Does anyone knows how can I find this gradient?

Question

enter image description here

Does anyone know how can I start to find the gradient of this function? Any help or hint would be appreciated.

Answer 1

The explicit form of the function is:

$$ f(x_1,x_2,...,x_n) = \sum_{i=1}^m \sqrt{ \left( \sum_{k=1}^n A_{ik} \thinspace x_k \right)^2 + \epsilon}$$

Applying the chain rule we have:

$$(\nabla f(x_1,x_2,...,x_n))_j = \sum_{i=1}^m - \frac{2 \thinspace A_{ij} \left( \sum_{k=1}^n A_{ik} \thinspace x_{k} \right)}{2 \sqrt{ \left( \sum_{k=1}^n A_{ik} \thinspace x_k \right)^2 + \epsilon}} \quad \forall \thinspace j \in \{1,...,n\}$$