A (μ, λ)-ES or (μ+λ)-ES where each individual carries its own step size σi (and optionally an individual rotation angle θi in the correlated variant). Mutation: x′ = x + σ·𝒩(0, R(θ)ΣR(θ)ᵀ), with σ′ = σ·exp(τ·𝒩(0,1)). Because σi is heritable, selection indirectly favors individuals whose step sizes are productive — self-adaptation emerges without any global update rule. Dot size encodes each individual's σ. The σ distribution panel shows how step sizes evolve across generations. On a rotated ellipse, watch ES struggle to align its search with the landscape axes — it has no covariance matrix to exploit. This explorer minimizes f(x,y) — darker = lower (better).
Fitness Landscape · minimizing f(x,y)
●best sample
●best-ever ★
⊕optimum
○parent σ range
Fitness History
σ Distribution (step sizes)
3-D View

Controls

Population σ Stats

σ̄ (mean)
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σ min
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σ max
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best f (gen)
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worst f (gen)
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selected μ
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Fitness Function

ES Variant

Mutation Mode

Display

Strategy Parameters

20
5
0.50
0.005

Mutation Parameters

1.00×
1.00×
0.10
20

Reference

σiPer-individual step size, carried on the genotype alongside xi and mutated each generation. Dot size on the landscape encodes σi.
θiPer-individual rotation angle (correlated variant). Rotates the mutation ellipse, allowing axes to align with the landscape. Ignored in isotropic mode.
τ, τ′Global and local learning rates for σ adaptation. σ′ = σ · exp(τ·N(0,1) + τ′·N(0,1)). Hansen recommends τ ∝ 1/√(2n), τ′ ∝ 1/√(2√n).
βStep size for angle mutation (correlated variant). θ′ = θ + β·N(0,1), clipped to [−π/2, π/2].
(μ, λ)Comma selection: parents die each generation; only offspring survive. Forces σ adaptation to work on a one-generation timescale. λ > μ required.
(μ + λ)Plus selection: parents compete with offspring. Elitist — best-ever is preserved — but σ can get stuck at a small value that prevents escape from local minima.
σ distributionHistogram of all λ individuals' σ values. Watch it shift and narrow as the algorithm converges — or widen if the landscape forces exploration.
Rotated Ellipsef(x,y) = (x cos θ + y sin θ)² + 10(−x sin θ + y cos θ)². 10:1 axis ratio at 30°. Isotropic ES struggles; correlated ES (with θi) can adapt.