Bent cigar
The problem
The bent cigar squares the first gene, and the others a million times more:
f(x) = x₁² + 10⁶ Σᵢ₌₂ⁿ xᵢ², each xᵢ in [−100, 100]
Its minimum is 0, at the origin. Here n = 30. It's BBOB's f12 (Hansen et al. 2009), which bends it with an asymmetric transformation and rotates it twice; this plain form and the bounds are the CEC 2014 (Liang, Qu and Suganthan 2013, function 2) and CEC 2017 (Awad et al. 2016, function 1) reports' basic function, which those suites shift and rotate.
What makes it hard
The function is a ridge: low only near the line along the first axis, a thousand times narrower than long. A search has to find the ridge, then follow it to the minimum, with steps a thousand times longer along it than across it, in a single direction. As it is, that direction is a gene's axis. Shifted and rotated, it's a random one, and the search has to learn it.
Representation
A Real genome of 30 genes: the point x itself. The fitness is f(x), to minimize. The function is
genoxide's problems::BentCigar, which brings its bounds and its minimum, and the shifted and
rotated instance problems::Rotated::new(problems::Shifted::new(function, 1), 1), which keeps them.
Algorithm
Five algorithms, each with a budget of 10,000 evaluations per dimension, 300,000 in all, and a target of 1e-8, from seed 1:
- CMA-ES (Hansen and Ostermeier, 2001, Evolutionary Computation 9(2): 159-195), which samples a population of 14 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;
- sep-CMA-ES (Ros and Hansen, 2008, PPSN X: 296-305), the same with a diagonal covariance matrix: a scale per gene but no correlations;
- differential evolution with genoxide's defaults, SHADE (Tanabe and Fukunaga, CEC 2013), with a population of 100;
- 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 1/30 per gene.
The second table runs the same algorithms on the function shifted and rotated, with genoxide's
problems::Shifted and problems::Rotated and seed 1: the minimum moves to a random point in the
middle 80% of the box, and an orthogonal matrix, drawn from normal numbers made orthonormal by
Gram-Schmidt as BBOB draws its rotations, turns the function about it. That's how the CEC and BBOB
suites use the function, with their own data; genoxide generates its instances instead.
Output
The first line gives the dimension and the budget. Then two tables, the function as it is and
shifted and rotated: a row per algorithm, the evaluations it had used when its best error first
reached each value of the heading, and the best error it found, to two significant digits. A dash is
an error not reached. 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 on the function in 2 dimensions, x₁² + 10⁶ x₂², not rotated (in 2 dimensions, it's the high-conditioned elliptic function), so that the population can be drawn on its contour. It meets the target after 756 evaluations.
Good results
The minimum is 0. As it is, sep-CMA-ES reaches 1e-8 after 8,316 evaluations and CMA-ES after 13,482; SHADE takes 42,800 and PSO 47,600. The genetic algorithm ends at 210.
Shifted and rotated, CMA-ES takes 13,902 evaluations, about as many: it learns the ridge's direction. SHADE gets there after 171,700, four times as many as before, and the others fail: sep-CMA-ES ends at 680, the genetic algorithm at 470 and PSO at 5,700.
Known optimum: 0 (at the origin)
Source: examples/bent_cigar
Interactive run: tachsin.gr/projects/genoxide/examples/bent-cigar
cargo run --release --example bent_cigar
//! Bent cigar: minimize a narrow ridge, a thousand times longer than wide, in 30 dimensions,
//! as it is and shifted and rotated, as in the CEC 2014 and 2017 suites.
//!
//! Compares how fast CMA-ES, with a full and with a diagonal covariance matrix (sep-CMA-ES),
//! differential evolution, particle swarm optimization and a real-coded genetic algorithm close in
//! on the minimum, 0 at the origin: the evaluations each takes until its error is at most 1, 1e-2, 1e-4, 1e-6 and 1e-8.
//! The function is genoxide's `problems::BentCigar`. Then the same on the function shifted and rotated, with genoxide's `problems::Shifted`
//! and `problems::Rotated`, as the CEC and BBOB suites transform it.
//!
