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.