Sub-populations¶

The population \(\mathbf{P}\) is divided into \(n_s\) sub-populations \(\mathbf{S}_{i=1,\dots,n_s}\) of size \(n_{cs}=n_c/n_s\). The division is obtained by sorting the candidate solutions by fitness and then growing each sub-population around a seed (elite) candidate by adding its nearest neighbours in scaled space.

Scaled distance¶

Neighbours are measured by the Euclidean distance in the scaled search space, so that every dimension contributes equally regardless of its bound width:

\[ d(\mathbf{x}^{1}, \mathbf{x}^{2}) = \sqrt{\sum_{iD=1}^{n_D}\left(\overline{x}^{\,1}_{iD} - \overline{x}^{\,2}_{iD}\right)^2}, \qquad \overline{x}_{iD} = \frac{x_{iD}-LB_{iD}}{UB_{iD}-LB_{iD}} . \]

Sub-population formation¶

The leading candidate solutions of the sorted population \(\mathbf{P}^{sort}\) become the elite "seeds" \(c_{E,S_i}\). Each sub-population is then filled with the \(n_{cs}-1\) closest remaining candidates (by the distance above), which are removed from the pool so they are assigned exactly once.

Algorithm 1 — Dynamic sub-population formation

Input: number of sub-populations \(n_s\) Output: list of sub-populations

for i = 1 .. n_s:
    create empty sub-population S_i
    identify the best CS in P_sort as elite c_{E,S_i}
    add c_{E,S_i} to S_i and remove it from P_sort
    compute distances d of the personal-best positions of the
        remaining CS in P_sort to x_{E,PB,S_i}            (scaled, Eq. 1)
    add the n_cs - 1 closest CS to S_i
    remove those n_cs - 1 CS from P_sort

Why sub-populations help¶

Splitting the population creates several semi-independent search fronts. The sub-population that happens to sit near the global best becomes highly exploitative (its diversity collapses quickly), while sub-populations farther away keep a higher diversity and continue to explore. This division of labour is what the dual-attractor position updates exploit.

Number of sub-populations \(n_s\)¶

DEEM (and DEEMI) allow the number of sub-populations to change over the run, controlled by nswarm_max / nswarm_min (npop_max / npop_min in DEEMI) and the reduction schedule method_subswarm_reduction. The empirical study in the paper found:

\(n_s = 1\) performs worst — sub-populations matter.
For \(n_D = 10\) the algorithm is fairly insensitive to \(n_s\).
For \(n_D = 30\) a linear decrease \(n_s = 10 \rightarrow 4\) is the best overall choice and is the recommended default.

Available reduction schedules (method_subswarm_reduction): constant, linear, exponential, sigmoid-2, sigmoid-3. Available creation strategies (method_subswarm_creation): equally-distributed, fitness-focused.