//! 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 bent_cigar
//! ```
mod trace;
use genoxide::observer::Snapshot;
use genoxide::prelude::*;
use genoxide::problems::{BentCigar, Problem, Rotated, Shifted};
const DIMENSIONS: usize = 30;
const BUDGET: u64 = 10_000 * DIMENSIONS as u64;
// the errors at which the table gives each run's evaluations
const ERRORS: [f64; 5] = [1e0, 1e-2, 1e-4, 1e-6, 1e-8];
const COLUMNS: [&str; 5] = ["1", "1e-2", "1e-4", "1e-6", "1e-8"];
fn main() -> Result<()> {
println!("Bent cigar in {DIMENSIONS} dimensions, {BUDGET} evaluations at most");
compare("Bent cigar", &BentCigar::new(DIMENSIONS))?;
// the CEC 2014 and 2017 suites' form, with genoxide's own shift and rotation
let rotated = Rotated::new(Shifted::new(BentCigar::new(DIMENSIONS), 1), 1);
compare("Shifted and rotated (seed 1)", &rotated)?;
// 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(())
}
// the table of the five algorithms on `problem`, after a line that names it
fn compare<P>(name: &str, problem: &P) -> Result<()>
where
P: Problem<Representation = Real> + FitnessFunction<Reals, Output = f64> + Clone,
{
let minimum = problem.optimum().expect("known").value();
let stop = || Stop::target(minimum + 1e-8).or(Stop::evaluations(BUDGET));
println!("{name}: evaluations until the error is at most");
print!("{:<10}", "algorithm");
COLUMNS.iter().for_each(|column| print!("{column:>9}"));
println!("{:>9}", "best");
for (name, covariance) in [
("CMA-ES", cmaes::Covariance::Full),
("sep-CMA-ES", cmaes::Covariance::Diagonal),
] {
let cmaes = Cmaes::builder(problem.representation())
.covariance(covariance)
.minimize()
.seed(1)
.build()?;
let mut reached = Reached::new(minimum);
let outcome = Engine::new(cmaes, problem.clone())
.stop_when(stop())
.on_generation(|snapshot| reached.record(snapshot))
.run()?;
reached.print(name, &outcome);
}
let de = De::builder(problem.representation())
.minimize()
.seed(1)
.build()?;
let mut reached = Reached::new(minimum);
let outcome = Engine::new(de, problem.clone())
.stop_when(stop())
.on_generation(|snapshot| reached.record(snapshot))
.run()?;
reached.print("DE", &outcome);
let pso = Pso::builder(problem.representation())
.population_size(40)
.minimize()
.seed(1)
.build()?;
let mut reached = Reached::new(minimum);
let outcome = Engine::new(pso, problem.clone())
.stop_when(stop())
.on_generation(|snapshot| reached.record(snapshot))
.run()?;
reached.print("PSO", &outcome);
let ga = Ga::builder(problem.representation())
.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(1)
.build()?;
let mut reached = Reached::new(minimum);
let outcome = Engine::new(ga, problem.clone())
.stop_when(stop())
.on_generation(|snapshot| reached.record(snapshot))
.run()?;
reached.print("GA", &outcome);
Ok(())
}
// the evaluations after the first generation whose best error was at most each of ERRORS, for a
// function whose minimum is `minimum`
struct Reached {
minimum: f64,
evaluations: [Option<u64>; 5],
}
impl Reached {
fn new(minimum: f64) -> Self {
let evaluations = [None; 5];
Self {
minimum,
evaluations,
}
}
fn record(&mut self, snapshot: &Snapshot<'_, Reals>) {
let progress = snapshot.progress();
let Some(best) = progress.best().and_then(Fitness::score) else {
return;
};
let error = best - self.minimum;
for (reached, bound) in self.evaluations.iter_mut().zip(ERRORS) {
if reached.is_none() && error <= bound {
*reached = Some(progress.evaluations());
}
}
}
// a row of the table: the evaluations, "-" for an error not reached, and the best error
fn print(&self, name: &str, outcome: &Outcome<Reals>) {
print!("{name:<10}");
for reached in self.evaluations {
let reached = reached.map_or("-".to_string(), |evaluations| evaluations.to_string());
print!("{reached:>9}");
}
// rounding can put a solution a few ulps below the minimum
let best = outcome.best_fitness().score().expect("valid");
let error = (best - self.minimum).max(0.0);
println!("{:>9}", format!("{error:.1e}"));
}
}
python examples/bent_cigar/main.py
"""Bent cigar: minimize a narrow ridge, a thousand times longer than wide, in 30 dimensions, as it
is and shifted and rotated, as in the CEC 2014 and 2017 suites.
