Bilevel Optimization: Theory, Algorithms, Applications and a Bibliography

Dempe, Stephan

doi:10.1007/978-3-030-52119-6_20

Cited by 88 publications

(90 citation statements)

References 1,083 publications

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“…The pessimistic approach assumes that the lower level chooses an optimal solution that is the worst for the upper-level objective as in [1] or with respect to the upper-level constraints as in [4].…”

Section: Multilevel Optimization and Near-optimality Robustnessmentioning

confidence: 99%

“…They are well suited to model sequential decision-making processes, where a first decision-maker, the leader intrinsically integrates the reaction of another decision-maker, the follower, into their decision-making problem. In recent years, most of the research focuses on the study and design of effi-cient solution methods for the case of two levels, namely bilevel problems [1], which fostered a growing range of applications.…”

Section: Introductionmentioning

confidence: 99%

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Complexity of near-optimal robust versions of multilevel optimization problems

2021

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Near-optimality robustness extends multilevel optimization with a limited deviation of a lower level from its optimal solution, anticipated by higher levels. We analyze the complexity of near-optimal robust multilevel problems, where near-optimal robustness is modelled through additional adversarial decision-makers. Near-optimal robust versions of multilevel problems are shown to remain in the same complexity class as the problem without near-optimality robustness under general conditions.

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Section: Multilevel Optimization and Near-optimality Robustnessmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Complexity of near-optimal robust versions of multilevel optimization problems

2021

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“…L (X,Y ) denotes the set of all continuous linear functions from X to Y , and T : K → L (X,Y ) is an operator. These hierarchical vector optimization problems are also particular cases of bilevel vector optimization problems (see, for instance, [37,38], and the references therein). Firstly let us recall some definitions (see, for instance [39,40,41]) Definition 4.1.…”

Section: Pareto Vector Optimization Over An Efficient Setmentioning

confidence: 99%

Existence of solutions of bilevel strong vector equilibrium problems and their applications

2021

J. Nonlinear Var. Anal.

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In this paper, a bilevel strong vector equilibrium problem is introduced and investigated by using the Fan's lemma. Two approaches for finding the solutions of this problem are considered. The applications provide sufficient conditions for the existence of the solutions of a Pareto vector optimization over set constraints and the existence of strong vector saddle points for a generalized Nash two-person noncooperative games where the payoff function is vector-valued.

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“…For most lower-level problems, there are several optimal solutions (different solutions yielding the same optimal value of the objective). Several methodologies have been developed for such cases, the two primary approaches being the optimistic and pessimistic bilevel formulations (Dempe, 2018), regularizing the set-valued problem by guaranteeing the uniqueness of the lower-level solution. The approach used in BilevelOptimization.jl is the optimistic one.…”

Section: Bilevel Optimizationmentioning

confidence: 99%

A Julia package for bilevel optimization problems

Besançon¹

2019

JOSS

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Mathematical optimization is the discipline dealing with the determination of the best (or almost best) decision with respect to a specific cost function and to a set of constraints on the decision. Bilevel optimization is a class of mathematical optimization problems with the optimality conditions of a lower-level problem embedded in the constraints. BilevelOptimization.jl is a toolbox built on top of the JuMP.jl ecosystem for mathematical optimization (Dunning, Huchette, & Lubin, 2017). Bilevel optimization is used to tackle various problems in areas such as power systems, security applications, network design or market equilibria. See Dempe (2018) for an overview of applications and recent formulations and theoretical progress.

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Bilevel Optimization: Theory, Algorithms, Applications and a Bibliography

Cited by 88 publications

References 1,083 publications

Complexity of near-optimal robust versions of multilevel optimization problems

Complexity of near-optimal robust versions of multilevel optimization problems

Existence of solutions of bilevel strong vector equilibrium problems and their applications

A Julia package for bilevel optimization problems

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