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I have the following optimal problem. Let $u_i \in \mathbb{C}$. Let $\lambda_i$ be unknowns and $\sum_{i=1}^n \lambda_i = F$ for some given $F \in \mathbb{C}$. Find $\lambda_1, \ldots, \lambda_n$ such that $\prod_{i=1}^{n} (u_i-\lambda_i)$ is maximal. I try to use Lagrange multiplier method. Let $G(\lambda_1, \ldots, \lambda_n, \lambda) = \prod_{i=1}^n (u_i-\lambda_i) - \lambda (\sum_{i=1}^n -F)$ and compute derivatives. But I cannot solve the equations. Do I need to use the function $\sum_{i=1}^{n} \log (u_i-\lambda_i)$ instead of $\prod_{i=1}^{n} (u_i-\lambda_i)$? Thank you very much.

Edit. $u_i, F \in \mathbb{R}$ are given.

The problem is solved.

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Notice $\prod_i(u_i-\lambda_i)$ is a complex-valued function, so it doesn't make sense to speak of it being maximal. Perhaps you're missing absolute value signs? Either way, I feel a simple argument could be used to show there is no maximum. –  anon May 14 '12 at 15:22
    
@anon, sorry, I made a mistake. They are reals. –  LJR May 14 '12 at 15:25
    
So you want all of $u_i,\lambda_i,F$ to be real, right? Still, it's a (multivariable) polynomial. They are only bounded if they're constant. –  anon May 14 '12 at 15:28

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