# Optimization of a function of a positive definite matrix and its inverse

This question is a little ill-posed, but I've been playing with some equations and am just wondering if this resembles any known problems that have been solved.

Suppose I have two real, positive definite (square) matrices $\mathbf{A}$ and $\mathbf{C}$, and I wish to find another real, positive definite matrix $\mathbf{B}$ such that $\mathbf{A B} + \mathbf{B}^{-1}\mathbf{C}$ is as close as possible to identity.

I'll entertain any reasonable definition of "close" that makes the problem tractable. Maybe minimization of $\| \mathbf{A B} + \mathbf{B}^{-1}\mathbf{C} - \mathbb{I} \|$ for some choice of norm.

Does anyone have any insight or experience with such a problem?

Thanks,

Nick

• It reminds me of Sylvester's equation. There however we have no inverse. – user35593 Mar 8 '16 at 20:48
• I wonder if the minimization problem could be cast in terms of a Frobenius norm. – Nick Mar 22 '16 at 3:28

You are actually looking to solve the continuous algebraic Riccati equation. For convenience, I will write your $B$ as $X=X^T$. Then the equation you're trying to solve is simply
$$X - XAX + (-C) = 0$$
Or even more explicitly, writing the Cholesky factorization of $A=BB^T$
$$\left( \frac{1}{2}I \right)^TX + X\left( \frac{1}{2}I \right) - XBB^TX + (-C) = 0$$
• Excellent point. In order to get a real symmetric posdef $X$ "close" to solving the ARE, we would have to reformulate the problem into a semidefinite program. – Richard Zhang Mar 11 '16 at 19:20