Minimize the following function with integer values with the given constraint.

Question

$$f(b_1,b_2,\ldots,b_m)=(b_1)^2+(b_2)^2+\cdots+(b_m)^2$$ such that $$ b_1+b_2+\cdots+b_m=l$$

where $b_i$'s are nonnegative. $m$ and $l$ are fixed positive integers. We want to minimize this function with integer values.

Answer 1

It's not clear to me that the problem differs from minimizing $b_1^2+b_2^2+\cdots+b_m^2$ (I assume that the $b_3$ in the question is a typo for $b_m$) subject to $b_1+b_2+\cdots+b_m=N$. And it's clear you get the minimum by letting some of the $b_j$ be $[N/m]$ and, if $N/m$ isn't an integer, letting some of the $b_j$ be $1+[N/m]$. This follows from $2a^2<(a-r)^2+(a+r)^2$ and $a^2+(a+1)^2<(a-r)^2+(a+r+1)^2$.

But maybe the idea is that $m$ isn't fixed, and you're allowed to minimize over it, too. So, please edit the body of the question to clarify what it is that you want.