I'm currently struggling with an optimization problem. My objective is to find the rotation $X$ that minimize the $l_1$ norm of $Q_{ijk}$ defined as:
$\tilde{Q}_{ijk} = \sum_{p,q,r = 1}^{N} X_{pi}Q_{pqr}X_{qj}X_{rk}$
Thus the objective function is : $|| \tilde{Q}_{ijk}||_1 $
So far I managed to evaluate numerically the gradient of the objective function with respect $X$. Since this operation is quite costly and does not allow to scale the algorithm to larger problems I was wondering how to find an analytical expression of the gradient of the $l_1$ norm of $\tilde{Q}$ with respect $X$.
Many thanks for any help!
Riccardo