Talk:Presentation
Combinatorial APS?
For some problems that are at least partly discrete, the basic APS applies a quantisation operator. If the quantum is $q_{d}$ for
the dimension $d$, the operator is defined by
$$
x_{i,d}\leftarrow q_{d}\left\lfloor \left(0.5+x_{i,d}/q_{d}\right)\right.\tag{1}
$$
where $\left\lfloor \right.$ is the floor operator.
Now, for combinatorial problems, it is certainly better to define specific operators for Expansion, Contraction, and Local search, similarly to what has been done, for example, for PSO. It means that we have to define a distance. Let us consider the canonical case in which the search space is the set of permutations of $D$ different elements. As we know, to transform a given permutation $X$ into another one $Y$, there exists a minimum number of transpositions. So we define the quasi-distance between $X$ and $Y$ as this minimum number. Here are a few ideas of how the specific operators could be defined. In what follows, we keep the same adaptive probability $p$ threshold than in standard APS. It means that we have to modify only
Expansion (reflection)
We have three "positions" (i.e. permutations), $X_{best}$, $X_{worst}$, and $X_{worst2}$ (the second worst), we have to combine them. In standard this combination is $$ Y=X_{worst2}+(X_{worst}-X_{best}\tag{2}$ $$
It is formally equivalent to the PSO operator "new_position = position + velocity", for the "velocity" is anyway the difference of two positions. We already know how to compute the difference of two permutations, and how to add two permutations. We could apply here these two operators. However, in APS, the above formula is applied in fact only for some dimensions, not for all, according to the probability threshold. In order to "simulate" the same method, we have to apply the combinatorial equivalent formula only on a subset of the dimensions.
Contraction
We "add" the three permutations $X_{best}$, $X_{worst}, and $X_{worst2}$, and we "multiply" the result by 1/3.
Local search
The easiest one: we just have to apply a "small" number of transpositions.
