How have you proven the objective function is concave and the constraints convex?
According to my calculations, the objective function is indefinite. Let X be 2 by 2, R = ones(2), which is psd with diagonal elements = 1, and is rank deficient. The Hessian of log(det(X'*R*X + eye(2))) evaluated at X = ones(2) has 2 zero eigenvalues, one positive eigenvalue, and one negative eigenvalue. Hence, is indefinite.