BMI with Yalmip

[Anh Lam Do]

Dear everyone, Dear Yohan Löfberg,

I have two (different) BMI problems as follows (P, K and \gamma are unknown)

Problem 1

Or

Problem 2

I would like to know if we could solve this kind of problem with Yalmip?

Or, anyone knows a trick to turn them into LMI ones?

Thank you very much in advance.

Kind regards,

Lam

[Johan Löfberg]

First one looks like standard case where you apply a congruence transformation followed by a variable change. I recommend you to read the standard book http://stanford.edu/~boyd/lmibook/ as a start

The second case looks worse and will probably not be applicable to those tricks

[Anh Lam Do]

Thank you for your prompt reply.

It means also that these problems can not be solved by Yalmip as I understand.

Kind regards,

Lam

[Johan Löfberg]

The first one is (probably) trivially solved using YALMIP+LMI solver as it (probably) can be written as an LMI

The second one can be attacked using YALMIP also, but since it is a BMI you have absolutely no guarantees as it is a hard problem. YALMIP has support for the local BMI solver penlab, and has a built in global solver for nonconvex problems called bmibnb, or you can write your own heuristics in yalmip using linearization, iterative optimization, trust-regions, path-following or what ever you can come up with.

[Anh Lam Do]

Thank you very much. I'll try the BMI solvers with yalmip.

By the way, I tried to use congruence transformation but it seems to be not easy. Both of the following don't work.

 or

[Johan Löfberg]

Congruence with (P^-1,I,I) works as far as I can see. This is basically standard case from the book.

[Anh Lam Do]

Dear Yohann,

After pre and post multiplying the first BMI with   (P^-1,I,I)

I obtain the following for which there is no appropriate variable change to obtain an LMI. 

[Johan Löfberg]

My guess is your semidefinite constraint is wrong to begin with. Aren't you doing something like L2 state feedback synthesis, i.e. (7.24)

[Anh Lam Do]

You're right. But this is (H infinitive) L2 synthesis of state feedback with (all states') measurement errors. I think because of this, I have difficulty obtaining an LMI.

[Johan Löfberg]

Huh? What is the difference?   As a start to find your error, you would never start from Problem 1. What did yo apply a Schur complement on to arrive at Problem 1.

[Anh Lam Do]

In fact, i have a problem like this:

To guarantee a bounded Hinfiniti norm of Transfer function from e_x to z, one has

At (14), i'm stuck with a BMI.

Thanks in advance for your advice.

[Johan Löfberg]

ok, that looks nasty.

[Anh Lam Do]

totally agree!