DC2-DTLZ1 with 3 objectives
The problem
Li, Chen, Fu and Yao (2019) proposed C-TAEA, a two-archive algorithm for constrained problems, and, to test it, the DC-DTLZ problems: DTLZ1 and DTLZ3 with constraints on the decision variables, where Jain and Deb's C-DTLZ problems constrain the objectives. Type 1 (DC1) constrains the first variable and cuts the front into cones from the origin; type 2 (DC2) constrains the distance function g and leaves almost nothing but the front feasible, with local optima of the violation on the way; type 3 (DC3) does both, on every position variable. The paper's supplement defines them (section 1.2) with parameters a and b, and gives a = 3 and b = 0.5 for DC1 only; genoxide takes the others from the code of the authors' laboratory, EMOC, and the docs of multi::problems::Dc1Dtlz1 say what else the supplement leaves open and how genoxide settles it.
DC2-DTLZ1 with 3 objectives and 7 variables in [0, 1]:
minimize f₁ = ½ x₁ x₂ (1 + g)
f₂ = ½ x₁ (1 − x₂) (1 + g)
f₃ = ½ (1 − x₁) (1 + g)
g = 100 (5 + Σᵢ₌₃⁷ ((xᵢ − 0.5)² − cos(20π (xᵢ − 0.5))))
subject to cos(aπg) ≥ b
e^(−g) ≥ b, a = 3, b = 0.9
DTLZ1's front is the triangle where the objectives sum to 1/2, reached with the distance variables at 0.5, where g = 0. With b = 0.9 the constraints ask g ≤ 0.048, in the first band of the cosine (the second constraint asks g ≤ −ln 0.9 ≈ 0.105): feasible solutions lie within a thin shell of the front, and the front is DTLZ1's, whole. Its ideal point is (0, 0, 0) and its nadir point (1/2, 1/2, 1/2). The supplement gives no front; genoxide derives it (g = 0 is feasible under every constraint, and the position constraints don't involve g, so DTLZ1's front points are optimal wherever their position variables are feasible, and no other point is), and it agrees with the sampled front that EMOC ships.
What makes it hard
The violation: b − cos(aπg) rises and falls with g, back to its least, b − 1, at g = 2/3, 4/3, …, where e^(−g) is the only constraint still broken. Constrained dominance ranks infeasible solutions by their violation alone, so a population that reaches one of these local minima of the violation stays there: to go on, it must climb out first. Almost nothing but the front is feasible, so it never finds a feasible solution to pull it down.
Representation
A Real genome of 7 genes in [0, 1]. The problem is genoxide's Dc2Dtlz1, whose fitness is the three objectives and the total constraint violation, 0 when it is feasible; in Python, gx.problems.Dc2Dtlz1(). Solutions compare by constrained dominance: a feasible solution beats an infeasible one, of two infeasible ones the smaller violation wins, and of two feasible ones Pareto dominance decides.
Algorithm
Two runs of NSGA-III, with a population of 92 for 1,000 generations each:
- NSGA-III (Deb and Jain, 2014; Jain and Deb, 2014, IEEE Transactions on Evolutionary Computation 18(4): 577-601 and 602-622) with the settings that the C-TAEA paper gives its C-NSGA-III (supplement, tables 3 and 4): the 91 reference directions of Das and Dennis's method with 12 divisions and a population of 92, simulated binary crossover with η = 30 at a rate of 1, and polynomial mutation with η = 20 at a rate of 1/n per gene, for 1,000 generations (the supplement gives no budget for the DC-DTLZ problems; 2,000 on DTLZ3, whose local fronts take longer), with constrained dominance;
- the same on DTLZ1 itself, without the constraints, its final front then scored by DC2-DTLZ1: the front is DTLZ1's, and a population that converges to it is feasible.
Output
A line per run: the size of its final front, how many of its solutions the problem finds feasible, their IGD+ and hypervolume, and the hypervolume as a share of that of a sample of the optimal front with at least as many points as the population. Then the hypervolumes of the whole front and of the sample.
