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Data-driven interdiction with asymmetric cost uncertainty: a distributionally robust optimization approach

Sergey S. Ketkov, Oleg A. Prokopyev

TL;DR

This work forms a distributionally robust interdiction (DRI) model, where both decision-makers solve conventional distributionally robust optimization problems based on the Wasserstein metric, and proves asymptotic consistency and derives a polynomial-size mixed-integer linear programming (MILP) reformulation.

Abstract

We consider a class of stochastic interdiction games between an upper-level decision-maker (the leader) and a lower-level decision-maker (the follower), where uncertainty lies in the follower's objective function coefficients. Specifically, the follower's profits (or costs) in our model comprise a random vector, whose probability distribution is estimated independently by the leader and the follower, based on their own data. To address the distributional uncertainty, we formulate a distributionally robust interdiction (DRI) model, where both decision-makers solve conventional distributionally robust optimization problems based on the Wasserstein metric. For this model, we prove asymptotic consistency and derive a polynomial-size mixed-integer linear programming (MILP) reformulation. Furthermore, in our bilevel optimization context, the leader may face uncertainty due to its incomplete knowledge of the follower's data. In this regard, we propose two distinct approximations of the true DRI model, where the leader has incomplete or no information about the follower's data. The first approach employs a pessimistic approximation, which turns out to be computationally challenging and requires a specialized reformulation amenable to a Benders-type decomposition algorithm. The second approach leverages a robust optimization approach from the leader's perspective. To address the resulting problem, we propose a scenario-based approximation that admits a potentially large single-level MILP reformulation and satisfies asymptotic robustness guarantees. Finally, for a class of randomly generated instances of the packing interdiction problem, we evaluate numerically how the information asymmetry and the decision-makers' risk preferences affect the models' out-of-sample performance.

Data-driven interdiction with asymmetric cost uncertainty: a distributionally robust optimization approach

TL;DR

This work forms a distributionally robust interdiction (DRI) model, where both decision-makers solve conventional distributionally robust optimization problems based on the Wasserstein metric, and proves asymptotic consistency and derives a polynomial-size mixed-integer linear programming (MILP) reformulation.

Abstract

We consider a class of stochastic interdiction games between an upper-level decision-maker (the leader) and a lower-level decision-maker (the follower), where uncertainty lies in the follower's objective function coefficients. Specifically, the follower's profits (or costs) in our model comprise a random vector, whose probability distribution is estimated independently by the leader and the follower, based on their own data. To address the distributional uncertainty, we formulate a distributionally robust interdiction (DRI) model, where both decision-makers solve conventional distributionally robust optimization problems based on the Wasserstein metric. For this model, we prove asymptotic consistency and derive a polynomial-size mixed-integer linear programming (MILP) reformulation. Furthermore, in our bilevel optimization context, the leader may face uncertainty due to its incomplete knowledge of the follower's data. In this regard, we propose two distinct approximations of the true DRI model, where the leader has incomplete or no information about the follower's data. The first approach employs a pessimistic approximation, which turns out to be computationally challenging and requires a specialized reformulation amenable to a Benders-type decomposition algorithm. The second approach leverages a robust optimization approach from the leader's perspective. To address the resulting problem, we propose a scenario-based approximation that admits a potentially large single-level MILP reformulation and satisfies asymptotic robustness guarantees. Finally, for a class of randomly generated instances of the packing interdiction problem, we evaluate numerically how the information asymmetry and the decision-makers' risk preferences affect the models' out-of-sample performance.
Paper Structure (23 sections, 12 theorems, 93 equations, 7 figures, 3 tables, 1 algorithm)

This paper contains 23 sections, 12 theorems, 93 equations, 7 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related literature
  3. Our approach and contributions
  4. Model setup and assumptions
  5. Basic DRI model
  6. Problem formulation and solution approach
  7. Convergence analysis
  8. Convergence for the follower
  9. Convergence for the leader
  10. Pessimistic approximation
  11. Problem formulation and complexity
  12. Benders decomposition-based algorithm for the ambiguity-free problem
  13. Benders decomposition-based algorithm for the risk-neutral problem
  14. Semi-pessimistic approximation
  15. Problem formulation and complexity
  16. ...and 8 more sections

Key Result

Lemma 1

Let Assumptions A1-A3 hold, and the ambiguity sets in (ambiguity set) be defined in terms of $p$-norm with $p \in \{1, \infty\}$. Then, for any fixed $\mathbf{x} \in X$, the follower's problem can be equivalently reformulated as a linear programming problem of the form: where $\boldsymbol{\Delta}_f^{\text{\tiny (k)}} = \mathbf{b} - \mathbf{B} \, \hat{\mathbf{c}}_f^{\text{\tiny (k)}} \geq \mathb

Figures (7)

  • Figure 1: The average relative out-of-sample loss (with MADs) of the follower (\ref{['eq: relative loss follower']}) (a) and the leader (\ref{['eq: relative loss leader']}) (b) as a function of $\delta_f$ and $\delta_l$, respectively, evaluated over 100 random test instances.
  • Figure 2: The average relative out-of-sample loss (with MADs) of the follower (\ref{['eq: relative loss follower']}) (a) and the leader (\ref{['eq: relative loss leader']}) (b) as a function of their respective sample sizes, evaluated over 100 random test instances.
  • Figure 3: The average relative in-sample (\ref{['eq: relative loss leader pessimistic']}) (a) and out-of-sample (\ref{['eq: relative in-sample loss']}) (b) loss (with MADs) as a function of the number of scenarios, $r_l$, for $k_l = k_f = 30$, $\delta_l = \delta_f = 0.1$ and $k_{lf} = 20$, evaluated over 100 random test instances. The dashed lines in (a) correspond to the empirical $5\%$ percentile of the relative in-sample loss.
  • Figure 4: The average relative in-sample (\ref{['eq: relative loss leader pessimistic']}) (a) and out-of-sample (\ref{['eq: relative in-sample loss']}) (b) loss of the ambiguity-free leader (with MADs) as a function of the common sample size, $k_{lf}$, for $k_l = k_f = 30$, $\delta_l = \delta_f = 0.1$ and $\alpha_l = 0.9$, evaluated over 100 random test instances. The follower is assumed to be risk-averse. The dashed line corresponds to the empirical $5\%$ percentile of the relative in-sample loss for the scenario-based semi-pessimistic approximation.
  • Figure 5: The average relative in-sample (\ref{['eq: relative loss leader pessimistic']}) (a, c) and out-of-sample (\ref{['eq: relative in-sample loss']}) (b, d) loss of the risk-neutral leader (with MADs) as a function of the common sample size, $k_{lf}$, for $k_l = k_f = 30$ and $\delta_l = \delta_f = 0.1$, evaluated over 100 random test instances. The follower is assumed to be risk-averse (a, b) and risk-neutral (c, d). The dashed line corresponds to the empirical $5\%$ percentile of the relative in-sample loss for the scenario-based semi-pessimistic approximation.
  • ...and 2 more figures

Theorems & Definitions (26)

  • Example 1
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 2
  • ...and 16 more