I'm looking for standard approach to reformulate this objective function. The aim is to find values of $x_i$ that are close to either $y_i$ or $-y_i$ ($y_i$s are known) in a least-squares sense:
$argmin_{x_i \in \mathbf{x}} \sum\| |x_i| - |y_i| \|^2$
Specifically, can the inner absolute value be rewritten in a simpler way to convert it into a QP?
One way would be to approximate the objective as: $argmin_{x_i \in \mathbf{x}} \sum\| x_i^2 - y_i^2\|^2$ but it is no longer quadratic.