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The Robust Bilevel Selection Problem

Dorothee Henke

TL;DR

The paper investigates robust bilevel optimization through the Bilevel Selection Problem, showing polynomial-time solvability in the non-robust case and rich complexity behavior under uncertainty. It develops polynomial-time algorithms for BSP and then analyzes robustness under discrete, interval, and discrete uncorrelated uncertainty, distinguishing disjoint and general item-set cases. Key contributions include NP-hardness in the general robust BSP, a 2-approximation for the overlapping-set case, and exponential-time exact approaches, plus polynomial-time algorithms for robust continuous variants leveraging piecewise-linear leader objectives. The results deepen understanding of how uncertainty interacts with bilevel structure, yielding both practical algorithms and theoretical insights with implications for robust decision-making in hierarchical optimization.

Abstract

In bilevel optimization problems, a leader and a follower make their decisions in a hierarchy, and both decisions may influence each other. Usually one assumes that both players have full knowledge also of the other player's data. In a more realistic model, uncertainty can be quantified, e.g., using the robust optimization approach: We assume that the leader does not know the follower's objective precisely, but only up to some uncertainty set, and her aim is to optimize the worst case of the corresponding scenarios. Now the question arises how the complexity of bilevel optimization changes under the additional complications of this uncertainty. We make a further step towards answering this question by examining an easy bilevel problem. In the Bilevel Selection Problem (BSP), the leader and the follower each select some items from their own item set, while a common number of items to select in total is given, and each player minimizes the total costs of the selected items, according to different sets of item costs. We show that the BSP can be solved in polynomial time and then investigate its robust version. If the two players' item sets are disjoint, it can still be solved in polynomial time for several types of uncertainty sets. Otherwise, we show that the Robust BSP is NP-hard and present a 2-approximation algorithm and exact exponential-time approaches. Furthermore, we investigate variants of the BSP where one or both of the two players take a continuous decision. One variant leads to an example of a bilevel optimization problem whose optimal value may not be attained. For the Robust Continuous BSP, where all variables are continuous, we also develop a new approach for the setting of discrete uncorrelated uncertainty, which gives a polynomial-time algorithm for the Robust Continuous BSP and a pseudopolynomial-time algorithm for the Robust Bilevel Continuous Knapsack Problem.

The Robust Bilevel Selection Problem

TL;DR

The paper investigates robust bilevel optimization through the Bilevel Selection Problem, showing polynomial-time solvability in the non-robust case and rich complexity behavior under uncertainty. It develops polynomial-time algorithms for BSP and then analyzes robustness under discrete, interval, and discrete uncorrelated uncertainty, distinguishing disjoint and general item-set cases. Key contributions include NP-hardness in the general robust BSP, a 2-approximation for the overlapping-set case, and exponential-time exact approaches, plus polynomial-time algorithms for robust continuous variants leveraging piecewise-linear leader objectives. The results deepen understanding of how uncertainty interacts with bilevel structure, yielding both practical algorithms and theoretical insights with implications for robust decision-making in hierarchical optimization.

Abstract

In bilevel optimization problems, a leader and a follower make their decisions in a hierarchy, and both decisions may influence each other. Usually one assumes that both players have full knowledge also of the other player's data. In a more realistic model, uncertainty can be quantified, e.g., using the robust optimization approach: We assume that the leader does not know the follower's objective precisely, but only up to some uncertainty set, and her aim is to optimize the worst case of the corresponding scenarios. Now the question arises how the complexity of bilevel optimization changes under the additional complications of this uncertainty. We make a further step towards answering this question by examining an easy bilevel problem. In the Bilevel Selection Problem (BSP), the leader and the follower each select some items from their own item set, while a common number of items to select in total is given, and each player minimizes the total costs of the selected items, according to different sets of item costs. We show that the BSP can be solved in polynomial time and then investigate its robust version. If the two players' item sets are disjoint, it can still be solved in polynomial time for several types of uncertainty sets. Otherwise, we show that the Robust BSP is NP-hard and present a 2-approximation algorithm and exact exponential-time approaches. Furthermore, we investigate variants of the BSP where one or both of the two players take a continuous decision. One variant leads to an example of a bilevel optimization problem whose optimal value may not be attained. For the Robust Continuous BSP, where all variables are continuous, we also develop a new approach for the setting of discrete uncorrelated uncertainty, which gives a polynomial-time algorithm for the Robust Continuous BSP and a pseudopolynomial-time algorithm for the Robust Bilevel Continuous Knapsack Problem.
Paper Structure (29 sections, 26 theorems, 7 equations, 3 figures, 8 algorithms)

This paper contains 29 sections, 26 theorems, 7 equations, 3 figures, 8 algorithms.

