A Survey on Mixed-Integer Programming Techniques in Bilevel Optimization

•We survey mixed-integer programming techniques as they are applied in bilevel optimization.•We focus on bilevel problems with convex or linear lower-level problems as well as on problems with mixed-integer lower levels.•Special attention is given to bilevel pricing problems, Stackelberg games, and...

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Vydáno v:EURO journal on computational optimization Ročník 9; s. 100007
Hlavní autoři: Kleinert, Thomas, Labbé, Martine, Ljubić, Ivana, Schmidt, Martin
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier Ltd 2021
Springer
Elsevier
Témata:
90-02
90-08
90Bxx
90C11
90C26
91A65
Applications
Bilevel optimization
Branch-and-bound
Branch-and-cut
Computer Science
Mixed-integer programming
Operations Research
Survey
90-02
Applications
90-08
90C26
Branch-and-cut
Branch-and-bound
90Bxx
Bilevel optimization
Survey
91A65
90C46
Mixed-integer programming
90C11
90C33
90C90
ISSN:2192-4406, 2192-4414
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Abstract •We survey mixed-integer programming techniques as they are applied in bilevel optimization.•We focus on bilevel problems with convex or linear lower-level problems as well as on problems with mixed-integer lower levels.•Special attention is given to bilevel pricing problems, Stackelberg games, and interdiction problems.•The survey contains more than 250 references. Bilevel optimization is a field of mathematical programming in which some variables are constrained to be the solution of another optimization problem. As a consequence, bilevel optimization is able to model hierarchical decision processes. This is appealing for modeling real-world problems, but it also makes the resulting optimization models hard to solve in theory and practice. The scientific interest in computational bilevel optimization increased a lot over the last decade and is still growing. Independent of whether the bilevel problem itself contains integer variables or not, many state-of-the-art solution approaches for bilevel optimization make use of techniques that originate from mixed-integer programming. These techniques include branch-and-bound methods, cutting planes and, thus, branch-and-cut approaches, or problem-specific decomposition methods. In this survey article, we review bilevel-tailored approaches that exploit these mixed-integer programming techniques to solve bilevel optimization problems. To this end, we first consider bilevel problems with convex or, in particular, linear lower-level problems. The discussed solution methods in this field stem from original works from the 1980’s but, on the other hand, are still actively researched today. Second, we review modern algorithmic approaches to solve mixed-integer bilevel problems that contain integrality constraints in the lower level. Moreover, we also briefly discuss the area of mixed-integer nonlinear bilevel problems. Third, we devote some attention to more specific fields such as pricing or interdiction models that genuinely contain bilinear and thus nonconvex aspects. Finally, we sketch a list of open questions from the areas of algorithmic and computational bilevel optimization, which may lead to interesting future research that will further propel this fascinating and active field of research.
AbstractList Bilevel optimization is a field of mathematical programming in which some variables are constrained to be the solution of another optimization problem. As a consequence, bilevel optimization is able to model hierarchical decision processes. This is appealing for modeling real-world problems, but it also makes the resulting optimization models hard to solve in theory and practice. The scientific interest in computational bilevel optimization increased a lot over the last decade and is still growing. Independent of whether the bilevel problem itself contains integer variables or not, many state-of-the-art solution approaches for bilevel optimization make use of techniques that originate from mixed-integer programming. These techniques include branch-and-bound methods, cutting planes and, thus, branch-and-cut approaches, or problemspecific decomposition methods. In this survey article, we review bilevel-tailored approaches that exploit these mixed-integer programming techniques to solve bilevel optimization problems. To this end, we first consider bilevel problems with convex or, in particular, linear lower-level problems. The discussed solution methods in this field stem from original works from the 1980's but, on the other hand, are still actively researched today. Second, we review modern algorithmic approaches to solve mixed-integer bilevel problems that contain integrality constraints in the lower level. Moreover, we also briefly discuss the area of mixed-integer nonlinear bilevel problems. Third, we devote some attention to more specific fields such as pricing or interdiction models that genuinely contain bilinear and thus nonconvex aspects. Finally, we sketch a list of open questions from the areas of algorithmic and computational bilevel optimization, which may lead to interesting future research that will further propel this fascinating and active field of research.
Bilevel optimization is a field of mathematical programming in which some variables are constrained to be the solution of another optimization problem. As a consequence, bilevel optimization is able to model hierarchical decision processes. This is appealing for modeling real-world problems, but it also makes the resulting optimization models hard to solve in theory and practice. The scientific interest in computational bilevel optimization increased a lot over the last decade and is still growing. Independent of whether the bilevel problem itself contains integer variables or not, many state-of-the-art solution approaches for bilevel optimization make use of techniques that originate from mixed-integer programming. These techniques include branch-and-bound methods, cutting planes and, thus, branch-and-cut approaches, or problem-specific decomposition methods. In this survey article, we review bilevel-tailored approaches that exploit these mixed-integer programming techniques to solve bilevel optimization problems. To this end, we first consider bilevel problems with convex or, in particular, linear lower-level problems. The discussed solution methods in this field stem from original works from the 1980’s but, on the other hand, are still actively researched today. Second, we review modern algorithmic approaches to solve mixed-integer bilevel problems that contain integrality constraints in the lower level. Moreover, we also briefly discuss the area of mixed-integer nonlinear bilevel problems. Third, we devote some attention to more specific fields such as pricing or interdiction models that genuinely contain bilinear and thus nonconvex aspects. Finally, we sketch a list of open questions from the areas of algorithmic and computational bilevel optimization, which may lead to interesting future research that will further propel this fascinating and active field of research.
