SDP trace minimization with affine constraints: help to formulate the problem

Specific Domains Optimization (Mathematical)
6

I would not work with the cholesky decomposition. The semidefinite programming are there to handle matrices that possess a cholesky decomposition in a convex way (by seeing them as semidefinite matrices whose cone have a self-concordant barrier). I would not expect an NLP solver to work better.

The dual of
min tr(X)
A(x) = b
X is PSD
is
max b^T y
I - sum A_i y_i is PSD
y free
so for modeling the dual, do

model = Model(...)
@variable(model, y[1:size(Ω, 2)])
using LinearAlgebra
@constraint(Symmetric(I - sum(Ω[:,i] * Ω[:,i]' * y[i] for i in 1:size(Ω, 2))) in PSDCone())
@objective(model, Max, b' * y)
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