Hi!
As you are trying an optimization problem where you have the conditions A – BC – 1D < 0, C > 0 as a constraint. You are interested to know how you can convert it into LMI form using the Schur complement.
The Schur complement is a powerful tool for dealing with matrix inequalities and can be used to convert your constraint into Linear Matrix Inequality (LMI) form.
Given a block matrix of the form:
[A B]
[C D]
where A is invertible, the Schur complement of A in this matrix is defined as D−CA^-1B.
In your case, you have the inequality A−BC^−1D < 0, which can be rewritten as A−BC^−1D=−S < 0, where 'S' is the 'Schur' complement.
The inequality C > 0 ensures that C is positive definite, which is a common requirement in LMI problems.
So, your constraints can be written in LMI form as S > 0 and C > 0.
Try checking the below links to know more about:
- Schur complement for linear matrix inequality (LMI): https://math.stackexchange.com/questions/4221853/schur-complement-for-linear-matrix-inequality-lmi
- Linear Matrix Inequalities: https://web.stanford.edu/class/ee363/sessions/s4notes.pdf
