Merge branch 'main' of github.com:PdIPS/CBXpy

README.md

@@ -22,13 +22,7 @@ A Documentation is available at [https://pdips.github.io/CBXpy](https://pdips.gi
 
 # What is CBX?
 
-For an ensemble of particles $x = (x_1, \ldots, x_N)$ the basic update step reads
-
-$$
-x_i \gets x_i - dt\, (x_i - c(x))
-$$
-
-CBXPy is a python package implementing consensus based particle schemes. Originally designed for optimization problems
+Originally designed for optimization problems
 
 $$
    \min_{x \in \mathbb{R}^n} f(x),
@@ -37,12 +31,16 @@ $$
 the scheme was introduced as CBO (Consensus Based Optimization). Given an ensemble of points $x = (x_1, \ldots, x_N)$, the update reads
 
 $$
-x_i \gets x_i - \lambda\, dt\, (x_i - c(x)) + \sigma\, \sqrt{dt} |x_i - c(x)| \xi_i
+x_i \gets x_i - \lambda\ dt\ (x_i - c(x)) + \sigma\ \sqrt{dt} |x_i - c(x)| \xi_i
 $$
 
 where $\xi_i$ are i.i.d. standard normal random variables. The core element is the consensus point
 
-$$c(x) = \frac{\sum_{i=1}^N x_i\, \exp(-\alpha\, f(x_i))}{\sum_{i=1}^N \exp(-\alpha\, f(x_i))}.$$
+$$
+\begin{align*}
+c(x) = \left(\sum_{i=1}^{N} x_i\ \exp(-\alpha\ f(x_i))\right)\bigg/\left(\sum_{i=1}^N \exp(-\alpha\ f(x_i))\right).
+\end{align*}
+$$
 
 with a parameter $\alpha>0$. The scheme can be extended to sampling problems  known as CBS, clustering problems and opinion dynamics, which motivates the acronym 
 **CBX**, indicating the flexibility of the scheme.