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

**URL:** <https://discourse.julialang.org/t/how-to-optimize-trace-of-matrix-inverse-with-jump-or-convex/94167>\
**Category:** Optimization (Mathematical)\
**Tags:** question, optimization\
**Created:** [February 6, 2023, 9:25pm UTC](https://discourse.julialang.org/t/how-to-optimize-trace-of-matrix-inverse-with-jump-or-convex/94167 "2023-02-06T21:25:24Z")\
**Posts on this page:** 1\
**Showing post:** 3

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**Author:** ![biona001](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/biona001/32/16497_2.png) [@biona001](https://discourse.julialang.org/u/biona001)\
**Post date:** [February 6, 2023, 11:46pm UTC](https://discourse.julialang.org/t/how-to-optimize-trace-of-matrix-inverse-with-jump-or-convex/94167/3 "2023-02-06T23:46:08Z")

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Thank you for your quick response. I found [this post](http://ask.cvxr.com/t/generalizing-trace-inv-for-matrix-quadratic-forms/5280) which recommends I do the following

```julia
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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