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Distributionally Robust Optimization with Decision-Dependent Information Discovery

Qing Jin, Angelos Georghiou, Phebe Vayanos, Grani A. Hanasusanto

TL;DR

This paper addresses two-stage distributionally robust optimization when information discovery is decision-dependent (DDID), i.e., observing parts of uncertainty requires costly first-stage actions. It casts the problem as a min–max–min–max formulation and adopts a K-adaptability approach to limit recourse policies, enabling tractable approximations and exact solutions via a two-layer decomposition. The outer layer uses an L-shaped scheme over measurement decisions, while the inner layer solves an evaluation problem with a branch-and-cut algorithm augmented by Benders and strengthened cuts; convergence guarantees are provided. Numerical experiments on Best Box and R&D portfolio problems show that DDID-DRO with K-adaptability yields substantial improvements over pure RO and that the proposed decomposition methods outperform monolithic MINLO on larger instances, highlighting practical value for decision-making under endogenous information and distributional uncertainty.

Abstract

We study two-stage distributionally robust optimization (DRO) problems with decision-dependent information discovery (DDID) wherein (a portion of) the uncertain parameters are revealed only if an (often costly) investment is made in the first stage. This class of problems finds many important applications in selection problems (e.g., in hiring, project portfolio optimization, or optimal sensor location). Despite the wide applicability of the problem, it has not been previously studied. We propose a framework for modeling and approximately solving DRO problems with DDID. We formulate the problem as a min-max-min-max problem and adopt the popular K-adaptability approximation scheme, which chooses K candidate recourse actions here-and-now and implements the best of those actions after the uncertain parameters that were chosen to be observed are revealed. We then present a decomposition algorithm that solves the K-adaptable formulation exactly. In particular, we devise a cutting plane algorithm that iteratively solves a relaxed version of the problem, evaluates the true objective value of the corresponding solution, generates valid cuts, and imposes them in the relaxed problem. For the evaluation problem, we develop a branch-and-cut algorithm that provably converges to an optimal solution. We showcase the effectiveness of our framework on the R&D project portfolio optimization problem and the best box problem.

Distributionally Robust Optimization with Decision-Dependent Information Discovery

TL;DR

This paper addresses two-stage distributionally robust optimization when information discovery is decision-dependent (DDID), i.e., observing parts of uncertainty requires costly first-stage actions. It casts the problem as a min–max–min–max formulation and adopts a K-adaptability approach to limit recourse policies, enabling tractable approximations and exact solutions via a two-layer decomposition. The outer layer uses an L-shaped scheme over measurement decisions, while the inner layer solves an evaluation problem with a branch-and-cut algorithm augmented by Benders and strengthened cuts; convergence guarantees are provided. Numerical experiments on Best Box and R&D portfolio problems show that DDID-DRO with K-adaptability yields substantial improvements over pure RO and that the proposed decomposition methods outperform monolithic MINLO on larger instances, highlighting practical value for decision-making under endogenous information and distributional uncertainty.

Abstract

We study two-stage distributionally robust optimization (DRO) problems with decision-dependent information discovery (DDID) wherein (a portion of) the uncertain parameters are revealed only if an (often costly) investment is made in the first stage. This class of problems finds many important applications in selection problems (e.g., in hiring, project portfolio optimization, or optimal sensor location). Despite the wide applicability of the problem, it has not been previously studied. We propose a framework for modeling and approximately solving DRO problems with DDID. We formulate the problem as a min-max-min-max problem and adopt the popular K-adaptability approximation scheme, which chooses K candidate recourse actions here-and-now and implements the best of those actions after the uncertain parameters that were chosen to be observed are revealed. We then present a decomposition algorithm that solves the K-adaptable formulation exactly. In particular, we devise a cutting plane algorithm that iteratively solves a relaxed version of the problem, evaluates the true objective value of the corresponding solution, generates valid cuts, and imposes them in the relaxed problem. For the evaluation problem, we develop a branch-and-cut algorithm that provably converges to an optimal solution. We showcase the effectiveness of our framework on the R&D project portfolio optimization problem and the best box problem.
Paper Structure (37 sections, 11 theorems, 68 equations, 4 figures, 12 tables, 3 algorithms)

This paper contains 37 sections, 11 theorems, 68 equations, 4 figures, 12 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background & Motivation
  3. Literature Review
  4. Proposed Approach & Contributions
  5. Organization of the Paper and Notations
  6. DRO with Decision-Dependent Information Discovery
  7. Decision Rule Formulation
  8. An Equivalent min-max-min-max Reformulation
  9. Decomposition Algorithm
  10. Constraint Generation Algorithm
  11. A Branch-and-Cut Algorithm.
  12. Convergence Analysis
  13. Benders' Cut
  14. Strengthened Feasibility Cut
  15. Strengthened Optimality Cut
  16. ...and 22 more sections

Key Result

Theorem 1

The objective value and set of optimal solutions $(\bm x,\bm w)$ to problem eq:endo_whole are equal to those of the min-max-min-max problem where $\Xi(\boldsymbol{w}, \overline{\boldsymbol{\xi}}):=\{\boldsymbol{\xi} \in \Xi: \boldsymbol{w} \circ \boldsymbol{\xi}=\boldsymbol{w} \circ \overline{\boldsymbol{\xi}}\}.$

Figures (4)

  • Figure 1: Overview of our proposed decomposition Algorithm \ref{['alg:l-shaped']}
  • Figure 2: Diagram of Algorithm \ref{['alg:bnc']} for solving problem \ref{['eq:Phi(w)']}
  • Figure B.1: Average improvement over the static solution in the best box problem as $K$ increases
  • Figure B.2: Average improvement over static solution in the R&D project portfolio optimization problem as $K$ increases

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • Theorem 5
  • Theorem 6
  • Lemma 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Lemma 1.2
  • ...and 1 more