# why sum of squares equals to assign the variable evenly in linear programming

I have a quesiton about linear programming.

My objective is trying to make the utilization evenly.

Example

$U_{1}, U_{2},...,U_{k}$

$\sum_{i=1..k} U_{i} = C$

$C$ is some constant. $U_{i}$ is our assignment.

$Objective:$ I want $U_{i}$ are as even as possible.

My question is that is the Objective be the same as minimizing $\sum_{i=1...k} (U_{i})^{2}$

If it does, then why? Is there proof of this?

Reference: My original problem is Load balancing in Multi-commodity flow problem https://en.wikipedia.org/wiki/Multi-commodity_flow_problem

• The function you propose does not fit the linear framework. You could minimize $\sum_i |U_i - C/k|$, which can be rephased with linear inequalities. – LinAlg Jun 21 '18 at 21:54