Büche-Rastrigin
The problem
The Büche-Rastrigin function is Rastrigin's function of scaled and slightly bent genes, with a penalty outside the box:
f(x) = 10 (n − Σ cos 2πzᵢ) + Σ zᵢ² + 100 Σ max(0, |xᵢ| − 5)², zᵢ = sᵢ T_osz(xᵢ)
each xᵢ in [−5, 5]
T_osz is BBOB's oscillation, sign(x) exp(x̂ + 0.049 (sin c₁x̂ + sin c₂x̂)) with x̂ = ln |x|, c₁ =
10 and c₂ = 7.9 for positive x, 5.5 and 3.1 otherwise: the identity, with small smooth wiggles. The
scale sᵢ grows from 1 to √10 along the genes, and is ten times larger where xᵢ > 0 and i is odd. Its
minimum is 0, at the origin. Here n = 10. It's BBOB's f4 (Hansen et al. 2009), with its optimum at
the origin and no offset, and BBOB's search domain.
What makes it hard
Rastrigin's function has a local minimum near every integer point, roughly 10ⁿ of them in the box. Here, on the positive side of the odd genes, the scale is ten times larger: the wells are ten times narrower and the slope ten times steeper, so the landscape is lopsided. BBOB built it to deceive search operators that are symmetric about the current point, which expect the minimum's basin to look the same on both sides.
Representation
A Real genome of 10 genes, each in [−5, 5]: the point x itself. The fitness is f(x), to minimize.
The function is genoxide's problems::BucheRastrigin, which brings its bounds and its minimum.
Algorithm
Five algorithms, each from seeds 1 to 10, with a budget of 10,000 evaluations per dimension, 100,000 per run, and a target of 1e-8:
- CMA-ES (Hansen and Ostermeier, 2001, Evolutionary Computation 9(2): 159-195), which samples a population of 10 from a normal distribution and adapts its mean, its step size and its covariance matrix, from a step size of 0.3 of each gene's range and a random start;
- the same with IPOP restarts (Auger and Hansen, 2005, IEEE CEC 2005: 1769-1776): a run that has converged starts again from a random point with twice the population;
- differential evolution with genoxide's defaults, SHADE (Tanabe and Fukunaga, CEC 2013), with a population of 100 and its restarts on stagnation;
- particle swarm optimization (Kennedy and Eberhart, 1995), 40 particles with Clerc and Kennedy's constriction coefficients and a global topology;
- a real-coded genetic algorithm: a population of 100, tournaments of 3, simulated binary crossover (Deb and Agrawal, 1995) with η = 15 and polynomial mutation with η = 20 at a rate of 1/10 per gene.
Output
The first line gives the dimension, the seeds and the budget. Then a row per algorithm: how many of
its 10 runs reached the minimum, to within 1e-8, the median of their evaluations (a dash if none
did), and the median of every run's best error, to two significant digits. The function is evaluated
with genoxide's portable math, so the runs are the same on every platform, and in Python, run
evaluates it in Rust, so both versions print the same.
The project page plays back another run: CMA-ES with IPOP restarts on the function in 2 dimensions, so that the population can be drawn on its contour. Within 20,000 evaluations it restarts 6 times, up to a population of 384, and ends at an error of 0.36: it doesn't reach the minimum in 2 dimensions either.
Good results
The minimum is 0. SHADE reaches it in all 10 runs, after a median of 66,650 evaluations: its differences between members of the population take the wells' spacing, and its restarts on stagnation free it from the local minima it falls into. No other algorithm reaches it. CMA-ES ends in a local minimum with a median error of 18, and 6.0 with IPOP restarts; PSO's median run ends at 7.5. The genetic algorithm, whose mutation changes one gene at a time, gets close, to a median error of 2.2e-4, but doesn't reach 1e-8 within the budget.
Known optimum: 0 (at the origin)
Source: examples/buche_rastrigin
Interactive run: tachsin.gr/projects/genoxide/examples/buche-rastrigin
cargo run --release --example buche_rastrigin
//! Büche-Rastrigin: minimize BBOB's Büche-Rastrigin function, an asymmetric Rastrigin, in 10
//! dimensions.
//!
