Cuckoo search is a bit cuckoo

Excuses for the lame title.

This could be the start of a very extended series of papers.

References

1. Christian L. Camacho-Villalón, Marco Dorigo, Thomas Stützle, An analysis of why cuckoo search does not bring any novel ideas to optimization, Computers & Operations Research, Volume 142, 2022, https://www.sciencedirect.com/science/article/pii/S0305054822000442.
2. Aranha, C., Camacho Villalón, C.L., Campelo, F. et al. Metaphor-based metaheuristics, a call for action: the elephant in the room. Swarm Intell 16, 1–6 (2022)

A strange series

In [1] a question is posed about a somewhat strange series. Let \(i \in \{ 0,\dots,n\}\). Then we require:

• \(\color{darkred}x_i \in \{0,1,2,\dots\}\),
• \(\color{darkred}x_i\) is equal to the number of times the value \(i\) occurs in the series. In mathematical terms, one could say: \[\color{darkred}x_i = |\{j:\color{darkred}x_j=i\}|\] (here \(|S|\) is the cardinality of the set \(S\)).

An example for \(n=4\) is: 

index  0 1 2 3 4 

value  2 1 2 0 0

sum of K largest

Consider a vector of variables \(\color{darkred}x_i\). We want to impose the constraint: \[\text{sum of $K$ largest $\color{darkred}x_i$}  \le 0.5\sum_i \color{darkred}x_i\] I.e. the sum of the largest \(K\) values are less than half of the total. Sometimes this is expressed mathematically as \[\sum_{i=1}^K \color{darkred}x_{[i]} \le 0.5\sum_i \color{darkred}x_i\] where \(\color{darkred}x_{[i]}\) is an ordered version of \(\color{darkred}x_i\) with \(\color{darkred}x_{[1]}\ge\color{darkred}x_{[2]} \ge \color{darkred}x_{[3]} \dots \). There are several formulations for this constraint, all of them interesting, I think.