If I have a constraint like $y_1 + y_2 +\dots + y_n = k$ for positive integers $x_i$, how would I minimize $$\quad\frac{x_1 }{ y_1} + \frac{x_2 }{ y_2} + \frac{x_3 }{ y_3} + \dots + \frac{x_n }{ y_n}$$ if all of the $a_i$ are given?

One can assume that $n\le5\cdot10^5$ and $k\le10^{12}$.

If I have a constraint like $y_1 + y_2 +\dots + y_n = k$ for positive integers $y_i$, how would I minimize $$\quad\frac{x_1 }{ y_1} + \frac{x_2 }{ y_2} + \frac{x_3 }{ y_3} + \dots + \frac{x_n }{ y_n}$$ if all of the $x_i$ are given?

One can assume that $n\le5\cdot10^5$ and $k\le10^{12}$.

If I have a constraint like $x_1 + x_2 +\dots + x_n = k$ for positive integers $x_i$, how would I minimize $$\text{minimize}\quad\frac{a_1 }{ x_1} + \frac{a_2 }{ x_2} + \frac{a_3 }{ x_3} + \dots + \frac{a_n }{ x_n}$$ if all of the $a_i$ are given?

One can assume that $n\le5\cdot10^5$ and $k\le10^{12}$.

If I have a constraint like $y_1 + y_2 +\dots + y_n = k$ for positive integers $x_i$, how would I minimize $$\quad\frac{x_1 }{ y_1} + \frac{x_2 }{ y_2} + \frac{x_3 }{ y_3} + \dots + \frac{x_n }{ y_n}$$ if all of the $a_i$ are given?

One can assume that $n\le5\cdot10^5$ and $k\le10^{12}$.

If I have a constraint like $y_1 + y_2 +\dots + y_n = k$ for positive integers $y_i$, how would I minimize $x_1 / y_1 + x_2 / y_2 + x_3 / y_3 + \dots + x_n / y_n$, if all of the $x_i$ are given?

If I have a constraint like $x_1 + x_2 +\dots + x_n = k$ for positive integers $x_i$, how would I minimize $$\text{minimize}\quad\frac{a_1 }{ x_1} + \frac{a_2 }{ x_2} + \frac{a_3 }{ x_3} + \dots + \frac{a_n }{ x_n}$$ if all of the $a_i$ are given?

One can assume that $n\le5\cdot10^5$ and $k\le10^{12}$.