On using trace_inv function


My objective function is convex, It is trace of inverse of symmetric positive definite matrix, Can someone help me in writing it in DCP format. Below is my function, P is a diagonal matrix (TxT), Q is a (TxN) matrix. I have to minimize the function over P and Q. Here α is a scalar and I is an NxN identity matrix. R is a Hermitian matrix (R’=R). I have two constraints norm(P,‘fro’)^2<=P_total and norm(Q,‘fro’)^2<=Q_total.
P is also an hermittian matrix.

I was getting “Only scalar quadratic forms can be specified in CVX” error.

That is neither convex nor concave. The product of variables ruins convexity. Note that trace_inv(X) is convex for an affine positive semidefintie argument, which is not the case for your expression.

Consider the scalar case. Let R = \alpha = 1. The expression is then 1/(1+P^2*Q^2); That is neither convex nor concave. For example, its Hessian at P = 1, Q = 0.5 has one positive eigenvalue and one negative eigenvalue.

Thank you very much. If the scalar case is not convex, can we always say that the function in higher dimension is also not convex? and for my problem I am allowed to find a convex upper bound, Are there any references for that?

If something is not convex in 1 dimension, then why should it be convex in higher dimensions?