Grasshopper Optimization Algorithm I am currently reading a paper on a meta-heuristic called 'Grasshopper Optimization Algorithm'.
The main idea of the algorithm is to utilize the social behavior of grasshoppers in a swarm to solve optimization problems. The distance to other grasshoppers in the swarm is used to determine, if a grashopper is repelled (exploration) or attracted (exploitation) from grasshoppers in its proximity.
However I am not sure if I understood the first part of the proposed formula correctly:
$$X_i^d = c\left(\sum\limits_{j=1,j\neq i}^Nc\frac{ub_d-lb_d}{2}s\left(\lvert x_j^d-x_i^d\rvert\right)\frac{x_j-x_i}{d_{ij}}\right) + \widehat{T}_d$$
$X_i^d$ calculates the next position of a search agent (grasshopper) with respect to the postion of the other search agents and the best found solution so far. So the next position of a grasshopper depends on its own position, the positions of all other grasshoppers in the swarm and the best solution found so far.
$\widehat{T}_d$ denotes the best solution found so far.
$c$ is a decreasing coefficient that regulates if the search agents explore or exploit in the search space. With increasing iteration count  $c$ decreases and the search agents tend to exploitation.
$s\left(\lvert x_j^d-x_i^d\rvert\right)$
The part in brackets calculates the distance between grashoppers.
$s$ is a formula which takes this value and determines if the search agents should explore or exploit.
$\frac{x_j-x_i}{d_{ij}}$ is a unit vector, where $d_{ij}$ denotes the distance between two grasshoppers.
It states that $ub_d$ is the upper bound and $lb_d$ is the lower bound in the $d$-th dimension.
It further states that the part $c\frac{ub_d-lb_d}{2}$ linearly decreases the space that grasshoppers (search agents) should explore and exploit.
I dont undertsand what is meant with 'upper and lower bound in the $d$-th dimension'.
Upper and lower bound for what exactly?
Can somebody help?
Information extracted from this paper:
Grashopper Optimization Algorithm
I suspect they probably mean
$$ub_d = \max_i x_i^d$$
where $x_i^d$ is the $d$h coordinate of $x_i$; and similarly for $lb_d$, but using $\min$ instead of $\max$.  But I don't know -- I am just speculating based on context.