Local minimum of the function

Question

We have a function

$$\sum_{i=1}^{n}x_{i}^{2}=min,\quad\mbox{ subject to }\sum_{i=1}^{n}x_{i}=c,$$

which should have the minimizer

$$\frac{c^{2}}{n}.$$

However, from the Lagrangian

$$\Lambda\left(x_{i},\lambda\right)=\sum_{i=1}^{n}x_{i}^{2}+\lambda\left(\sum_{i=1}^{n}x_{i}-c\right),$$

is

$$x_{i}=-\frac{\lambda}{2}$$

which gives me the different minimizer

$$\frac{c}{n}.$$

Thanks for your help.

Answer 1

Indeed $x_i=\frac{c}{n}$ for each $i=1,2,\ldots n$. But then $$\sum_{i=1}^{n} x_i^2 =\sum_{i=1}^{n} \left(\frac{c}{n}\right)^2=n\cdot \frac{c^2}{n^2}=\frac{c^2}{n}$$ so that you have the correct answer already!

Answer 2

There is no conflict. The minimum value is $\frac{c^2}{n}$, attained at $x_i=\frac{c}{n}$.