I am trying to calculate argmax of the following function w.r.t. $x$:
$$ f(x) = \sum \limits_{i=1}^{N} \Big( max (-||y_i - x||_2^2 + \alpha, 0)\Big) $$
where $x$ and $y_i$ are vectors. I have tried to find the x* i.e the argmax by evaluating the derivative and setting it equal to 0, but no luck. I followed this approach:
$$ \frac{\partial f(x)}{\partial x} = \sum \limits_{i=1}^{N} \Big(2(y_i - x) \mathbf{1}(||y_i - x||_2^2 < \alpha)\Big) $$ where $\mathbf{1}(.)$ is the indicator function. By putting the above equation equal to $0$, I should get the argmax, but the above equation seems tricky because of the indicator function.
Any help would be appreciated!
It's better to solve this problem as constrained optimization problem. You can consider the norm squared as the objective function and the constraint as the norm squared should be greater than $\alpha$.
Using KKT multipliers should be straight forward.