Does anyone have experience in moving from Matlab’s LMI toolbox to the various JuliaOpt packages? I’m taking a nonlinear dynamics course (AAE666 at Purdue) and we’ve just been introduced to linear matrix inequalities in the context of quadratic stability. As such, I’m still trying to grapple with the toolbox and LMIs in general and I’m not sure how to translate such a problem to a JuliaOPT package. Any resources or thoughts would be much appreciated.
Did you have a look at JuMP, especially the Section Semidefinite Constraints or the Example Example minimal Ellipse?
I have no experience so far in LMI’s but I think this would be a good starting point.
For example solving the Lyapunov Inequality for a continuous system could look like this:
using JuMP
using MosekTools
using LinearAlgebra
A = [0.5 1; 0 0.3]; n = size(A,1); Q = [1 0; 0 1]
model = Model(with_optimizer(Mosek.Optimizer))
@variable(model, P[i=1:n, j=1:n], PSD)
@SDconstraint(model, P ⪰ 0)
@SDconstraint(model, A'*P + P*A ⪯ Q)
@SDconstraint(model, A'*P + P*A ⪰ Q)
optimize!(model)
P = value.(P)
# Test result:
A'*P + P*A ≈ Q
I wanted to try the example in the above post with a different optimizer. But, I got a StactOverflowError
. Any ideas what I am doing wrong?
using JuMP
using SCS
using LinearAlgebra
A = [0.5 1; 0 0.3]
n = size(A,1)
Q = [1 0; 0 1.]
model = Model(with_optimizer(SCS.Optimizer))
@variable(model, P[1:n, 1:n], PSD)
@SDconstraint(model, P ⪰ 0)
@SDconstraint(model, A'*P + P*A ≥ Q)
@SDconstraint(model, A'*P + P*A ≤ Q)
optimize!(model)
P = value.(P)
# Test result:
A'*P + P*A ≈ Q
ERROR: LoadError: StackOverflowError:
Stacktrace:
[1] mutable_operate!(::typeof(MutableArithmetics.sub_mul), ::Matrix{AffExpr}, ::Matrix{VariableRef}, ::Matrix{Float64}) (repeats 79984 times)
@ MutableArithmetics ~/.julia/packages/MutableArithmetics/bPWR4/src/linear_algebra.jl:132
How about using Convex.jl?
julia> using Convex, SCS, LinearAlgebra
julia> A = [0.5 1; 0 0.3]
2×2 Matrix{Float64}:
0.5 1.0
0.0 0.3
julia> n = size(A,1)
2
julia> eigvals(A)
2element Vector{Float64}:
0.5
0.3
julia> X = Semidefinite(n)
Variable
size: (2, 2)
sign: real
vexity: affine
id: 404…993
julia> constraint = A'*X + X*A ⪯ Matrix{Float64}(I, n, n)
sdp constraint (affine)
└─ + (affine; real)
├─ 2×2 Matrix{Float64}
└─  (affine; real)
└─ + (affine; real)
├─ …
└─ …
julia> problem = satisfy(constraint)
minimize
└─ 0
subject to
└─ sdp constraint (affine)
└─ + (affine; real)
├─ 2×2 Matrix{Float64}
└─  (affine; real)
└─ …
status: `solve!` not called yet
julia> solve!(problem,() > SCS.Optimizer(verbose=false))
julia> problem.status
OPTIMAL::TerminationStatusCode = 1
julia> X.value
2×2 Matrix{Float64}:
1.83577 2.88281
2.88281 11.9403
julia> eigvals(X.value)
2element Vector{Float64}:
1.071158392495276
12.704883497869783
Someone asked this yesterday on StackOverflow: optimization  Julia JuMP StackOverflow Error when defining a constraint on linear matrix inequalities  Stack Overflow
It’s a bug: https://github.com/jumpdev/MutableArithmetics.jl/issues/86
As a workaround, split it into an expression first:
@expression(model, ex, A' * P + P * A)
@SDconstraint(model, ex <= Q)
If you update your packages, is this fixed now? If not, can you provide the output of ] st m
.
The problem is fixed after the update. After the update, MutableArithmetics
package is upgraded from v.0.2.14 to v.0.2.15. The problem is fixed already, but in case you are interested, here is the output of ]st m
(@devenv) pkg> st m
Status `~/.julia/environments/devenv/Manifest.toml`
[c3fe647b] AbstractAlgebra v0.15.1
[621f4979] AbstractFFTs v1.0.1
[537997a7] AbstractPlotting v0.15.27
[1520ce14] AbstractTrees v0.3.4
[79e6a3ab] Adapt v3.3.0
[27a7e980] Animations v0.4.1
[c7e460c6] ArgParse v1.1.3
[ec485272] ArnoldiMethod v0.1.0
[4fba245c] ArrayInterface v3.1.7
[4c555306] ArrayLayouts v0.6.5
[bf4720bc] AssetRegistry v0.1.0
[13072b0f] AxisAlgorithms v1.0.0
[aae01518] BandedMatrices v0.16.8
[6e4b80f9] BenchmarkTools v0.5.0
[9e28174c] BinDeps v1.0.2
[b99e7846] BinaryProvider v0.5.10
[ad839575] Blink v0.12.5
[8e7c35d0] BlockArrays v0.15.2
[a74b3585] Blosc v0.7.0
[764a87c0] BoundaryValueDiffEq v2.7.1
[e1450e63] BufferedStreams v1.0.0
[fa961155] CEnum v0.4.1
[15fcbb24] CGAL v0.4.0
[5d742f6a] CSVFiles v1.0.0
[49dc2e85] Calculus v0.5.1
[324d7699] CategoricalArrays v0.9.5
[c45f814d] Causal v0.3.0 `~/.julia/dev/Causal`
[d360d2e6] ChainRulesCore v0.9.37
[fe2786b3] ChaoticCommunications v0.1.0 `~/.julia/dev/ChaoticCommunications`
[aaaa29a8] Clustering v0.14.2
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[c3611d14] ColorVectorSpace v0.8.7
[5ae59095] Colors v0.12.7
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[bbf7d656] CommonSubexpressions v0.3.0
[34da2185] Compat v3.27.0
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(@devenv) pkg>
Great! I had fixed this a while ago, it just needed a new release.
