The capital allocation framework proposed by [] presents capital allocation principles as solutions to particular optimization problems and provides a general solution of the quadratic allocation problem via a geometric proof. However, the widely used haircut allocation principle is not reconcilable with that optimization setting. In this paper we provide an alternative proof of the quadratic allocation problem based on the Lagrange multipliers method to reach the general solution. We show that the haircut allocation principle can be accommodated to the optimization setting with the quadratic optimization criterion if one of the original conditions is relaxed. Two examples are provided to illustrate the accommodation of this allocation principle.