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Negative semidefinteness and schur complement

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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?

Answers (1)

Manikanta Aditya
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.
Try checking the below links to know more about:

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