Penalized 2
The problem
Yao, Liu and Lin's second generalized penalized function has sines of three times the genes, and a penalty beyond ±5:
f(x) = 0.1 {sin²(3πx₁) + Σᵢ₌₁ⁿ⁻¹ (xᵢ − 1)² [1 + sin²(3πxᵢ₊₁)] + (xₙ − 1)² [1 + sin²(2πxₙ)]}
+ Σ u(xᵢ, 5, 100, 4), each xᵢ in [−50, 50]
u(x, a, k, m) = k (|x| − a)^m if |x| > a, and 0 otherwise
Its minimum is 0, at (1, …, 1), where every term is 0. Here n = 30. It's Yao, Liu and Lin's (1999) f13, whose appendix gives this definition, the bounds and the dimension; their table I drops the square of the last term's (xₙ − 1), without which the function would have no minimum there. The function is usually credited to Levy and Montalvo's tunneling papers (1985), which weren't read.
What makes it hard
The sines put a local minimum near every point where their arguments are whole multiples of π: a grid of shallow wells over the box, the more of them the more genes. The wells' depth scales with the squares in front of the sines, so they are shallower the nearer the minimum, and the squares lead towards it. The penalty u is 0 inside [−10, 10] (Penalized 1) or [−5, 5] (Penalized 2), and rises as a fourth power outside: a wall that keeps the search away from the bounds of [−50, 50].
Representation
A Real genome of 30 genes, each in [−50, 50]: the point x itself. The fitness is f(x), to
minimize. The function is genoxide's problems::Penalized2, 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, 300,000 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 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;
- 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 1/30 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. It meets the target after 396 evaluations.
Good results
The minimum is 0. CMA-ES with IPOP restarts reaches it in all 10 runs, after a median of 6,776 evaluations, and without restarts in 9 of 10, after 6,762. SHADE reaches it every time, after 30,250, and PSO 7 times, after 26,840. The genetic algorithm never does: its median run ends at 1.1e-5.
Known optimum: 0 (at (1, …, 1))
Source: examples/penalized2
Interactive run: tachsin.gr/projects/genoxide/examples/penalized2
cargo run --release --example penalized2
//! Penalized 2: minimize Yao, Liu and Lin's second penalized function in 30 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 (1, …, 1), to within
//! 1e-8. The function is genoxide's `problems::Penalized2`.
//!
//! 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 penalized2
//! ```
mod trace;
use genoxide::prelude::*;
use genoxide::problems::{Penalized2, Problem};
const DIMENSIONS: usize = 30;
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 = Penalized2::new(DIMENSIONS);
let minimum = problem.optimum().expect("known").value();
println!(
"Penalized 2 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: Penalized2, 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/penalized2/main.py
"""Penalized 2: minimize Yao, Liu and Lin's second penalized function in 30 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 (1, …, 1), to within 1e-8. The function is
genoxide's `problems::Penalized2`, 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/penalized2/main.py
"""
import genoxide as gx
from trace import record_small
DIMENSIONS = 30
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.Penalized2(DIMENSIONS)
minimum = problem.optimum.value
print(
f"Penalized 2 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:
Penalized 2 in 30 dimensions, 10 seeds, 300000 evaluations at most per run
algorithm at min evaluations median error
CMA-ES 9/10 6762 9.0e-9
CMA-ES with IPOP 10/10 6776 8.8e-9
DE 10/10 30250 9.4e-9
PSO 7/10 26840 9.8e-9
GA 0/10 - 1.1e-5
evaluations: the median of the runs that reach the minimum