Every CMA strategy accepts low / up (scalar or length-dim)
and a bound_mode of "clip" or "resample", covering the
box-constrained CMA request. Bounds are applied to the
sampled vector before ind_init. Resample rejects the whole
vector; after resample_limit failed draws the offspring is
clipped so generate always returns lamb individuals. Both
modes are constraint-handling approximations. Under "clip",
Strategy and StrategySeparable evaluate the clipped point but
learn the mean and covariance from the unclipped draw, then clip
the new centroid into the box, so a corner optimum does not zero
the variance across a box face.
MO-CMA's rank-one covariance update gates on \(\lVert w \rVert\),
not on w.max(). A negative evolution path no longer skips the
update. When \(w \approx 0\) the factors still scale by
\(\sqrt{\alpha}\).
The stall-case \(\alpha\) on MO-CMA includes the \(c_{\mathrm{cov}}\)
factor, so repeated stall generations do not inflate the
covariance.
MO-CMA writes one step-size trial per parent when
\(\lambda \neq \mu\), so several children of the same parent do not
stack \(\sigma\) updates. generate samples every parent if any
parent fitness is invalid; update ranks only valid fitnesses.
When fewer than \(\lambda\) parents remain, children are sampled
from the available set instead of indexing past the last parent.
RestartStrategy wraps standard, separable, \((1+\lambda)\), or
MO-CMA with IPOP or BIPOP restart scheduling. ea_generate_update_restarts
runs the generate/update loop and logs restart regime, population
size, and evaluation budget each generation. Hyperparameters given
to the constructor, compute_params, or reset_state stay pinned
across restarts and leftover-budget resizes until a later call
overrides them; only what depends on \(\lambda\) and is not pinned
is re-derived.
sample_centroid replaces a non-finite box end with the
unbounded default \([-5, 5]\), so a one-sided bound no longer
writes inf or NaN as an IPOP/BIPOP restart mean.
Restart TolFun needs Hansen's \(10 + 30n/\lambda\) history, then
stops only if that window's best-of-generation range is below
tol_fun. Two equal generation-bests no longer terminate at
generation 2.
StrategySeparable is separable CMA (Ros and Hansen, 2008): \(C\)
stays a length-\(n\) diagonal, sampling and the update are \(O(n)\),
and default \(c_1\) / \(c_\mu\) are the full-matrix rates scaled by
\((n + 2) / 3\). The generate / update surface, including
low / up, matches Strategy. RestartStrategy can wrap it.
\((1+\lambda)\) CMA does not treat an unevaluated parent as worse
than every offspring. After reset_state / an IPOP restart the
first update adopts the best child without a fake
\(p_{\mathrm{succ}}=1\) step-size blow-up.
BIPOP small-regime \(\sigma\) is \(\sigma_{\mathrm{large}}\cdot
10^{-2U[0,1]}\), not a hardcoded \(2.0\). A custom first-run step
size no longer launches small restarts in \([0.02, 2]\).
Default \(\mu\) is at least \(1\) when \(\lambda\ge 1\).
\(\mu=\lfloor\lambda/2\rfloor\) is no longer \(0\) at \(\lambda=1\),
so Strategy / StrategySeparable and a RestartStrategy
leftover budget of one evaluation no longer raise
ZeroDivisionError on an empty weight vector.
Strategy.compute_params rebuilds \(C\), \(B\), and \(D\) only on
first init or when cm_init is in kwargs. A later
compute_params(offsprings=…) no longer wipes the learned
covariance.
A leftover-budget CMA batch keeps
\(\mu=\min(\mu_{\mathrm{prev}},\lambda)\).
resize_offsprings no longer resets survivors to
\(\lfloor\lambda/2\rfloor\), so a \(\lambda=8\), \(\mu=4\) run
with 3 evaluations left does not finish at \(\mu=1\). MO-CMA
keeps its parent count the same way: a short last batch no
longer collapses the parent set, and restart survivors grow
back with \(\lambda\) instead of only ratcheting down.
Strategy symmetrizes \(C\) before the eigendecomposition and
floors its eigenvalues at \(10^{-14}\) times the largest one. A
singular or nearly singular covariance no longer gives a tiny
negative eigenvalue, so diag_d is not NaN and cond is not
negative. DEAP took the square root of the raw eigh output.
Strategy.update computes the rank-μ term from
\((x_i - m)/\sigma\), so it stays finite after \(\sigma^2\)
underflows. When \(\sigma\) really collapses, the state turns
non-finite, or the eigendecomposition of the new \(C\) fails or
is non-finite, update restores the state it had before the
call and raises FloatingPointError, and RestartStrategy
restarts the run; StrategySeparable does the same instead of
going NaN.
DEAP divided by sigma ** 2 and called eigh on the resulting
non-finite \(C\), which raised a raw LinAlgError with the
strategy half-updated.