# Matrix inverse as objective

**URL:** <https://discourse.julialang.org/t/matrix-inverse-as-objective/129162>\
**Category:** Optimization (Mathematical)\
**Tags:** jump, optim\
**Created:** [May 20, 2025, 4:27am UTC](https://discourse.julialang.org/t/matrix-inverse-as-objective/129162 "2025-05-20T04:27:23Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![Rick](https://avatars.discourse-cdn.com/v4/letter/r/9f8e36/32.png) [@Rick](https://discourse.julialang.org/u/Rick)\
**Post date:** [May 20, 2025, 4:27am UTC](https://discourse.julialang.org/t/matrix-inverse-as-objective/129162/1 "2025-05-20T04:27:23Z")

</div>

Hi everyone,

I am trying to solve a semi-definite program with an objective that contains a matrix inverse.

More precisely, let a \geq 0, and A be a positive definite matrix of size d \times d. \mathbf{1} below is a vector of d ones. I want to solve the following program

\max\_{X}\quad [a + \mathbf{1}'(A +X)^{-1} \mathbf{1}]^{-1} \qquad \text{s.t.} \qquad \text{$X$ is a $d\times d$ positive semidefinite matrix}

Is there a way to solve this using the existing functions in Julia and JuMP?

Maybe I have missed it, but I tried looking into the [Standard form · MathOptInterface](https://jump.dev/MathOptInterface.jl/stable/reference/standard_form/) functions and could not find something helpful for my problem.

Any thoughts are appreciated. Thank you!

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<div class="post-metadata">

**Author:** ![odow](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/odow/32/28685_2.png) [@odow](https://discourse.julialang.org/u/odow)\
**Post date:** [May 20, 2025, 4:46am UTC](https://discourse.julialang.org/t/matrix-inverse-as-objective/129162/2 "2025-05-20T04:46:17Z")

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Matrix inverse + PSD means there’s usually a trick with Schur’s lemma: [6 Semidefinite optimization — MOSEK Modeling Cookbook 3.3.1](https://docs.mosek.com/modeling-cookbook/sdo.html#matrix-inequalities-from-schurs-lemma)

I assume you have some other constraints, so that your objective is always `>= 0`.

Then it’s equivalent to minimizing the term without the outer inverse.

`a` is constant so it can be dropped.

You want \min\_{X\succeq 0}: 1^\top(A + X)^{-1} 1.

Rewrite to epigraph form: \min\_{X\succeq 0,C}: C such that C \succeq 1^\top(A + X)^{-1} 1.

By Schur’s lemma \min\_{X\succeq 0,C}: C such that \begin{bmatrix}C & 1^\top\\ 1 &(A+X)\end{bmatrix} \succeq 0.

In JuMP:

```julia
julia> using JuMP, SCS

julia> A = [3 -2; -2 4]
2×2 Matrix{Int64}:
  3 -2
 -2 4

julia> d = size(A, 1)
2

julia> model = Model(SCS.Optimizer)
A JuMP Model
├ solver: SCS
├ objective_sense: FEASIBILITY_SENSE
├ num_variables: 0
├ num_constraints: 0
└ Names registered in the model: none

julia> @variable(model, X[1:d, 1:d], PSD)
2×2 Symmetric{VariableRef, Matrix{VariableRef}}:
 X[1,1] X[1,2]
 X[1,2] X[2,2]

julia> @constraint(model, LinearAlgebra.tr(X) == 1) # Or something?
X[1,1] + X[2,2] = 1

julia> @variable(model, C)
C

julia> @objective(model, Min, C)
C

julia> @constraint(model, [C ones(d)'; ones(d) (A + X)] >= 0, PSDCone())
[C 1 1
 1 X[1,1] + 3 X[1,2] - 2
 1 X[1,2] - 2 X[2,2] + 4] ∈ PSDCone()

julia> optimize!(model)
------------------------------------------------------------------
	       SCS v3.2.7 - Splitting Conic Solver
	(c) Brendan O'Donoghue, Stanford University, 2012
------------------------------------------------------------------
problem: variables n: 4, constraints m: 10
cones: z: primal zero / dual free vars: 1
	  s: psd vars: 9, ssize: 2
settings: eps_abs: 1.0e-04, eps_rel: 1.0e-04, eps_infeas: 1.0e-07
	  alpha: 1.50, scale: 1.00e-01, adaptive_scale: 1
	  max_iters: 100000, normalize: 1, rho_x: 1.00e-06
	  acceleration_lookback: 10, acceleration_interval: 10
	  compiled with openmp parallelization enabled
lin-sys: sparse-direct-amd-qdldl
	  nnz(A): 9, nnz(P): 0
------------------------------------------------------------------
 iter | pri res | dua res | gap | obj | scale | time (s)
------------------------------------------------------------------
     0| 1.74e+01 1.08e+00 1.90e+01 -7.86e+00 1.00e-01 3.13e-04 
    50| 6.41e-05 1.28e-05 7.62e-05 8.12e-01 1.00e-01 8.29e-04 
------------------------------------------------------------------
status: solved
timings: total: 8.41e-04s = setup: 1.17e-04s + solve: 7.23e-04s
	 lin-sys: 2.50e-05s, cones: 2.79e-04s, accel: 5.37e-06s
------------------------------------------------------------------
objective = 0.812454
------------------------------------------------------------------

julia> value(C)
0.8124158521710856

julia> ones(d)' * inv(A + value(X)) * ones(d)
0.812469395065637

```

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<div class="post-metadata">

**Author:** ![Rick](https://avatars.discourse-cdn.com/v4/letter/r/9f8e36/32.png) [@Rick](https://discourse.julialang.org/u/Rick)\
**Post date:** [May 20, 2025, 5:51am UTC](https://discourse.julialang.org/t/matrix-inverse-as-objective/129162/3 "2025-05-20T05:51:58Z")

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Thank you!!!