minimax optimization

Answered

CHAYMAE MAKRI

November 08, 2022 08:43

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Hello,

How can I solve the multi-objective problem?
F= min ( P *x + max (Q*x ) )
where x is the decision variable, and P and Q are constants.

Thanks

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• Official comment

  

  Simranjit Kaur

  • Gurobi Staff

  November 16, 2025 23:25

  This post is more than three years old. Some information may not be up to date. For current information, please check the Gurobi Documentation or Knowledge Base. If you need more help, please create a new post in the community forum, or try Gurobot, our chatbot interface offering instant, expert-level support.
• 

  Jaromił Najman

  • Gurobi Staff

  November 08, 2022 13:15 Edited

  Hi,

  Are you sure that you want to solve a multi-objective problem? Because then your model should rather be something like

  \[\begin{align*}
  &\min (Px, -Qx) \\
  &\text{s.t. some constraints}
  \end{align*}\]

  If you indeed want to solve a multi-objective model, please refer to our Documentation of Working with Multiple Objectives.

  The formulation you wrote looks more like bilevel optimization. There is plently of literature on how to solve (mixed-integer) linear bilevel optimization problems. Most of them require the dualization of the second level. One good overview is provided by Ben-Ayed 1993.

  Best regards, 
  Jaromił

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