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arxiv: 2503.01563 · v3 · submitted 2025-03-03 · 🧮 math.OC

On Coupling Constraints in Pessimistic Linear Bilevel Optimization

Dorothee Henke , Henri Lefebvre , Martin Schmidt , Johannes Th\"urauf This is my paper

Pith reviewed 2026-05-23 01:44 UTC · model grok-4.3

classification 🧮 math.OC
keywords pessimistic bilevel optimizationcoupling constraintsoptimistic bilevel optimizationlinear bilevel programsproblem reformulationsolution equivalence
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The pith

Pessimistic linear bilevel problems with coupling constraints are equivalent to pessimistic problems without them.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that any pessimistic linear bilevel optimization problem with coupling constraints can be turned into an equivalent pessimistic problem without those constraints while keeping exactly the same globally optimal solutions. It also proves that the original problem is equivalent to an optimistic problem without coupling constraints. The equivalence unifies the three problem classes so that theory and algorithms developed for any one apply to the others. Readers care because it removes the long-held view that coupling constraints make pessimistic problems distinctly harder to solve.

Core claim

Given a pessimistic linear bilevel program with coupling constraints, one can construct a pessimistic linear bilevel program without coupling constraints that possesses precisely the same set of globally optimal solutions. The results also establish equivalence to an optimistic linear bilevel program without coupling constraints, enabling the transfer of theory and solution techniques across these problem classes.

What carries the argument

A reformulation that produces an equivalent pessimistic bilevel problem without coupling constraints while preserving the set of globally optimal solutions.

Load-bearing premise

The standard definitions of optimistic and pessimistic solutions remain valid after the linear reformulation that removes coupling constraints.

What would settle it

A concrete linear bilevel instance with coupling constraints whose optimal solution set differs from that of the derived uncoupled pessimistic version would disprove the claimed equivalence.

Figures

Figures reproduced from arXiv: 2503.01563 by Dorothee Henke, Henri Lefebvre, Johannes Th\"urauf, Martin Schmidt.

Figure 1
Figure 1. Figure 1: Models and Reformulations The first assumption is sufficient to guarantee that S(x) ̸= ∅ holds for all x ∈ X. The latter is also used in Dempe et al. (2014) and Aussel and Svensson (2019). The second assumption is required for applying the results from Henke et al. (2025) and the third one ensures solvability of the overall problem. Before we move on to our main results, let us briefly discuss what we cons… view at source ↗
read the original abstract

The literature on pessimistic linear bilevel optimization with coupling constraints is rather scarce and it has been common sense that these problems are harder to tackle than pessimistic bilevel problems without coupling constraints. In this note, we show that this is not the case. To this end, given a pessimistic problem with coupling constraints, we derive a pessimistic problem without coupling constraints that has the same set of globally optimal solutions. Moreover, our results also show that one can equivalently replace a pessimistic problem with such constraints with an optimistic problem without coupling constraints. This paves the way of both transferring theory and solution techniques from any type of these problems to any other one.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The paper claims that a pessimistic linear bilevel program with coupling constraints can be reformulated as an equivalent pessimistic linear bilevel program without coupling constraints that has exactly the same set of globally optimal leader solutions. It further shows that the original problem is equivalent to an optimistic linear bilevel program without coupling constraints. The result is presented as an identity on the solution set that holds under the standard definitions of optimistic and pessimistic solutions for linear programs, with the goal of allowing transfer of theory and algorithms across these problem classes.

Significance. If the derivation is valid, the result is significant because it removes the perceived additional difficulty of coupling constraints in the pessimistic linear case. The direct equivalence (rather than an approximation or relaxation) within the standard linear bilevel setting, without extra technical conditions such as compactness or attainment beyond those already implicit in the pessimistic value function, allows existing results and solvers for non-coupling or optimistic problems to be applied immediately. This is a clean structural observation that strengthens the literature on bilevel optimization.

