# How do I write this LMI constraint in cvx format?

`(A*inv(Ps) + B*Y)' * (pi_ss*Ps + pi_su*Pu) * (A*inv(Ps) + B*Y) - inv(Ps) < 0` for Y (1x2 matrix)
Where A is a 2x2 matrix , B is a 2x1 matrix l, Ps and Put are positive definite matrices and pi_ss and pi_su are 0.1 and 0.9 resp.

How would I write this expression as a constraint in cvx format?

What are the optimization variables?
Edited: See below.

my variable is Y
here is my code:

cvx_begin sdp quiet
variable Y(1,2)
(A/Ps + BY)’(pi_ssPs+pi_suPu)(A/Ps+BY) - inv(Ps)<= -eps*eye(2);
cvx_end

I believe that my quadratic form is a matrix and not a scalar that why im getting the error message. however i do not know how to proceed further.

A, B, Ps, Pu, pi_ss and pi_su are known. Therefore it is a LMI

Edited: See below. …

Then could you please suggest how I would be able to solve this inequality problem? I’m really having trouble figuring that out.

if you have a BMI, which is non-convex, you can try using PENLAB or BMIBNB under YALMIP.

Okay…thank you so much!

can the inequality be written in schur complement form as such:

[inv(pi_ssPs + pi_suPu) (A/Ps + BY) ; (A/Ps + BY)’ inv(Ps)] > 0

to get (A/Ps + BY)’ * (pi_ssPs + pi_suPu) * (A/Ps + BY) - inv(Ps) < 0
because Ps as well as (pi_ssPs + pi_suPu) are symmetric matrices

I did not get enough sleep. Yes, I believe so.