Bilevel Programming Problems

Dempe, Stephan; Kalashnikov, Vyacheslav V.; Pérez-Valdés, Gerardo A.; Kalashnykova, Nataliya I.

doi:10.1007/978-3-662-45827-3

Cited by 233 publications

(115 citation statements)

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“…Remark 1). [14] describes three possible options to cope with this problem. The optimistic bilevel optimization problem assumes a cooperative strategy of leader and follower, i.e., in case of multiple solutions the follower tries to minimize the upper level objective.…”

Section: Related Workmentioning

confidence: 99%

“…Dempe et al [14] classifies the following optimality conditions 2 . The primal Karush-Kuhn-Tucker (KKT) transformation replaces the lower level problem by the necessary and sufficient optimality condition for a convex function.…”

Section: Related Workmentioning

confidence: 99%

“…The classical KKT transformation substitutes the lower level problem with the classical KKT conditions. Due to the extra variable, the problems are not fully equivalent anymore (see [14]). This approach, which leads to a non-smooth mathematical problem with complementary constraints (MPEC), is the most frequently used one.…”

Section: Related Workmentioning

confidence: 99%

“…The linear structure of the lower level problem allows the construc- 2 The classification in [14] applies to the optimistic bilevel problem.…”

Section: Related Workmentioning

confidence: 99%

“…If the lower level problem does not provide a unique solution, the loss function L must be defined on the power set of R N and a different notion of optimality must be introduced. Since, this results in problems beyond the scope of this paper, we refer to [14]. A common circumvention is to consider the corresponding optimistic bilevel problem.…”

Section: Remarkmentioning

confidence: 99%

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Techniques for Gradient-Based Bilevel Optimization with Non-smooth Lower Level Problems

et al. 2016

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We propose techniques for approximating bilevel optimization problems with non-smooth lower level problems that can have a non-unique solution. To this end, we substitute the expression of a minimizer of the lower level minimization problem with an iterative algorithm that is guaranteed to converge to a minimizer of the problem. Using suitable non-linear proximal distance functions, the update mappings of such an iterative algorithm can be differentiable, notwithstanding the fact that the minimization problem is non-smooth.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

“…The linear structure of the lower level problem allows the construc- 2 The classification in [14] applies to the optimistic bilevel problem.…”

Section: Related Workmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

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Techniques for Gradient-Based Bilevel Optimization with Non-smooth Lower Level Problems

et al. 2016

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Simultaneous design and NMPC control under uncertainty and structural decisions: A discrete‐steepest descent algorithm

Palma‐Flores,

Ricardez‐Sandoval,

Biegler

2023

AIChE Journal

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In this article, we address the integration of design and nonlinear model‐based control under uncertainty and structural decisions for naturally ordered structures. We propose an algorithmic framework to determine the optimal location of process units or streams over an ordered discrete set that operates in closed‐loop with a model‐based controller. The formulation corresponds to a mixed‐integer bilevel problem (MIBLP) that is transformed into a single‐level mixed‐integer nonlinear problem (MINLP) using a KKT transformation strategy. In our methodology, the integer decisions are partitioned into subsets called external variables, such that the MINLP is decomposed into an integer‐based master problem and primal subproblems with fixed discrete variables. The master and primal problems are solved using a Discrete‐Steepest Descent Algorithm (D‐SDA). We illustrate the discrete‐based methodology in a case study for a binary distillation column. The D‐SDA showed an improved performance compared to a benchmark continuous‐based formulation using differentiable distribution functions (DDFs).

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Reduction of the bilevel stochastic optimization problem with quantile objective function to a mixed‐integer problem

Dempe

Ivanov

Наумов

2017

Appl Stoch Models Bus & Ind

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The paper is devoted to the stochastic optimistic bilevel optimization problem with quantile criterion in the upper level problem. If the probability distribution is finite, the problem can be transformed into a mixed-integer nonlinear optimization problem. We formulate assumptions guaranteeing that an optimal solution exists. A production planning problem is used to illustrate usefulness of the model.

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Bilevel Programming Problems

Cited by 233 publications

References 0 publications

Techniques for Gradient-Based Bilevel Optimization with Non-smooth Lower Level Problems

Techniques for Gradient-Based Bilevel Optimization with Non-smooth Lower Level Problems

Simultaneous design and NMPC control under uncertainty and structural decisions: A discrete‐steepest descent algorithm

Reduction of the bilevel stochastic optimization problem with quantile objective function to a mixed‐integer problem

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