//! Runs CMA-ES without and with IPOP restarts (a population that doubles at each restart),
//! differential evolution (SHADE), particle swarm optimization and a real-coded genetic algorithm
//! from 10 seeds each, and counts the runs that reach the minimum, 0 at the origin, to within
//! 1e-8. The function is genoxide's `problems::BucheRastrigin`.
//!
//! With `GENOXIDE_TRACE=<file>`, it also writes a trace of a run for the plot on the example's
//! page, with `trace.rs`.
//!
//! ```text
//! cargo run --release --example buche_rastrigin
//! ```
mod trace;
use genoxide::prelude::*;
use genoxide::problems::{BucheRastrigin, Problem};
const DIMENSIONS: usize = 10;
const SEEDS: u64 = 10;
const BUDGET: u64 = 10_000 * DIMENSIONS as u64;
// a run stops once its error to the minimum is at most this
const ERROR: f64 = 1e-8;
const ALGORITHMS: [&str; 5] = ["CMA-ES", "CMA-ES with IPOP", "DE", "PSO", "GA"];
fn main() -> Result<()> {
let problem = BucheRastrigin::new(DIMENSIONS);
let minimum = problem.optimum().expect("known").value();
println!(
"Büche-Rastrigin in {DIMENSIONS} dimensions, {SEEDS} seeds, {BUDGET} evaluations at most per run"
);
println!("algorithm at min evaluations median error");
for algorithm in ALGORITHMS {
// the evaluations of the runs that reach the minimum, and every run's best error
let mut evaluations = Vec::new();
let mut errors = Vec::new();
for seed in 1..=SEEDS {
let outcome = run(algorithm, problem, seed, minimum + ERROR)?;
if outcome.stop_reason() == StopReason::Target {
evaluations.push(outcome.evaluations() as f64);
}
// rounding can put a solution a few ulps below the minimum
let best = outcome.best_fitness().score().expect("valid");
errors.push((best - minimum).max(0.0));
}
let reached = format!("{}/{SEEDS}", evaluations.len());
let evaluations = median(evaluations).map_or("-".to_string(), |e| format!("{e:.0}"));
let error = median(errors).expect("a run");
println!(
"{algorithm:<16} {reached:>6} {evaluations:>11} {:>12}",
format!("{error:.1e}")
);
}
println!("evaluations: the median of the runs that reach the minimum");
// with GENOXIDE_TRACE=<file>, a trace for the plot on the example's page, of a separate
// run in 2 dimensions: the plot is the function's contour
trace::record_small()?;
Ok(())
}
// a run of `algorithm` from `seed`, until its best is at most `target` or it has used BUDGET
// evaluations
fn run(algorithm: &str, problem: BucheRastrigin, seed: u64, target: f64) -> Result<Outcome<Reals>> {
let real = problem.representation();
let stop = Stop::target(target).or(Stop::evaluations(BUDGET));
match algorithm {
"CMA-ES" | "CMA-ES with IPOP" => {
let restarts = if algorithm == "CMA-ES" {
cmaes::Restarts::Never
} else {
cmaes::Restarts::Ipop
};
let cmaes = Cmaes::builder(real)
.restarts(restarts)
.minimize()
.seed(seed)
.build()?;
Engine::new(cmaes, problem).stop_when(stop).run()
}
"DE" => {
let de = De::builder(real).minimize().seed(seed).build()?;
Engine::new(de, problem).stop_when(stop).run()
}
"PSO" => {
let pso = Pso::builder(real)
.population_size(40)
.minimize()
.seed(seed)
.build()?;
Engine::new(pso, problem).stop_when(stop).run()
}
_ => {
let ga = Ga::builder(real)
.population_size(100)
.select(Tournament::new(3)?)
.crossover(SimulatedBinaryCrossover::new(15.0)?)
.mutate(PolynomialMutation::per_gene(1.0 / DIMENSIONS as f64, 20.0)?)
