# Rank1 PSD constraint dual variables
**URL:** https://discourse.julialang.org/t/rank1-psd-constraint-dual-variables/80348
**Category:** Optimization (Mathematical)
**Tags:** jump, optimization
**Created:** [May 2, 2022, 12:13am UTC](https://discourse.julialang.org/t/rank1-psd-constraint-dual-variables/80348 "2022-05-02T00:13:24Z")
**Posts on this page:** 9
**Page:** 1
### Author:  [@HWS](https://discourse.julialang.org/u/HWS)
#### Post date: [May 2, 2022, 12:13am UTC](https://discourse.julialang.org/t/rank1-psd-constraint-dual-variables/80348/1 "2022-05-02T00:13:24Z")
Hi,
I ran the following code in JuMP,
```julia
@variable(m, x[1:n]>=0)
@variable(m, X[1:n,1:n], Symmetric)
@constraint(m, X_xx_cons, [1 x'; x X] in PSDCone() )
@objective(m, Min, tr(Q'*X)+dot(b,x))
optimize!(m)
A = dual(X_xx_cons)
```
and after solving it, the dual variable A is an upper triangular matrix. How can I recover the dual variable of constraint X\_xx\_cons that is a positive semidefinite matrix ? ( Is it because of symmetry, the off diagonal of A is two times what it supposed to be?) Thank you.
---
### Author:  [@odow](https://discourse.julialang.org/u/odow)
#### Post date: [May 2, 2022, 12:27am UTC](https://discourse.julialang.org/t/rank1-psd-constraint-dual-variables/80348/2 "2022-05-02T00:27:27Z")
Hi there,
What is `n`, `Q`, and `b`?
> the dual variable A is an upper triangular matrix
Hmm. That seems like a bug. Try explicitly wrapping the `[1 x'; x X]` in `Symmetric:
```julia
using JuMP, SCS, LinearAlgebra
n = 3
b = rand(n)
Q = [1 0.5 0; 0.5 1 0.5; 0 0.5 1]
model = Model(SCS.Optimizer)
@variable(model, x[1:n] >= 0)
@variable(model, X[1:n, 1:n], Symmetric)
c = @constraint(model, Symmetric([1 x'; x X]) in PSDCone())
@objective(model, Min, LinearAlgebra.tr(Q * X) + b' * x)
optimize!(model)
julia> A = dual(c)
4×4 Symmetric{Float64, Matrix{Float64}}:
2.06697e-12 9.3909e-7 1.32841e-6 9.38912e-7
9.3909e-7 1.0 0.5 -8.0519e-14
1.32841e-6 0.5 1.0 0.5
9.38912e-7 -8.0519e-14 0.5 1.0
```
---
### Author:  [@odow](https://discourse.julialang.org/u/odow)
#### Post date: [May 2, 2022, 12:39am UTC](https://discourse.julialang.org/t/rank1-psd-constraint-dual-variables/80348/3 "2022-05-02T00:39:39Z")
I’ve opened an issue: [Bug in dual of PositiveSemidefiniteConeSquare · Issue #1830 · jump-dev/MathOptInterface.jl · GitHub](https://github.com/jump-dev/MathOptInterface.jl/issues/1830)
---
### Author:  [@HWS](https://discourse.julialang.org/u/HWS)
#### Post date: [May 2, 2022, 12:40am UTC](https://discourse.julialang.org/t/rank1-psd-constraint-dual-variables/80348/4 "2022-05-02T00:40:18Z")
Hi,
Thank you for the reply. I think in my code, I did not have Symmetric() in the constraint.
---
### Author:  [@HWS](https://discourse.julialang.org/u/HWS)
#### Post date: [May 2, 2022, 12:46am UTC](https://discourse.julialang.org/t/rank1-psd-constraint-dual-variables/80348/5 "2022-05-02T00:46:30Z")
In fact, if I delete Symmetric(), I got this
```julia
n = 3
b = rand(n)
Q = [1 0.5 0; 0.5 1 0.5; 0 0.5 1]
model = Model(Mosek.Optimizer)
@variable(model, x[1:n] >= 0)
@variable(model, X[1:n, 1:n], Symmetric)
c = @constraint(model, [1 x'; x X] in PSDCone())
@objective(model, Min, LinearAlgebra.tr(Q * X) + b' * x)
optimize!(model)
julia> A = dual(c)
4×4 Matrix{Float64}:
4.79303e-9 2.15225e-5 5.44024e-5 5.29432e-5
0.0 1.0 1.0 3.88365e-10
0.0 0.0 1.0 1.0
0.0 0.0 0.0 1.0
```
---
### Author:  [@odow](https://discourse.julialang.org/u/odow)
#### Post date: [May 2, 2022, 12:47am UTC](https://discourse.julialang.org/t/rank1-psd-constraint-dual-variables/80348/6 "2022-05-02T00:47:37Z")
Yes, I think this is a bug. You can work-around it by using `Symmetric`.
---
### Author:  [@HWS](https://discourse.julialang.org/u/HWS)
#### Post date: [May 2, 2022, 1:03am UTC](https://discourse.julialang.org/t/rank1-psd-constraint-dual-variables/80348/7 "2022-05-02T01:03:04Z")
Hi Oscar,
One follow up question, in JuMP, are there alternative ways to express constraint X-x\*x^top \succeq 0? I have tried the following but it does not work, is it because JuMP only allows affine expression in PSDCone()?
```julia
n = 3
b = rand(n)
Q = [1 0.5 0; 0.5 1 0.5; 0 0.5 1]
model = Model(Mosek.Optimizer)
@variable(model, x[1:n] >= 0)
@variable(model, X[1:n, 1:n], Symmetric)
c = @constraint(model, X-x*x' in PSDCone() )
@objective(model, Min, LinearAlgebra.tr(Q * X) + b' * x)
optimize!(model)
ERROR: MathOptInterface.UnsupportedConstraint{MathOptInterface.ScalarQuadraticFunction{Float64}, MathOptInterface.LessThan{Float64}}: `MathOptInterface.ScalarQuadraticFunction{Float64}`-in-`MathOptInterface.LessThan{Float64}` constraint is not supported by the model: Unable to transform a quadratic constraint into a second-order cone constraint because the quadratic constraint is not strongly convex.
Convex constraints that are not strongly convex (i.e., the matrix is positive semidefinite but not positive definite) are not supported yet.
Note that a quadratic equality constraint is non-convex.
```
Thanks!
---
### Author:  [@odow](https://discourse.julialang.org/u/odow)
#### Post date: [May 2, 2022, 1:24am UTC](https://discourse.julialang.org/t/rank1-psd-constraint-dual-variables/80348/8 "2022-05-02T01:24:48Z")
> [@HWS](#):
>
> is it because JuMP only allows affine expression in PSDCone()?
It’s actually the solver that doesn’t support this, and JuMP can’t reformulate the problem into something that the solver supports because `X - x * x'` is non convex.
---
### Author:  [@HWS](https://discourse.julialang.org/u/HWS)
#### Post date: [May 2, 2022, 1:47am UTC](https://discourse.julialang.org/t/rank1-psd-constraint-dual-variables/80348/9 "2022-05-02T01:47:07Z")
Thank you.