how to formulate a KKT complementarity condition for second order cone constraint?

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john zhang

2024年04月12日 05:28

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I have a second order cone constrint, i.e. P^2 + Q^2 <= vl.

Then I am trying to write a KKT complementarity condition, i.e. \lambda * (P^2 + Q^2 - vl) == 0. However, gurobi seems like not supporting this kind of formulation, as it not supports multiplying three variables.

How can I solve this issue? Should I introduce other variables and constraints? Such as tmp1 = P^2, tmp2 = Q^2, tmp3 = vl, then \lambda * (tmp1 + tmp2 - tmp3) = 0?

Or I use another formulation for complementarity condition, i.e. let y = [2P 2Q v-l v+l]^T, and \lambda^T \cdot y = 0, and \lambda belongs to dual cone. But in this case, the number of dual variables is different from the number of primal constraints.

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  Simranjit Kaur

  • Gurobi Staff

  2026年11月13日 01:26

  This post is more than two 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.
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  john zhang

  2024年04月12日 06:10

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  Another question is that, is it possible to get the dual variable value with respect to the second order cone constraint?

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  Jaromił Najman

  • Gurobi Staff

  2024年04月12日 09:51

  Hi John,

  How can I solve this issue? Should I introduce other variables and constraints? Such as tmp1 = P^2, tmp2 = Q^2, tmp3 = vl, then \lambda * (tmp1 + tmp2 - tmp3) = 0?

  You are correct. You have to introduce auxiliary variables. This is also described in the Knowledge Base article How do I model multilinear terms in Gurobi?

  Another question is that, is it possible to get the dual variable value with respect to the second order cone constraint?

  You can get dual variable values from Gurobi as long as your model is convex. You have to set the QCPDual parameter for this. If your model is non-convex, then you cannot get any dual values from Gurobi. Your complementarity conditions are non-convex, thus you cannot retrieve any dual information from Gurobi after a successful solve. We describe a way for how to obtain dual values for MIPs in the Knowledge Base article How do I retrieve the (dual) Pi values for a MIP problem? The same idea applies to non-convex continuous models.

  Best regards, 
  Jaromił

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