`(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 + B*Y)’*(pi_ss*Ps+pi_su*Pu)*(A/Ps+B*Y) - 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

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.

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

[inv(pi_ss*Ps + pi_su*Pu) (A/Ps + B*Y) ; (A/Ps + B*Y)’ inv(Ps)] > 0

to get (A/Ps + B*Y)’ * (pi_ss*Ps + pi_su*Pu) * (A/Ps + B*Y) - inv(Ps) < 0

because Ps as well as (pi_ss*Ps + pi_su*Pu) are symmetric matrices

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