Maximizing the logarithm of the determinant plus the trace of the inverse of a matrix How can I solve the following unconstrained optimization problem in $\alpha \in \mathbb R$?
$$\text{maximize} \quad \log \det( \alpha A+I) + \mbox{tr} \left((\alpha A + I)^{-1} \alpha A \right)$$
where $A$ is a positive semidefinite matrix and $I$ is the identity matrix.
 A: I assume you want $\alpha$ to be non-negative?  It looks like your objective function is concave.  If so, the minimum is achieved at the extreme points of the allowed range for $\alpha$.  That is, at $\alpha=0$ or $\alpha=\infty$.  Obviously $\alpha=\infty$ isn't it, so the answer is $\alpha=0.$
As a Rodrigo de Azevedo  points out, the maximum is not attained.  If your matrix has eigenvalues $\lambda_i$, the  objective function is 
$$ \sum_{i=1}^n \left( \log(\alpha \lambda_i+1) + \frac{\alpha \lambda_i}{\alpha \lambda_i + 1} \right) .$$ 
Its derivative, a rational function, is easy to write down, and is manifestly positive-valued for all $\alpha$.
A: Define a new matrix variable $$X=I+\alpha A$$
and note the following relationships
$$\eqalign{
\alpha A &= X-I \cr
\alpha AX^{-1} &= (X-I)X^{-1} = I-X^{-1} \cr
 dX &= A\,d\alpha \cr
}$$
Now write the objective function in terms of $X$
$$\eqalign{
 f &= \log\det X + {\rm tr}(\alpha AX^{-1}) \cr
   &= \log\det X + {\rm tr}(I) - {\rm tr}(X^{-1}) \cr
}$$ The function is unbounded as $\alpha\to\infty\,\,$ since the trace term tend to zero while the log-det term increases.
Now find the differential and gradient
$$\eqalign{
df &= d{\rm tr}(\log X-X^{-1}) \cr
  &= {\rm tr}((X^{-1}-X^{-2})\,dX) \cr
  &= {\rm tr}((X^{-1}-X^{-2})A)\,d\alpha \cr
  &= {\rm tr}(\alpha A^2X^{-2})\,d\alpha \cr\cr
\frac{df}{d\alpha}&= \alpha\,{\rm tr}(A^2X^{-2}) \cr
}$$
Setting the gradient to zero, means either
$\alpha=0$ or the trace-term is zero.
Since 
$$\lim_{|\alpha|\to\infty} AX^{-1} = \lim_{|\alpha|\to\infty} I\alpha^{-1} = 0$$ one way to drive the trace-term to zero is to let $\alpha$ take on very large values.
It might be possible to find other real values for $\alpha$ which annihilate the trace, but those would depend on the properties of each specific $A$ matrix. 
