304.02338](https://arxiv.org/abs/2304.02338)  
**Date:** 2023-04-05  
**Summary page:** https://www.summarizepaper.com/en/paper/2304.02338v1/

## Key points

- The paper proposes a framework called Decision Programming for solving decision problems with endogenous uncertainty using mixed-integer programming.
- The framework uses influence diagrams to represent decision problems and formulates them as mixed-integer linear programming problems that can be solved using off-the-shelf mathematical optimization solvers.
- The authors incorporate conditional non-anticipativity constraints (C-NACs) into Decision Programming models to address conditionally observed information, which is a type of endogenous uncertainty that received less attention in the introduction of Decision Programming.
- The extended framework allows for simultaneous consideration of the two main types of endogenous uncertainty: decision-dependent information structure and decision-dependent probability distribution.
- A decomposition approach is presented that enables considering continuous decision variables in certain parts of the problem where the original formulation was restricted to discrete variables only.
- Two example problems are illustrated: an illustrative multiperiod game and a large-scale cost-benefit problem regarding climate change mitigation. Neither of these example problems could be solved with existing frameworks.
- The contributions of this paper include two different modeling approaches for considering conditionally observed information and a subdiagram decomposition approach to improve computational efficiency.
- According to Hellemo et al., Type 3 endogenously uncertain stochastic programming (T3ESP) problems have not been discussed in literature before, making this framework particularly valuable for capacity expansion contexts.
- Computational experiments show similar solution times for both approaches but suggest that C-NAC constraints may be more suitable for problems with decision-dependent probability distributions.
- Overall, the proposed framework provides a valuable tool for solving decision problems with endogenous uncertainty and has the potential to address challenging modeling and computational issues in various contexts.

## Summary

This paper proposes a framework for solving decision problems with endogenous uncertainty using mixed-integer programming. Despite recent methodological advances, modeling endogenous uncertainty remains challenging due to limitations in modeling capabilities and computational requirements. The proposed framework, called Decision Programming, uses influence diagrams to represent decision problems and formulates them as mixed-integer linear programming problems that can be solved using off-the-shelf mathematical optimization solvers. This paper focuses on the type of endogenous uncertainty that received less attention in the introduction of Decision Programming: conditionally observed information. To address this issue, the authors incorporate conditional non-anticipativity constraints (C-NACs) into Decision Programming models, which are used in multi-stage stochastic programming (MSSP) models to represent such uncertainties. This allows for simultaneous consideration of the two main types of endogenous uncertainty: decision-dependent information structure and decision-dependent probability distribution. Additionally, the authors present a decomposition approach that enables considering continuous decision variables in certain parts of the problem where the original formulation was restricted to discrete variables only. The extended framework is illustrated with two example problems: an illustrative multiperiod game and a large-scale cost-benefit problem regarding climate change mitigation. Neither of these example problems could be solved with existing frameworks. The contributions of this paper include two different modeling approaches for considering conditionally observed information and a subdiagram decomposition approach to improve computational efficiency. According to Hellemo et al., Type 3 endogenously uncertain stochastic programming (T3ESP) problems have not been discussed in literature before, making this framework particularly valuable for capacity expansion contexts. In practice, both approaches have their advantages and disadvantages. Adding observation nodes only requires modifying the influence diagram by adding new nodes but may result in larger problem sizes, while C-NAC constraints require additional computations but may lead to smaller problem sizes. Computational experiments show similar solution times for both approaches but suggest that C-NAC constraints may be more suitable for problems with decision-dependent probability distributions. Overall, the proposed framework provides a valuable tool for solving decision problems with endogenous uncertainty and has the potential to address challenging modeling and computational issues in various contexts.