You can use Penbmi, there is a Julia wrapper for using it with JuMP.
In the future, Alpine might be able to support bilinear matrix inequalities as well so as to provide guaranteed optimal solutions, using an SDP solver and an a local search BMI solver (like Penbmi) as subsolvers.
You could also create a Julia wrapper for BMISOLVER.
I have created a few JuMP interfaces for MATLAB solvers (SeDuMi.jl, CDCS.jl, SDPNAL.jl, SDPT3.jl), it’s not too much work, I can guide you through it if you’re interested.
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