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I have a simple optimization problem; I already know the answer but I don't know how to prove it! Assuming we have a series of positive real numbers, $$ x_1, x_2, x_3, ..., x_n, $$ we want to find a series $$ y_1, y_2, y_3, ..., y_n $$ that minimizes the goal function $$ \sum_{i=1}^{n} \frac{x_i^2}{y_i}$$ while ensuring $$ \sum_{i=1}^{n} y_i = 1 .$$

I already know the answer is $$ y_i = \cfrac{x_i}{ \displaystyle \sum_{j=1}^{n} x_j} ,$$

but I don't know the way to reach the answer. Any help would be appreciated.

Goentagen
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Amir Hashemi
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1 Answers1

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The lagrangian

$$ L = \sum_{i=1}^{n} \frac{x_i^2}{y_i}+\lambda\left(\sum_{i=1}^{n} y_i - 1\right) $$

has as stationary points, the solutions for

$$ \cases{ \nabla_{y_i}L = -\frac{x_i^2}{y_i^2}+\lambda=0\\ \sum_{i=1}^{n} y_i - 1 = 0 } $$

From the first equation we have

$$ -\frac{x_i^2}{y_i}+\lambda y_i = 0\Rightarrow -\sum_{i=1}^{n} \frac{x_i^2}{y_i}+\lambda = 0 $$

then

$$ -\frac{x_i^2}{y_i^2}+\sum_{k=1}^{n} \frac{x_k^2}{y_k}=0 $$

Now you can check if your solution works.

Cesareo
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