# Negative semidefinteness and schur complement

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Arjun M on 21 Apr 2022
Hello all,
I am trying an optimization problem where I have the condition A - BC-1D < 0, C > 0 as a constraint. How can I convert this into LMI form using schur complement?

Manikanta Aditya on 12 Jan 2024
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 DCA^-1B.
In your case, you have the inequality ABC^−1D < 0, which can be rewritten as ABC^−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.