# Minimize MAE for sum of powers

The model is $$y_i=\beta\big(x_{i1}^\alpha+x_{i2}^\alpha+...+x_{im_i}^\alpha\big)+\epsilon_i\textrm{ for }i=1, 2, ..., n$$
I want to minimize the MAE, i.e. $$\Sigma_{i=1}^n{\big|y_i-\beta\big(x_{i1}^\alpha+x_{i2}^\alpha+...+x_{im_i}^\alpha\big)\big|}$$
I tried gradient descent, but I'm worried that the solution is not a global minimum.
Could anyone help me?