Compares how fast CMA-ES, with a full and with a diagonal covariance matrix (sep-CMA-ES),
differential evolution, particle swarm optimization and a real-coded genetic algorithm close in on
the minimum, 0 at the origin: the evaluations each takes until its error is at most 1, 1e-2, 1e-4,
1e-6 and 1e-8. The function is genoxide's `problems::BentCigar`. Then the same on the function
shifted and rotated, with genoxide's `problems::Shifted` and `problems::Rotated`, as the CEC and
BBOB suites transform it.
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/bent_cigar/main.py
"""
import genoxide as gx
from trace import record_small
DIMENSIONS = 30
BUDGET = 10_000 * DIMENSIONS
# the errors at which the table gives each run's evaluations
ERRORS = [1e0, 1e-2, 1e-4, 1e-6, 1e-8]
COLUMNS = ["1", "1e-2", "1e-4", "1e-6", "1e-8"]
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)}"
class Reached:
"""The evaluations after the first generation whose best error was at most each of ERRORS,
for a function whose minimum is ``minimum``."""
def __init__(self, minimum):
self.minimum = minimum
self.evaluations = [None] * len(ERRORS)
def record(self, progress):
if progress.best_fitness is None:
return
error = progress.best_fitness - self.minimum
for i, bound in enumerate(ERRORS):
if self.evaluations[i] is None and error <= bound:
self.evaluations[i] = progress.evaluations
def print(self, name, result):
"""A row of the table: the evaluations, "-" for an error not reached, and the best
error."""
cells = ["-" if reached is None else str(reached) for reached in self.evaluations]
# rounding can put a solution a few ulps below the minimum
cells.append(error_text(max(result.best_fitness - self.minimum, 0.0)))
print(f"{name:<10}" + "".join(f"{cell:>9}" for cell in cells))
def compare(name, problem):
"""The table of the five algorithms on ``problem``, after a line that names it."""
minimum = problem.optimum.value
print(f"{name}: evaluations until the error is at most")
print(f"{'algorithm':<10}" + "".join(f"{column:>9}" for column in COLUMNS) + f"{'best':>9}")
genome = problem.genome
for label, algorithm in (
("CMA-ES", gx.Cmaes(genome, objective="minimize", seed=1)),
("sep-CMA-ES", gx.Cmaes(genome, covariance="diagonal", objective="minimize", seed=1)),
("DE", gx.De(genome, objective="minimize", seed=1)),
("PSO", gx.Pso(genome, population_size=40, objective="minimize", seed=1)),
(
"GA",
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=1,
),
),
):
reached = Reached(minimum)
result = algorithm.run(
problem, target=minimum + 1e-8, evaluations=BUDGET, on_generation=reached.record
)
reached.print(label, result)
print(f"Bent cigar in {DIMENSIONS} dimensions, {BUDGET} evaluations at most")
compare("Bent cigar", gx.problems.BentCigar(DIMENSIONS))
# the CEC 2014 and 2017 suites' form, with genoxide's own shift and rotation
rotated = gx.problems.Rotated(
gx.problems.Shifted(gx.problems.BentCigar(DIMENSIONS), seed=1), seed=1
)
compare("Shifted and rotated (seed 1)", rotated)
# 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:
Bent cigar in 30 dimensions, 300000 evaluations at most
Bent cigar: evaluations until the error is at most
algorithm 1 1e-2 1e-4 1e-6 1e-8 best
CMA-ES 10318 11144 11830 12600 13482 8.8e-9
sep-CMA-ES 5460 6118 6804 7602 8316 9.5e-9
DE 25200 29600 33900 38300 42800 9.0e-9
PSO 22200 28080 35240 40720 47600 9.6e-9
GA - - - - - 2.1e2
Shifted and rotated (seed 1): evaluations until the error is at most
algorithm 1 1e-2 1e-4 1e-6 1e-8 best
CMA-ES 10570 11452 12292 13034 13902 5.6e-9
sep-CMA-ES - - - - - 6.8e2
DE 111700 124400 138800 154500 171700 9.9e-9
PSO - - - - - 5.7e3
GA - - - - - 4.7e2