Quadratic form with inverse of a matrix as objective

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Matteo Croci

2021年11月23日 00:41

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I have a convex problem which, in simplified form, reads like this:

find m_1,m_2 that minimize invA[0,0], where invA = A^-1, and A = m_1*B_1 + m_2*B_2, where B_1 and B_2 are square symmetric positive definite matrices.

How can I implement this in Gurobipy?  I can introduce invA as an additional variable and impose invA*A = I, but this makes the problem non-convex.

Thanks!

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  Simranjit Kaur

  • Gurobi Staff

  2025年06月23日 02:44 編集済み

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  Jaromił Najman

  • Gurobi Staff

  2021年11月25日 10:33

  Unfortunately, the only way to keep the convexity of your problem would be to analytically derive a formulate for the values of \(A^{-1}\). This means that unless you can provide explicit equality constraints for \(A^{-1}=\dots\), you will have to introduce the non-convexity through the bilinear constraint enforcing the inverse equality constraint \(A^{-1}\cdot A = 1\). Note that introducing non-convexity does not mean that your problem becomes unsolvable. It highly depends on your numerics, model structure, and size. Thus, you should at least give it a try and test the non-convex formulation.

  Best regards,
  Jaromił

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