Dear Friends
I need to solve an LMI as follows
As I am a beginner to Julia I do not know how to solve it
(build it in Julia context) and which commands are appropriate.
here is the problem
I need to find P
I appreciate your help with this.
Well, I think that the reason why you were directed by @rafael.guerra from your original post to this post was not exactly to replicate your question here but rather to have a look at what had been discussed here. The discussion here turns out to be relevant for your need. Have you at least tried and struggled? Where exactly have you struggled?
But let me restate here for convenience: you have two options: either use JuMP or Convex to formulate your LMI feasibility problem. For this particular problem both do the job. The documentation for both contains a sufficient description how to define and solve semidefinite problems, in this case just LMI feasibility problems. Examples of usage for both packages are also above in this thread. It should not be difficult to massage those code snippets so that they address your problem.
I am eager to help you once you encounter problems.
At minimum, please, paste here a code for generating those matrices A
, B
, Q
and R
(inside triple quotes).
Two notes. First, as this LMI corresponds to the standard discretetime LQoptimal regulation (aka LQR) problem  it is just the discretetime algebraic Riccati equation (ARE) rewritten into an LMI format , it is quite vital to specify that the matrices Q
and R
are at least positive semidefinite (R
even positive definite), right? Without it the problem does not have much use, right? This is an important part of the problem statement.
Second, since the modeling (and solving) packages that I have listed above can combine individual LMIs, it may be more convenient to split that single LMI into two: the first will contain the left upper 2x2 block, the second will be just the constraint that P
is positive (semi?)definite. The softwares allow imposing constraints directly on the variables (see the examples above in this thread), hence you will only have to encode a single 2x2 matrix LMI constraint.
Well, although it is a different problem from what you originally posted, the approach to its solution remains the same  just assemble a large matrix from individual blocks. Below I show the code for the original problem:
using Convex
using SCS
n = 3
m = 2
A = rand(n,n)
B = rand(n,m)
Q = diagm([1.0, 10.0, 100.0])
R = diagm([1.0, 10.0])
P = Semidefinite(n)
constraint = ([R+B'*P*B B'*P*A; A'*P*B A'*P*A+QP] ⪰ 0)
problem = satisfy(constraint)
solve!(problem,SCS.Optimizer)
P.value
Note: the inequalities in the code are nonstrict while your problem asked for strict inequalities. This can be accomplished by including some nonzero term on the right hand side. Typically an identity matrix is used.
What worries me, however, is that in your new problem the matrix is not symmetric. And symmetry is typically assumed in LMIs. Is your matrix correct?
Thank you so much for your reply
Yes I have checked it but it seems to be strange obtaining
If this be the only result (I mean the obtained nonesymmetric matrix)
Then this means that it can not be solved by the current
LMI methods? As they are for symmetric matrices?
the negative definiteness of Q will not affect the solution or the solver?
I am afraid I do not quite get what you mean. So, you started with a matrix defining a positive (or negative, it does not matter, just multiply by 1) definite form and you factorized it and now you want to enforce the (semi)definiteness constraint on the factor? Why? What is actually your original problem?
Aha, it was a mistake I made while I was posting here the first problem was not my problem I am really sorry for this mistake.
In my problem, all the matrices except Q are semipositive definite (positive definite) and I need to find P. (being positive definite or semi positive definite does not hurt the result which I am going to make and it depends on the rest part that which one is better I can take them positive definite)
If we multiply Q by a 1 how will problem change?
So this asymmetric Matrix inequality can be solved by the LMI solvers offered by Julia?
My original problem is proving (negative definiteness of \Lambda)
So if P is positive definite Must I write :
P = Definite(n)
In the constraints
Must I put
constraint = (\Lambda<0)?
. I also thought of developing a numerical method if it is possible and if this matrix is not an LMI
but it is not my main problem during my study.
Thank you so much
I am afraid I am not able to follow you. Now you start talking about some \Lambda
variable, which appeared in none of your previously stated problems…
A few general remarks:

I can hardly think of a need to analyze positive (semi) definiteness of a matrix while not assuming a symmetry. At least in control theory (which is where your problems are obviously coming from), these matrices define quadratic matrix forms and even if you somehow start with a nonsymmetrix matrix
P
, you can symmetrize it through(P+P')/2
with no impact on the value of the quadratic form. Thats why you will barely encounter a discussion of nonsymmetric matrices in the LMI/SDP literature. 
Whether you can use the function
Positive()
, you can easily check by yourself. You do not need discussion forum for that, do you? Just read the documentation: for Convex.jl package at Basic Types · Convex.jl. For JuMP. jl package at Variables · JuMP. 
In order to increase your chances of being helped by somebody here, please read and follow the instructions at Please read: make it easier to help you. In particular (but not only) the item 2 on inserting the code into your posts.
My dear friend thank you so much for your notes.
As Julia is ratherly young programming language (but at the same time really effective) all sources
does not have examples that can cover almost every corner and this made me ask.
sorry if they are too primitive maybe.
Again I really appreciate your help.