Evolution Strategy (μ+λ)

Classical Evolutionary Optimization

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Evolution Strategy dates to Rechenberg and Schwefel's work in 1960s Berlin and is the conceptual ancestor of CMA-ES. The (μ+λ) variant maintains μ parents, generates λ Gaussian-perturbed offspring each generation, and selects the best μ individuals from the combined parent + offspring pool. HumpDay's EvolutionStrategy uses a fixed step size σ=0.2 (no covariance adaptation — that's CMA-ES's contribution).

Interactive 3D Visualization

See Evolution Strategy in action on 3D optimization surfaces:

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Requires WebGL support

Auto-selection metadata. Consulted by humpday.minimize() when picking a default algorithm.
  • Overhead tier 1: light (~100µs/iter). Eligible when measured eval_time ≥ 10 µs.
  • Dimensional cap: no cap — linear-in-dim or better.
  • Minimum trials: 10 — baseline minimum.

Implementation Details

Component Details Links
Original Algorithm Ingo Rechenberg & Hans-Paul Schwefel
Mutation-driven evolutionary search
(μ+λ) selection over combined parent+offspring pool
Published: 1965 (Rechenberg), 1975 (Schwefel) 		Wikipedia
Humpday Python Pure-Python Implementation
μ=10 parents, λ=min(30, n_trials/3) offspring
Gaussian mutation with fixed step size σ=0.2, clipped to the unit cube
No required dependencies
File: humpday/optimizers/evolutionary_algorithms.py 		Python Code
Humpday JavaScript Browser Implementation
Pure-JavaScript port of the (μ+λ)-ES for the in-browser demo
Class: EvolutionStrategy 		JavaScript Code

Performance Characteristics

  • Best for: Continuous unconstrained objectives where rotation-invariance isn't critical and a fixed-σ baseline suffices.
  • Worst for: Ill-conditioned problems — without covariance adaptation, σ can't match the basin geometry the way CMA-ES does.
  • Per generation: up to λ evaluations; sort by fitness, keep μ best.
  • Relative to mealpy OriginalES at n_trials=200, n_dim=2: HumpDay wins across sphere, Rosenbrock, and Ackley. See SOTA status for the live numbers.

Educational Resources

Algorithm Overview

  1. Initialize μ parents uniformly at random in the unit cube.
  2. Mutate: pick a parent at random, perturb by σ·𝒩(0,I), clip to the unit cube. Repeat λ times.
  3. Select: sort the combined (parent + offspring) pool by fitness and keep the best μ.
  4. Repeat until the evaluation budget is exhausted.

Notation: the "+" in (μ+λ) indicates elitism — parents compete against their own offspring. The alternative (μ,λ) variant discards parents each generation; HumpDay implements the elitist (μ+λ) form.