Problem with the solution of an LMI (Convex Optimization Problem) - No suitable solver found.

[Abdur Rehman]

Hello,

I am facing a problem with the solution of an LMI. 

Following is the LMI along with its detail.

Where, 

A(hat) = State matrix (it is of the size 6x6 in my case)

B(hat) = input matrix (it is of the size 6x2 in my case)

1. Lyapunov matrices:
       P = PT > 0  (as you can see it is symmetrical)

2. There are two slack variables:
       G and Y

3. a positive scaler:

       e(epsilon) > 0

Problem:

I tried to solve this LMI, but I think there might be a problem with my approach. I tried almost every solver i.e., sdpt3, mosek, fmincon, sedumi etc. Every time it says, "solver not applicable".

Here is my approach to solve this problem:

Code:

P=sdpvar(nx,nx,'symmetric');   %6x6

G=sdpvar(nx,nx,'full');   %6x6     Slack Variable

Y=sdpvar(nu,nx,'full');    %6x2     Slack Variable 

e=sdpvar(1);           %A scaler

%This is the LMI

F1 = [Ahat*G+G'*Ahat'+Bhat*Y+Y'*Bhat'  (P-G+e*(G'*Ahat'+Y'*Bhat'))';   

      P-G+e*(G'*Ahat'+Y'*Bhat')         -e*(G+G')];

Constraints = [e>=0.0000000001*1 F1<=.00000000001 -P<=0.000000001];       

 

options = sdpsettings('savesolveroutput',1,'verbose',1,'solver','mosek','debug',1);

optimize(Constraints,e,options);

    

I would be grateful to you, if you can identify my fault in the above approach.

Kind Regards

Abdur Rehman.

[Johan Löfberg]

you have a product between e and G, hence F1 is not an LMI.

https://yalmip.github.io/command/bisection/

https://yalmip.github.io/example/decayrate/

and you dont want F1<=eps, you want F1 <=eps*I etc.

[Johan Löfberg]

and I'm just guessing it is quasi-convex. You have to prove that to use bisection. otherwise, just grid in e

[Abdur Rehman]

Thank you for the help.

Actually e is a scaler here. So a positive scaler thing is being multiplied by Matrix G. I think that doesn't make it Bi-linear matrix then. 

Can you please guide me, How should I define a scaler here. 

I have used sdpvar command to define this scaler. (is it the wrong way to define the scaler?)

In code:

e=sdpvar(1);           %A scaler

Moreover, I also have guessed that I need to minimize e. Actually I need slack variables i.e., G and Y for my calculations. Thats why I put e it in place of my objective.
optimize(Constraints,e,options);

Regards

[Abdur Rehman]

This screen shot is from the Yalmip website.

[Johan Löfberg]

Then you think wrong.

[Johan Löfberg]

I have no idea what you are trying to say with that. A matrix of size 1x1 is called a scalar, yes.

[Abdur Rehman]

Thank You for your help sir,

I think I got a clue.

Please see the attached research paper, page 2, Lemma 4:

I have now modified the code and set 

e=1    as a fixed value. and now its working. 

Can you please suggest me, what could be the best value for e.

Once again Thank you.

Regards.

[Johan Löfberg]

As I said, you either use bisection (if quasi-convex), or you simply grid in e and look for the smallest possible which still renders the problem feasible

[Abdur Rehman]

Actually I don't know how to grid in e.
How can I grid in e?

The other option of using Bisection is ruled out, since I am sure the problem is not quasi-convex.

Regards

[Johan Löfberg]

you simply fix e and try different values, and pick the best. 

for e = 0:0.01:10

...

end

to speed up things, you can use the optimizer command with e as a parameter

https://yalmip.github.io/command/optimizer/