Table of Contents

  1. Introduction
  2. The Bilevel Selection Problem Without Uncertainty
  3. Problem Definition
  4. Problem Variants and Relation to Knapsack Problems
  5. Item Sets
  6. Continuous Variables
  7. Maximization
  8. Capacity Constraint
  9. Item Costs
  10. A Polynomial-Time Algorithm
  11. Definition of the Robust Bilevel Selection Problem
  12. The Adversary's Problem
  13. Discrete Uncertainty
  14. Interval Uncertainty
  15. Discrete Uncorrelated Uncertainty
  16. ...and 14 more sections

Key Result

Theorem 2

Algorithm alg_BSP solves the Bilevel Selection Problem in case of ${\mathcal{E}_l} \cap {\mathcal{E}_f} = \emptyset$ in time $O(n \log n)$.

Figures (3)

  • Figure 1: Example of leader's and follower's objective functions of the Continuous Bilevel Selection Problem, depending on the splitting of the total capacity $b$ into leader's and follower's capacities ${b_l}$ and ${b_f}$, respectively. Let ${\mathcal{E}_l} = \{e_1, e_2, e_3, e_4\}$, ${\mathcal{E}_f} = \{e_5, e_6, e_7, e_8\}$, $b = 5$, $c(e_1) = -1$, $c(e_2) = -1$, $c(e_3) = 0$, $c(e_4) = 3$, $c(e_5) = 1$, $c(e_6) = -3$, $c(e_7) = 2$, $c(e_8) = -1$, $d(e_5) = -2$, $d(e_6) = 0$, $d(e_7) = 1$, and $d(e_8) = 1$. We adopt the pessimistic setting, which is reflected in the order of the items $e_7$ and $e_8$. The terms $c(x)$, $c(y)$, and $d(y)$ are used as shortcuts for the objective function values $\sum_{e \in {\mathcal{E}_l}} c(e) x_e$, $\sum_{e \in {\mathcal{E}_f}} c(e) y_e$, and $\sum_{e \in {\mathcal{E}_f}} d(e) y_e$, where $x \in [0, 1]^{\mathcal{E}_l}$ and $y \in [0, 1]^{\mathcal{E}_f}$ are optimal leader's and follower's solutions for given values of ${b_l}$ and ${b_f}$, respectively. The dashed linear pieces are parts of the functions that correspond to infeasible overall solutions because both leader and follower have to select at least one item in order to reach the total desired capacity of $b = 5$. More formally, we have ${b_l^-} = {b_f^-} = 1$ and ${b_l^+} = {b_f^+} = 4$ here.
  • Figure 2: Example of the leader's objective function of the Robust Continuous Bilevel Selection Problem with discrete uncertainty, depending on the leader's capacity ${b_l}$. Let ${\mathcal{E}_l} = \{e_1, e_2, e_3, e_4\}$, ${\mathcal{E}_f} = \{e_5, e_6, e_7, e_8\}$, $b = 5$, $c(e_1) = -1$, $c(e_2) = -1$, $c(e_3) = 0$, $c(e_4) = 3$, $c(e_5) = 1$, $c(e_6) = -3$, $c(e_7) = 2$, $c(e_8) = -1$, and $\mathcal{U} = \{d_1, d_2\}$ with $d_1(e_5) = -2$, $d_1(e_6) = 0$, $d_1(e_7) = 1$, $d_1(e_8) = 1$, $d_2(e_5) = -1$, $d_2(e_6) = 3$, $d_2(e_7) = 0$, and $d_2(e_8) = -3$. This is the same instance as in Figure \ref{['fig_CBSP_leader_obj']} without uncertainty, but now with an additional second scenario $d_2$. We again adopt the pessimistic setting, which is reflected in the order of the items $e_7$ and $e_8$ in scenario $d_1$.
  • Figure 3: The leader's objective function of the Robust Bilevel Selection Problem instance described in Example \ref{['ex_RCBSP_not_equiv']}

Theorems & Definitions (49)

  • Remark 1
  • Theorem 2
  • proof
  • Remark 3
  • Theorem 4
  • proof
  • Corollary 5
  • Theorem 6
  • proof
  • Lemma 7
  • ...and 39 more