.minimize()
.seed(seed)
.build()?;
Engine::new(ga, problem).stop_when(stop).run()
}
}
}
// the median of `values`, None without any
fn median(mut values: Vec<f64>) -> Option<f64> {
values.sort_by(f64::total_cmp);
let middle = values.len() / 2;
match values.len() {
0 => None,
n if n % 2 == 1 => Some(values[middle]),
_ => Some((values[middle - 1] + values[middle]) / 2.0),
}
}
python examples/buche_rastrigin/main.py
"""Büche-Rastrigin: minimize BBOB's Büche-Rastrigin function, an asymmetric Rastrigin, in 10
dimensions.
Runs CMA-ES without and with IPOP restarts (a population that doubles at each restart), differential
evolution (SHADE), particle swarm optimization and a real-coded genetic algorithm from 10 seeds
each, and counts the runs that reach the minimum, 0 at the origin, to within 1e-8. The function is
genoxide's `problems::BucheRastrigin`, which run evaluates in Rust.
With ``GENOXIDE_TRACE=<file>``, it also writes a trace of a run for the plot on the example's page,
with trace.py.
python examples/buche_rastrigin/main.py
"""
import genoxide as gx
from trace import record_small
DIMENSIONS = 10
SEEDS = 10
BUDGET = 10_000 * DIMENSIONS
# a run stops once its error to the minimum is at most this
ERROR = 1e-8
ALGORITHMS = ["CMA-ES", "CMA-ES with IPOP", "DE", "PSO", "GA"]
def build(name, genome, seed):
"""The algorithm called ``name``, on ``genome``, from ``seed``."""
if name == "CMA-ES":
return gx.Cmaes(genome, objective="minimize", seed=seed)
if name == "CMA-ES with IPOP":
return gx.Cmaes(genome, restarts="ipop", objective="minimize", seed=seed)
if name == "DE":
return gx.De(genome, objective="minimize", seed=seed)
if name == "PSO":
return gx.Pso(genome, population_size=40, objective="minimize", seed=seed)
return gx.Ga(
genome,
population_size=100,
select=gx.Tournament(3),
crossover=gx.SimulatedBinaryCrossover(15.0),
mutation=gx.PolynomialMutation(20.0, rate=1 / DIMENSIONS),
objective="minimize",
seed=seed,
)
def median(values):
"""The median of ``values``, None without any."""
values = sorted(values)
middle = len(values) // 2
if not values:
return None
return values[middle] if len(values) % 2 else (values[middle - 1] + values[middle]) / 2
def error_text(error):
"""An error to two significant digits, as Rust writes it: 9.9e-9."""
mantissa, exponent = f"{error:.1e}".split("e")
return f"{mantissa}e{int(exponent)}"
problem = gx.problems.BucheRastrigin(DIMENSIONS)
minimum = problem.optimum.value
print(
f"Büche-Rastrigin in {DIMENSIONS} dimensions, {SEEDS} seeds, "
f"{BUDGET} evaluations at most per run"
)
print("algorithm at min evaluations median error")
for name in ALGORITHMS:
# the evaluations of the runs that reach the minimum, and every run's best error
evaluations, errors = [], []
for seed in range(1, SEEDS + 1):
result = build(name, problem.genome, seed).run(
problem, target=minimum + ERROR, evaluations=BUDGET
)
if result.stop_reason == "target":
evaluations.append(float(result.evaluations))
# rounding can put a solution a few ulps below the minimum
errors.append(max(result.best_fitness - minimum, 0.0))
reached = f"{len(evaluations)}/{SEEDS}"
middle = median(evaluations)
evaluations_text = "-" if middle is None else f"{middle:.0f}"
error = error_text(median(errors))
print(f"{name:<16} {reached:>6} {evaluations_text:>11} {error:>12}")
print("evaluations: the median of the runs that reach the minimum")
# with GENOXIDE_TRACE=<file>, a trace for the plot on the example's page, of a separate run in
# 2 dimensions: the plot is the function's contour
record_small()
What it prints, from a seeded run:
Büche-Rastrigin in 10 dimensions, 10 seeds, 100000 evaluations at most per run
algorithm at min evaluations median error
CMA-ES 0/10 - 1.8e1
CMA-ES with IPOP 0/10 - 6.0e0
DE 10/10 66650 9.4e-9
PSO 0/10 - 7.5e0
GA 0/10 - 2.2e-4
evaluations: the median of the runs that reach the minimum