How to optimize trace of matrix inverse with JuMP or Convex?

Specific Domains Optimization (Mathematical)
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3

Thank you for your quick response. I found this post which recommends I do the following

using Hypatia, JuMP, LinearAlgebra
p = 5
x = randn(p, p)
A = Symmetric(x' * x)

# model and constraints constraints
model = Model(() -> Hypatia.Optimizer())
@variable(model, S[1:p, 1:p], PSD)
@constraint(model, A >= S, PSDCone())

# introduce X and Y
@variable(model, X[1:p, 1:p])
@variable(model, Y[1:p, 1:p])
@constraint(model, [X I; I S] in PSDCone())
@constraint(model, [Y I; I A-S] in PSDCone())

# objective
@objective(model, Min, tr(X) + tr(Y))

# solve and return
JuMP.optimize!(model)

which seems to work. The idea is that instead of working with tr(S^{-1}) in the objective, we work with tr(X) where \begin{bmatrix}X & I\\ I & S\end{bmatrix} \succeq 0. Then by Schur complement argument, \begin{bmatrix}X & I\\ I & S\end{bmatrix} \succeq 0 \iff X - S^{-1} \succeq 0 \iff X \succeq S^{-1} \iff tr(X) \ge tr(S^{-1}).

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