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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?

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    Looking at [a plot of your function for $n=2$ and $M=e$](http://www.wolframalpha.com/input/?i=plot%20log%281%2blambda%5E2%29%20%2b%20log%281%2b%28e-lambda%29%5E2%29%20for%20lambda=0..e), it doesn't seem to me like there would be any way to cast this as a convex problem. What you might be able to do is prove that local maxima can only occur at $\big(\frac Mn, \frac Mn, \ldots, \frac Mn\big)$ and at permutations of $(M, 0, \ldots, 0)$, and then figure out which would be the global optimum for specified $M$ and $n$.2012-04-22
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    On the other hand, if you have strict positivity constraints on $\lambda_i$, then you may not always have a solution at all when $M$ is small. This is like minimizing $x$ subject to $x>0$: the global optimum cannot be attained because your domain is not closed.2012-04-23
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    Thanks, you are right. The problem in its original form is: "For a given $M$, assuming that we know the correct number $n$ of non-zero $\lambda_i$'s, maximize ...". So we can assume that $M$ has appropriate value.2012-04-23

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