I want to solve the following problem: Maximize $\sum_{i=1}^n\log(1+\lambda_i^2)$ subject to $\lambda_i >0$ and $\sum_{i=1}^n\lambda_i = M$. I was wondering how I could cast it as a convex problem.
One thought came to mind of treating $\lambda_i^2$ as variables instead of $\lambda_i$. To modify the sum constraint, I could only think of using the Cauchy-Schwarz inequality to get $\sum_{i=1}^n\lambda_i^2 \geq \frac{M^2}{n}$. (Additionally, we always have: $\sum_{i=1}^n\lambda_i^2 \leq M^2$.)
My guess (or hope) is that the solution is $\lambda_i = \frac{M}{n}$ for all $i$. Can anyone see this?