# Is sqrt(a'*X*A*X'*a + b + c) convex in X? If so, how may I eliminate the quadratic form?

Hello. I would like to know if `sqrt(a'*X*A*X'*a + b + c)` is convex in `X` where `a` is a column vector, `A` is a PSD matrix, `X` is my optimization variable (matrix), and `b` and `c` are positive scalars. If it is convex, is there a way to write this in CVX since I assume it will not like the quadratic form?

As you would know from reading the FAQ link previously provided to you, you should be telling us, not asking us, whether something is convex.

In any event, it is convex, and expressible as
`norm([sqrtm(A)*X'*a;sqrt(b+c)])`

Alternatively, if A is actually positive definite (so that `chol(A)` exists) and not merely PSD, you could use `chol(A)` instead of `sqrtm(A)`.

Mark, thanks. I have read that page before (it is probably the most linked page on this forum). Yes, if I had spent a few more hours on my convex analysis studying, I would have probably proven convexity of it and just asked the CVX specific question of: how do I eliminate the quadratic form?

However, as Boyd has said, convex optimization is an art (and science), and I had an artistic intuition that the expression was convex, but I went ahead and threw that question in there to make sure I could get an answer from you gurus.

Excuse me if that feels like exploitation.