# Reformulation of nonlinear matrix inequality

where matrices \(A_i, C_0\) and scalars \(\varepsilon\) and \(T\) are given, while the decision variables are:

- \(P_i\), required to be symmetric positive definite
- \(\lambda_{i,j}\), required to be positive semidefinite scalars

Note that this is a so-called "Lyapunov-Metzler" inequality. The symbol \(j \in \Omega_i\) can be translated as "for all \(j\) not equal to \(i\)". Of course, I am able to solve the LMI if I consider \(\lambda_{i,j}\) as given scalars. Considering \(\lambda_{i,j}\) as decision variables make the problem nonlinear and I was wondering if there still is the possibility to use Gurobi solve.

Thank you!

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