I have a problem that looks like this:
set of expressions of the form:
$p_i = ax_{i}+bx_{i}^{\frac{2}{3}} $ for $ 1 \leq i \leq n$
and I'm trying to find $ min_{\Sigma x_i=G}\{max_{1\leq i \leq n}|p_i|\} $.
In other words, if we denote $P \in \mathbb{R}^n $ s.t $ p_i = ax_{i}+bx_{i}^{\frac{2}{3}} $, I want to minimize the infinty norm of $ P $, under the constraint of $ \Sigma x_i=G $ and $x_i \geq 0$.
I want to have a close expression and not a numeric approach.
I have tried to solve this in many different ways, starting from lagrange multipliers and trying to proove this is an NP-hard problem (as a polynomial system), and I couldn't.
Any help - solution or a reference will be appreciate.