What is the minimum value of this summation with given constraints?

[lfdahl]

Find the minimum of the sum: \[\sum_{i=1}^{5}x_i\], where $x_i \ge 0$ and $\sum_{i<j}|x_i-x_j| = 1.$

 

[I like Serena]

Find the minimum of the sum: \[\sum_{i=1}^{5}x_i\], where $x_i \ge 0$ and $\sum_{i<j}|x_i-x_j| = 1.$

Hey lfdahl! ;)

Here's my attempt.

Spoiler

WLOG we can reorder any choice of variables, such that $x_1 \ge x_2 \ge x_3 \ge x_4 \ge x_5$.
Then we get:
$$\sum_{i<j}|x_i - x_j| = \sum_{i<j} x_i - x_j = 4x_1 + 2x_2 + 0x_3 - 2 x_4 - 4 x_5 = 1$$
To most effectively minimize $\sum x_i = x_1 + x_2 + x_3 + x_4 + x_5$, we pick $x_1=\frac 14$ and $x_2=x_3=x_4=x_5=0$.
So the requested minimum is $\frac 14$.

 

[lfdahl]

Hey lfdahl! ;)

Here's my attempt.

Spoiler

WLOG we can reorder any choice of variables, such that $x_1 \ge x_2 \ge x_3 \ge x_4 \ge x_5$.
Then we get:
$$\sum_{i<j}|x_i - x_j| = \sum_{i<j} x_i - x_j = 4x_1 + 2x_2 + 0x_3 - 2 x_4 - 4 x_5 = 1$$
To most effectively minimize $\sum x_i = x_1 + x_2 + x_3 + x_4 + x_5$, we pick $x_1=\frac 14$ and $x_2=x_3=x_4=x_5=0$.
So the requested minimum is $\frac 14$.

Thankyou, I like Serena, very much for your answer!