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Re-initialisation (restart)

To prevent the algorithm from becoming trapped in a local minimum, DEEM employs a diversity-based restart. When stagnation is detected, the positions of all candidate solutions — except the global best \(\mathbf{x}_{GB}\) — are reset to new locations in the least-visited regions of the search space, restoring exploration.

Stagnation criterion

A restart is triggered when either the search has spent too many iterations below the early-termination tolerance, or the diversity has collapsed:

\[ n_i^{TOL} > n_i^{res} \quad\vee\quad DIV < \varepsilon^{DIV}, \]

where \(n_i^{TOL}\) is the number of consecutive iterations with an improvement below \(TOL\), \(n_i^{res}\) is the restart threshold (default \(n_i^{max}/5\)), and \(\varepsilon^{DIV}\) is the diversity threshold (default \(10^{-2}\)). The normalised diversity \(DIV\) is defined on the position-updates page.

Density-based re-initialisation

New positions are placed in least-visited regions, tracked by a density grid:

  1. Each dimension \(iD\) of the search space is divided into a number of bins between \(LB_{iD}\) and \(UB_{iD}\). A density matrix records visit counts per bin.
  2. As candidate solutions traverse the search space, the bins corresponding to their positions are incremented.
  3. On restart, the algorithm finds the bins with the minimum visit counts and samples new positions within them, kept inside the bounds.

Because the global best is preserved, the elitism guarantees are not lost across a restart: the best solution found so far always survives.

The restart produces characteristic diversity spikes: at stagnation, diversity jumps up as the population is scattered into unexplored regions, then decays again over the next 20–50 iterations as the sub-populations re-cluster. Combined with the multi-population and elitism mechanisms, this both escapes local minima and keeps refining the best region.

DEEMI: an improved restart

The base restart is per-dimension and density-driven. The improved variant DEEMI offers an alternative hybrid restart (method_reset="hybrid") that keeps the global best exactly, re-seeds part of the population around the most distinct archive elites using a shrinkage covariance (so it respects parameter correlations), and fills the rest with a scrambled low-discrepancy (Sobol) sequence. The original behaviour remains available with method_reset="density".