Q1) Random Search

Recall :

\[ W^k=W^{k-1}+\alpha d^{k-1} \]

where \(||d^{k-1}||=1\) (ie. normalized) and we just sample from 0 to \(2\pi\) (0 to 360 degrees).

Meaning,

\[ d^{k-1}=\begin{bmatrix}\cos(\theta) \\ \sin(\theta)\end{bmatrix} \]

\[ X \sim \text{Unif}(0,2\pi) \]

Helper function) Directions

random_direction_radians <- 
  function(num_samp){runif(num_samp, min = 0, max = 2 * pi)}

random_direction_radians

## function(num_samp){runif(num_samp, min = 0, max = 2 * pi)}