minor comments (1)
  1. The abstract and introduction could explicitly state the precise definition of the pessimistic solution concept used (e.g., the leader's value function) to make the equivalence statement self-contained for readers unfamiliar with the authors' prior work.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation to accept. We appreciate the recognition that the equivalence result is a clean structural observation with potential to transfer theory and algorithms across problem classes.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives an equivalence result showing that a pessimistic linear bilevel program with coupling constraints has the same global optimal leader solutions as a related pessimistic (and also optimistic) program without such constraints. This is a direct mathematical identity on solution sets, obtained by algebraic reformulation of the standard optimistic/pessimistic value functions in the linear setting. No parameter fitting, self-definitional loops, or load-bearing self-citations appear in the provided abstract or description; the result rests on the problem definitions themselves rather than reducing to prior author work or renamed empirical patterns. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper is a short theoretical note deriving equivalences from standard definitions of pessimistic and optimistic bilevel problems; it introduces no free parameters, invented entities, or ad-hoc axioms beyond the linearity assumption implicit in the title.

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Works this paper leans on

20 extracted references · 20 canonical work pages

  1. [1]

    Links between linear bilevel and mixed 0–1 programming problems

    Audet, C., P. Hansen, B. Jaumard, and G. Savard (1997). “Links between linear bilevel and mixed 0–1 programming problems.” In:Journal of Optimization Theory and Applications93.2, pp. 273–300.doi:10.1023/A:1022645805569

    work page doi:10.1023/a:1022645805569 1997

  2. [2]

    Is Pessimistic Bilevel Programming a Special Case of a Mathematical Program with Complementarity Constraints?

    Aussel, D. and A. Svensson (2019). “Is Pessimistic Bilevel Programming a Special Case of a Mathematical Program with Complementarity Constraints?” In:Journal of Optimization Theory and Applications181.2, pp. 504–520.doi: 10.1007/ s10957-018-01467-7. REFERENCES 9

    work page 2019

  3. [3]

    A survey on bilevel optimization under uncertainty

    Beck, Y., I. Ljubić, and M. Schmidt (2023). “A survey on bilevel optimization under uncertainty.” In:European Journal of Operational Research311.2, pp. 401–426. doi:10.1016/j.ejor.2023.01.008

    work page doi:10.1016/j.ejor.2023.01.008 2023

  4. [4]

    Beck, Y. and M. Schmidt (2021).A Gentle and Incomplete Introduction to Bilevel Optimization. Lecture notes.url: http://www.optimization-online.org/DB_ FILE/2021/06/8450.pdf

    work page 2021

  5. [5]

    On the structure and properties of a linear multilevel programming problem

    Benson, H. P. (1989). “On the structure and properties of a linear multilevel programming problem.” In:Journal of Optimization Theory and Applications 60.3, pp. 353–373.doi:10.1007/BF00940342

    work page doi:10.1007/bf00940342 1989

  6. [6]

    Two-level linear programming

    Bialas, W. F. and M. H. Karwan (1984). “Two-level linear programming.” In: Management Science30.8, pp. 1004–1020.doi:10.1287/mnsc.30.8.1004

    work page doi:10.1287/mnsc.30.8.1004 1984

  7. [7]

    Mathematical programs with optimization problems in the constraints

    Bracken, J. and J. T. McGill (1973). “Mathematical programs with optimization problems in the constraints.” In:Operations Research21.1, pp. 37–44.doi: 10.1287/opre.21.1.37

    work page doi:10.1287/opre.21.1.37 1973

  8. [8]

    Bilevel linear optimization belongs to NP and admits polynomial-size KKT-based reformulations

    Buchheim, C. (2023). “Bilevel linear optimization belongs to NP and admits polynomial-size KKT-based reformulations.” In:Operations Research Letters 51.6, pp. 618–622.doi:10.1016/j.orl.2023.10.006

    work page doi:10.1016/j.orl.2023.10.006 2023

  9. [9]

    Candler, W. and R. Norton (1977).Multi-level programming. Discussion Papers, Development Research Center, International Bank for Reconstruction and Devel- opment. World Bank.url: http://documents.worldbank.org/curated/en/ 219041468315334935

    work page 1977

  10. [10]

