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Revealing Decision Conservativeness Through Inverse Distributionally Robust Optimization

Qi Li, Zhirui Liang, Andrey Bernstein, Yury Dvorkin

TL;DR

This work tackles the problem of inferring a decision-maker's conservativeness in Wasserstein DRO by observing the decisions produced by Forward DRO (F-DRO). It introduces Inverse DRO (I-DRO), a KKT-based bi-linear formulation that recovers the ambiguity radius $ε$ and provides existence, uniqueness, and necessary/sufficient conditions for exact recovery. The authors prove theoretical guarantees and identify three potential failure modes, then validate the approach on power-system DC-OPF problems, including an IEEE 5-bus case and NYISO 11-zone data, demonstrating accurate recovery and practical computation times. The results highlight the method's potential to forecast or replicate conservative decision-making in uncertain environments, with broader applicability beyond power systems when the forward model exhibits DRO-like structure.

Abstract

This paper introduces Inverse Distributionally Robust Optimization (I-DRO) as a method to infer the conservativeness level of a decision-maker, represented by the size of a Wasserstein metric-based ambiguity set, from the optimal decisions made using Forward Distributionally Robust Optimization (F-DRO). By leveraging the Karush-Kuhn-Tucker (KKT) conditions of the convex F-DRO model, we formulate I-DRO as a bi-linear program, which can be solved using off-the-shelf optimization solvers. Additionally, this formulation exhibits several advantageous properties. We demonstrate that I-DRO not only guarantees the existence and uniqueness of an optimal solution but also establishes the necessary and sufficient conditions for this optimal solution to accurately match the actual conservativeness level in F-DRO. Furthermore, we identify three extreme scenarios that may impact I-DRO effectiveness. Our case study applies F-DRO for power system scheduling under uncertainty and employs I-DRO to recover the conservativeness level of system operators. Numerical experiments based on an IEEE 5-bus system and a realistic NYISO 11-zone system demonstrate I-DRO performance in both normal and extreme scenarios.

Revealing Decision Conservativeness Through Inverse Distributionally Robust Optimization

TL;DR

This work tackles the problem of inferring a decision-maker's conservativeness in Wasserstein DRO by observing the decisions produced by Forward DRO (F-DRO). It introduces Inverse DRO (I-DRO), a KKT-based bi-linear formulation that recovers the ambiguity radius and provides existence, uniqueness, and necessary/sufficient conditions for exact recovery. The authors prove theoretical guarantees and identify three potential failure modes, then validate the approach on power-system DC-OPF problems, including an IEEE 5-bus case and NYISO 11-zone data, demonstrating accurate recovery and practical computation times. The results highlight the method's potential to forecast or replicate conservative decision-making in uncertain environments, with broader applicability beyond power systems when the forward model exhibits DRO-like structure.

Abstract

This paper introduces Inverse Distributionally Robust Optimization (I-DRO) as a method to infer the conservativeness level of a decision-maker, represented by the size of a Wasserstein metric-based ambiguity set, from the optimal decisions made using Forward Distributionally Robust Optimization (F-DRO). By leveraging the Karush-Kuhn-Tucker (KKT) conditions of the convex F-DRO model, we formulate I-DRO as a bi-linear program, which can be solved using off-the-shelf optimization solvers. Additionally, this formulation exhibits several advantageous properties. We demonstrate that I-DRO not only guarantees the existence and uniqueness of an optimal solution but also establishes the necessary and sufficient conditions for this optimal solution to accurately match the actual conservativeness level in F-DRO. Furthermore, we identify three extreme scenarios that may impact I-DRO effectiveness. Our case study applies F-DRO for power system scheduling under uncertainty and employs I-DRO to recover the conservativeness level of system operators. Numerical experiments based on an IEEE 5-bus system and a realistic NYISO 11-zone system demonstrate I-DRO performance in both normal and extreme scenarios.
Paper Structure (17 sections, 2 theorems, 16 equations, 1 figure, 4 tables)

This paper contains 17 sections, 2 theorems, 16 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Overview of Inverse Optimization
  3. Forward DRO Formulation
  4. Reformulation of Chance-Constrained Linear Program
  5. Convex DRO Formulation
  6. Inverse DRO Formulation
  7. Partially Constrained I-DRO Problem
  8. Properties of I-DRO Solutions
  9. Extensions of I-DRO Model
  10. Numerical Experiments
  11. Chance-constrained DC-OPF model
  12. Illustrative Example: IEEE 5-bus System
  13. Scenario 1: Overly Conservative Decision-Making
  14. Scenario 2: Chance Constraints Not Binding
  15. Scenario 3: Historical Samples Not Representative
  16. ...and 2 more sections

Key Result

Theorem 1

(Existence and Uniqueness of $\epsilon^{*}$) For each observed decision $x^0$, I-DRO in inverse DRO guarantees the existence of a unique optimal solution $\epsilon^{*}$, assuming that $x^0$ is the exact optimal solution of F-DRO in forward DRO and the value of $\bar{\epsilon}$ is set sufficiently la

Figures (1)

  • Figure 1: Forward and Inverse DRO Frameworks.

Theorems & Definitions (11)

  • Definition 1
  • Remark 1
  • Theorem 1
  • proof
  • Definition 2
  • Theorem 2
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • ...and 1 more