How to eliminate quadratic form of -log(det( X*A*X^T + B))

Hello. I have a term in my objective that is -log(det(X*A*X' + B)) where X is the optimization variable (matrix), and A and B are positive definite matrices. I know how to eliminate the quadratic form when B = 0, but not when B is nonzero. Any ideas?

This is non-convex and non-concave even in the scalar case.

Consider the special case where A = B = 1, the expression is then -log(X^2+1). Its 2nd derivative at X = 0.5 is 0.96, and at X = 2 is -0.24. So it’s neither convex nor concave.

So my idea is you should first prove convexity (or concavity), as discussed in
Why isn't CVX accepting my model? READ THIS FIRST! .

Ahh… thanks Mark. I guess it is only convex when B = 0. Ill have to look at formulating it differently then.