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Branch-and-Sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part II: Convergence analysis and numerical results

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  • Polyxeni-M. Kleniati
  • Claire Adjiman

Abstract

In the first part of this work, we presented a global optimization algorithm, Branch-and-Sandwich, for optimistic bilevel programming problems that satisfy a regularity condition in the inner problem (Kleniati and Adjiman in J Glob Optim, 2014 ). The proposed approach can be interpreted as the exploration of two solution spaces (corresponding to the inner and the outer problems) using a single branch-and-bound tree, where two pairs of lower and upper bounds are computed: one for the outer optimal objective value and the other for the inner value function. In the present paper, the theoretical properties of the proposed algorithm are investigated and finite $$\varepsilon $$ ε -convergence to a global solution of the bilevel problem is proved. Thirty-four problems from the literature are tackled successfully. Copyright The Author(s) 2014

Suggested Citation

  • Polyxeni-M. Kleniati & Claire Adjiman, 2014. "Branch-and-Sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part II: Convergence analysis and numerical results," Journal of Global Optimization, Springer, vol. 60(3), pages 459-481, November.
    • Handle: RePEc:spr:jglopt:v:60:y:2014:i:3:p:459-481
    DOI: 10.1007/s10898-013-0120-8
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    References listed on IDEAS

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    1. Stephan Dempe & Alain B. Zemkoho, 2011. "The Generalized Mangasarian-Fromowitz Constraint Qualification and Optimality Conditions for Bilevel Programs," Journal of Optimization Theory and Applications, Springer, vol. 148(1), pages 46-68, January.
    2. Polyxeni-Margarita Kleniati & Claire Adjiman, 2014. "Branch-and-Sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part I: Theoretical development," Journal of Global Optimization, Springer, vol. 60(3), pages 425-458, November.
    3. ,, 2000. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 16(2), pages 287-299, April.
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    1. Jörg Fliege & Andrey Tin & Alain Zemkoho, 2021. "Gauss–Newton-type methods for bilevel optimization," Computational Optimization and Applications, Springer, vol. 78(3), pages 793-824, April.
    2. Yasmine Beck & Daniel Bienstock & Martin Schmidt & Johannes Thürauf, 2023. "On a Computationally Ill-Behaved Bilevel Problem with a Continuous and Nonconvex Lower Level," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 428-447, July.
    3. R. Paulavičius & C. S. Adjiman, 2020. "New bounding schemes and algorithmic options for the Branch-and-Sandwich algorithm," Journal of Global Optimization, Springer, vol. 77(2), pages 197-225, June.
    4. Hatim Djelassi & Moll Glass & Alexander Mitsos, 2019. "Discretization-based algorithms for generalized semi-infinite and bilevel programs with coupling equality constraints," Journal of Global Optimization, Springer, vol. 75(2), pages 341-392, October.
    5. Boukouvala, Fani & Misener, Ruth & Floudas, Christodoulos A., 2016. "Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO," European Journal of Operational Research, Elsevier, vol. 252(3), pages 701-727.
    6. Dajun Yue & Jiyao Gao & Bo Zeng & Fengqi You, 2019. "A projection-based reformulation and decomposition algorithm for global optimization of a class of mixed integer bilevel linear programs," Journal of Global Optimization, Springer, vol. 73(1), pages 27-57, January.
    7. S. Dempe & S. Franke, 2016. "On the solution of convex bilevel optimization problems," Computational Optimization and Applications, Springer, vol. 63(3), pages 685-703, April.
    8. Burcu Beykal & Styliani Avraamidou & Ioannis P. E. Pistikopoulos & Melis Onel & Efstratios N. Pistikopoulos, 2020. "DOMINO: Data-driven Optimization of bi-level Mixed-Integer NOnlinear Problems," Journal of Global Optimization, Springer, vol. 78(1), pages 1-36, September.
    9. Florensa, Carlos & Garcia-Herreros, Pablo & Misra, Pratik & Arslan, Erdem & Mehta, Sanjay & Grossmann, Ignacio E., 2017. "Capacity planning with competitive decision-makers: Trilevel MILP formulation, degeneracy, and solution approaches," European Journal of Operational Research, Elsevier, vol. 262(2), pages 449-463.
    10. Maximilian Merkert & Galina Orlinskaya & Dieter Weninger, 2022. "An exact projection-based algorithm for bilevel mixed-integer problems with nonlinearities," Journal of Global Optimization, Springer, vol. 84(3), pages 607-650, November.
    11. Richard Oberdieck & Nikolaos A. Diangelakis & Styliani Avraamidou & Efstratios N. Pistikopoulos, 2017. "On unbounded and binary parameters in multi-parametric programming: applications to mixed-integer bilevel optimization and duality theory," Journal of Global Optimization, Springer, vol. 69(3), pages 587-606, November.
    12. Polyxeni-Margarita Kleniati & Claire Adjiman, 2014. "Branch-and-Sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part I: Theoretical development," Journal of Global Optimization, Springer, vol. 60(3), pages 425-458, November.

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