What is the name of the following algorithm? I am  using the following algorithm to solve an optimization problem I have. I was wondering if there is a proper name for this algorithm in the literature. 

Let $f(x)$ be my objective function.

I guess some $x_0$ and see have it performs.

Next, I randomly generate around $x_0$, some $\{x_k \}$ for $k \in [1:K]$  and see how the compare to $x_0$.  If I find  $x_n$ that perform better than $x_0$.
I perform the same procedure again with $x_n$.

I am only starting to learn about algorithms and would be grateful if you point me to the name and some reference on similar algorithms
This is a form of local search.  It is similar to hill climbing.  It's also reminiscent of stochastic hill climbing, as @Juho suggests.