Invertible Constraint in JuMP

How can I add a constraint in JuMP model with ProxSDP optimizer such that the solution is invertible?

Do you mean creating an SDP variable which is invertible?
You can do

@variable(model, Q[1:n, 1:n], PSD)
using LinearAlgebra
eps = 1e-5
X = Q + eps * I

X is then positive definite hence invertible

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