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A tutorial on solving single‐leader‐multi‐follower problems using SOS1 reformulations.

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Název: A tutorial on solving single‐leader‐multi‐follower problems using SOS1 reformulations.
Autoři: Aussel, Didier, Egea, Cécile, Schmidt, Martin
Zdroj: International Transactions in Operational Research; May2025, Vol. 32 Issue 3, p1227-1250, 24p
Témata: LINEAR programming, MATHEMATICAL reformulation, CONVEX sets, CONVEX functions, PROBLEM solving
Abstrakt: In this tutorial, we consider single‐leader‐multi‐follower games in which the models of the lower‐level players have polyhedral feasible sets and convex objective functions. This situation allows for classic Karush–Kuhn–Tucker reformulations of the separate lower‐level problems, which lead to challenging single‐level reformulations of Mathematical Programing with Complementarity Constraints (MPCC) type. The main contribution of this tutorial is to present a ready‐to‐use reformulation of this MPCC using special‐ordered‐sets of type 1 (SOS1) conditions. These conditions are readily available in all modern mixed‐integer linear optimization solvers that solve the single‐leader‐multi‐follower problem to optimality. After formally stating the problem class under consideration as well as deriving its reformulations, we present explicit Python code that shows how these techniques can be realized using the solver Gurobi. Finally, we also show the effect of the SOS1‐based reformulation using the real‐world example of industrial eco‐park modeling. [ABSTRACT FROM AUTHOR]
Copyright of International Transactions in Operational Research is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: A tutorial on solving single‐leader‐multi‐follower problems using SOS1 reformulations.
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  Data: <searchLink fieldCode="AR" term="%22Aussel%2C+Didier%22">Aussel, Didier</searchLink><br /><searchLink fieldCode="AR" term="%22Egea%2C+Cécile%22">Egea, Cécile</searchLink><br /><searchLink fieldCode="AR" term="%22Schmidt%2C+Martin%22">Schmidt, Martin</searchLink>
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  Data: International Transactions in Operational Research; May2025, Vol. 32 Issue 3, p1227-1250, 24p
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  Data: <searchLink fieldCode="DE" term="%22LINEAR+programming%22">LINEAR programming</searchLink><br /><searchLink fieldCode="DE" term="%22MATHEMATICAL+reformulation%22">MATHEMATICAL reformulation</searchLink><br /><searchLink fieldCode="DE" term="%22CONVEX+sets%22">CONVEX sets</searchLink><br /><searchLink fieldCode="DE" term="%22CONVEX+functions%22">CONVEX functions</searchLink><br /><searchLink fieldCode="DE" term="%22PROBLEM+solving%22">PROBLEM solving</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: In this tutorial, we consider single‐leader‐multi‐follower games in which the models of the lower‐level players have polyhedral feasible sets and convex objective functions. This situation allows for classic Karush–Kuhn–Tucker reformulations of the separate lower‐level problems, which lead to challenging single‐level reformulations of Mathematical Programing with Complementarity Constraints (MPCC) type. The main contribution of this tutorial is to present a ready‐to‐use reformulation of this MPCC using special‐ordered‐sets of type 1 (SOS1) conditions. These conditions are readily available in all modern mixed‐integer linear optimization solvers that solve the single‐leader‐multi‐follower problem to optimality. After formally stating the problem class under consideration as well as deriving its reformulations, we present explicit Python code that shows how these techniques can be realized using the solver Gurobi. Finally, we also show the effect of the SOS1‐based reformulation using the real‐world example of industrial eco‐park modeling. [ABSTRACT FROM AUTHOR]
– Name: Abstract
  Label:
  Group: Ab
  Data: <i>Copyright of International Transactions in Operational Research is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1111/itor.13466
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        Text: English
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        PageCount: 24
        StartPage: 1227
    Subjects:
      – SubjectFull: LINEAR programming
        Type: general
      – SubjectFull: MATHEMATICAL reformulation
        Type: general
      – SubjectFull: CONVEX sets
        Type: general
      – SubjectFull: CONVEX functions
        Type: general
      – SubjectFull: PROBLEM solving
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      – TitleFull: A tutorial on solving single‐leader‐multi‐follower problems using SOS1 reformulations.
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            – D: 01
              M: 05
              Text: May2025
              Type: published
              Y: 2025
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