    Necessary optimality condi- tions in pessimistic bilevel programming

    Dempe, S., B. Mordukhovich, and A. Zemkoho (2014). “Necessary optimality condi- tions in pessimistic bilevel programming.” In:Optimization63.4, pp. 505–533. doi:10.1080/02331934.2012.696641

    work page doi:10.1080/02331934.2012.696641 2014

  11. [11]

    (2002).Foundations of Bilevel Programming

    Dempe, S. (2002).Foundations of Bilevel Programming. Springer.doi:10.1007/ b101970

    work page 2002

  12. [12]

    Kalashnikov, G

    Dempe, S., V. Kalashnikov, G. A. Pérez-Valdés, and N. Kalashnykova (2015).Bilevel Programming Problems. Springer.doi:10.1007/978-3-662-45827-3

    work page doi:10.1007/978-3-662-45827-3 2015

  13. [13]

    Connections between RobustandBilevelOptimization

    Goerigk, M., J. Kurtz, M. Schmidt, and J. Thürauf (2025). “Connections between RobustandBilevelOptimization.” In:Open Journal of Mathematical Optimization 6, 2, pp. 1–17.doi:10.5802/ojmo.38

    work page doi:10.5802/ojmo.38 2025

  14. [14]

    On coupling constraints in linear bilevel optimization

    Henke, D., H. Lefebvre, M. Schmidt, and J. Thürauf (2025). “On coupling constraints in linear bilevel optimization.” In:Optimization Letters19.3, pp. 689–697.doi: 10.1007/s11590-024-02156-3

    work page doi:10.1007/s11590-024-02156-3 2025

  15. [15]

    Lefebvre, H. and M. Schmidt (2024).Exact augmented Lagrangian duality for nonconvex mixed-integer nonlinear optimization.url:https://optimization- online.org/?p=27046

    work page 2024

  16. [16]

    Bilevel Programming: A Combinatorial Per- spective

    Marcotte, P. and G. Savard (2005). “Bilevel Programming: A Combinatorial Per- spective.” In:Graph Theory and Combinatorial Optimization. Ed. by D. Avis, A. Hertz, and O. Marcotte. Boston, MA: Springer US, pp. 191–217.doi:10.1007/0- 387-25592-3_7

    work page doi:10.1007/0- 2005

  17. [17]

    A tutorial on properties of the epigraph reformulation

    Stein, O. (2025). “A tutorial on properties of the epigraph reformulation.” In:EURO Journal on Computational Optimization13, p. 100109.doi: 10.1016/j.ejco. 2025.100109. Tahernejad,S.,T.K.Ralphs,andS.T.DeNegre(2020).“Abranch-and-cutalgorithm for mixed integer bilevel linear optimization problems and its implementation.” In:Mathematical Programming Computat...

    work page doi:10.1016/j.ejco 2025

  18. [18]

    Discrete linear bilevel programming problem

    Vicente, L., G. Savard, and J. Júdice (1996). “Discrete linear bilevel programming problem.” In:Journal of Optimization Theory and Applications89.3, pp. 597–614. doi:10.1007/BF02275351. von Stackelberg, H. (1934).Marktform und Gleichgewicht. Springer. 10 REFERENCES von Stackelberg, H. (1952).Theory of the market economy. Oxford University Press

    work page doi:10.1007/bf02275351 1996

  19. [19]

    Pessimistic Bilevel Optimization

    Wiesemann, W., A. Tsoukalas, P.-M. Kleniati, and B. Rustem (2013). “Pessimistic Bilevel Optimization.” In:SIAM Journal on Optimization23.1, pp. 353–380.doi: 10.1137/120864015

    work page doi:10.1137/120864015 2013

  20. [20]

    A Practical Scheme to Compute the Pessimistic Bilevel Optimiza- tion Problem

    Zeng, B. (2020). “A Practical Scheme to Compute the Pessimistic Bilevel Optimiza- tion Problem.” In:INFORMS Journal on Computing32.4, pp. 1128–1142.doi: 10.1287/ijoc.2019.0927. (D. Henke)University of Passau, School of Business, Economics and Information

    work page doi:10.1287/ijoc.2019.0927 2020