TITLE: Constrained infinity norm minimization
QUESTION [5 upvotes]: I have a problem like this:
$$\min_x |Ax|_\infty  \text{ s.t. } \sum_i x_i = c$$
That is, I want to find the vector $x$ whose elements sum to a constant $c$ that minimized the infinity norm of $Ax$.
Does this problem have a particular name? I've been unable to find a function to do something like this in things like lpsolve, glpk, etc. Can anyone suggest a c/c++ solver for this problem?

REPLY [2 votes]: As mentioned in the comments, the problem can be restated as
$$\min y \\ s.t. \quad y \geq |Ax|_i \\ \quad \sum_i x_i = c$$
To provide this form to a solver (e.g. lpsolve), we can rewrite the constraint as
$$\min y \\s.t. \quad -y \leq (Ax)_i \leq y, \ i = 1, \dots, m, \\ \quad \sum_i x_i = c$$
that is,
$$\min y 
\\ s.t. -y \leq (Ax)_1 \leq y\\
\quad -y \leq (Ax)_2 \leq y\\
\dots\\
\quad -y \leq (Ax)_m \leq y.\\ 
\quad \sum_i x_i = c
$$
This form actually means that there are N+1 variables ($y \in \mathbb{R}$ and the original $x$'s).