•We survey mixed-integer programming techniques as they are applied in bilevel optimization.•We focus on bilevel problems with convex or linear lower-level problems as well as on problems with mixed-integer lower levels.•Special attention is given to bilevel pricing problems, Stackelberg games, and interdiction problems.•The survey contains more than 250 references. Bilevel optimization is a field of mathematical programming in which some variables are constrained to be the solution of another optimization problem. As a consequence, bilevel optimization is able to model hierarchical decision processes. This is appealing for modeling real-world problems, but it also makes the resulting optimization models hard to solve in theory and practice. The scientific interest in computational bilevel optimization increased a lot over the last decade and is still growing. Independent of whether the bilevel problem itself contains integer variables or not, many state-of-the-art solution approaches for bilevel optimization make use of techniques that originate from mixed-integer programming. These techniques include branch-and-bound methods, cutting planes and, thus, branch-and-cut approaches, or problem-specific decomposition methods. In this survey article, we review bilevel-tailored approaches that exploit these mixed-integer programming techniques to solve bilevel optimization problems. To this end, we first consider bilevel problems with convex or, in particular, linear lower-level problems. The discussed solution methods in this field stem from original works from the 1980’s but, on the other hand, are still actively researched today. Second, we review modern algorithmic approaches to solve mixed-integer bilevel problems that contain integrality constraints in the lower level. Moreover, we also briefly discuss the area of mixed-integer nonlinear bilevel problems. Third, we devote some attention to more specific fields such as pricing or interdiction models that genuinely contain bilinear and thus nonconvex aspects. Finally, we sketch a list of open questions from the areas of algorithmic and computational bilevel optimization, which may lead to interesting future research that will further propel this fascinating and active field of research.
ArticleNumber 100007
Author Schmidt, Martin
Labbé, Martine
Ljubić, Ivana
Kleinert, Thomas
Author_xml – sequence: 1
  givenname: Thomas
  surname: Kleinert
  fullname: Kleinert, Thomas
  email: thomas.kleinert@fau.de
  organization: Friedrich-Alexander-Universität Erlangen-Nürnberg, Discrete Optimization, Cauerstr. 11, Erlangen 91058, Germany
– sequence: 2
  givenname: Martine
  surname: Labbé
  fullname: Labbé, Martine
  email: martine.labbe@ulb.be
  organization: Université Libre de Bruxelles, Department of Computer Science, Boulevard du Triomphe, CP212, Brussels 1050, Belgium
– sequence: 3
  givenname: Ivana
  orcidid: 0000-0002-4834-6284
  surname: Ljubić
  fullname: Ljubić, Ivana
  email: ivana.ljubic@essec.edu
  organization: ESSEC Business School of Paris, 95021 Cergy-Pontoise, France
– sequence: 4
  givenname: Martin
  orcidid: 0000-0001-6208-5677
  surname: Schmidt
  fullname: Schmidt, Martin
  email: martin.schmidt@uni-trier.de
  organization: Trier University, Department of Mathematics, Universitätsring 15, Trier 54296, Germany
BackLink https://inria.hal.science/hal-03095139$$DView record in HAL
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Applications
90-08
90C26
Branch-and-cut
Branch-and-bound
90Bxx
Bilevel optimization
Survey
91A65
90C46
Mixed-integer programming
90C11
90C33
90C90
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  article-title: Supply chain competition on shelf space and pricing for soft drinks: A bilevel optimization approach
  publication-title: International Journal of Production Economics
  doi: 10.1016/j.ijpe.2018.12.018
– year: 1998
  ident: 10.1016/j.ejco.2021.100007_bib0267
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Snippet •We survey mixed-integer programming techniques as they are applied in bilevel optimization.•We focus on bilevel problems with convex or linear lower-level...
Bilevel optimization is a field of mathematical programming in which some variables are constrained to be the solution of another optimization problem. As a...
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SubjectTerms 90-02
90-08
90Bxx
90C11
90C26
91A65
Applications
Bilevel optimization
Branch-and-bound
Branch-and-cut
Computer Science
Mixed-integer programming
Operations Research
Survey
Title A Survey on Mixed-Integer Programming Techniques in Bilevel Optimization
URI https://dx.doi.org/10.1016/j.ejco.2021.100007
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