The indicators use the objectives normalized by the front's ideal and nadir points, so that the front spans [0, 1] in each. IGD+ (Ishibuchi et al., 2015, EMO 2015, LNCS 9019: 110-125) averages, over 2,000 points of the optimal front from genoxide's optimal_front, the distance to the nearest feasible solution of the found front, counting only the objectives in which the solution is worse. Smaller is better; a front of as many points as the population can't cover the 2,000 exactly, and the sample shows what it can. The hypervolume (Zitzler and Thiele, 1999, IEEE Transactions on Evolutionary Computation 3(4): 257-271) is the volume the feasible solutions dominate up to the reference point (1.1, 1.1, 1.1). Larger is better; the whole front's is computed from 3,000 of its points, and the sample's is what a front of that many points reaches. In Python, run evaluates the problem in Rust, so both versions print the same.
The page plays the runs back side by side, each population as the problem scores it: its feasible non-dominated solutions, its infeasible ones (hollow, at most 100 a frame), and the optimal front, sampled.
The project page plays these runs back.
Good results
The target: every solution feasible, and 99% of the hypervolume of the sample of 105 points of the front, about what a front of 92 solutions spread like it has.
The run with constrained dominance ends with 1 solution, none feasible, the least violation 0.8975: stuck at a local minimum of the violation. Without the constraints, it ends with 92 solutions, 92 feasible, IGD+ 0.0289, hypervolume 1.1203, 99.7% of the sample's. Over seeds 1 to 20, every run with constrained dominance ends with no feasible solution, and without the constraints IGD+ from 0.0288 to 0.0307 and 99.4% to 99.7% of the sample's hypervolume, every run reaching the target. The C-TAEA paper's C-NSGA-III ends with a median IGD of 17.6 (its table 3), the same failure, and C-TAEA with 0.0220. Ignoring the constraints works here because they leave the front whole, as Tanabe and Oyama (2017) note of C1-DTLZ1 and C1-DTLZ3.
Known optimum: DTLZ1's front, whole; ideal point (0, 0, 0), nadir point (1/2, 1/2, 1/2); hypervolume 1.1577 (normalized objectives, reference point (1.1, 1.1, 1.1))
Source: examples/dc2_dtlz1_3obj
Interactive run: tachsin.gr/projects/genoxide/examples/dc2-dtlz1-3obj
cargo run --release --example dc2_dtlz1_3obj
//! DC2-DTLZ1 with 3 objectives: minimize three objectives of DTLZ1 subject to two constraints on
//! its distance function, which leave almost nothing but the front feasible, with NSGA-III.
//!
//! From genoxide's `multi::problems::Dc2Dtlz1`. Runs NSGA-III with the settings of the C-TAEA
//! paper's C-NSGA-III, a population of 92 for 1,000 generations, twice: with constraint dominance,
//! and on DTLZ1 without the constraints. Prints each final front's size, how many of its solutions
//! are feasible, their IGD+ to 2,000 points of the optimal front and their hypervolume, with the
//! objectives normalized by the front's ideal and nadir points, as a share of that of a sample of
//! the front with at least as many points as the population; then the hypervolumes of the whole
//! front and of the sample.
//!
//! With `GENOXIDE_TRACE=<file>`, it also writes a trace of its runs for the plot on the example's
//! page, with `trace.rs`.
//!
//! ```text
//! cargo run --release --example dc2_dtlz1_3obj
//! ```
mod trace;
use genoxide::Objective::Minimize;
use genoxide::multi::MultiFitnessFunction;
use genoxide::multi::indicator::{hypervolume, igd_plus};
use genoxide::multi::problems::{Dc2Dtlz1, Dtlz1, MultiProblem};
use genoxide::prelude::*;
// the reference point of the hypervolume, with the objectives normalized by the front's ideal and
// nadir points
pub const REFERENCE: [f64; 3] = [1.1, 1.1, 1.1];
fn main() -> Result<()> {
let problem = Dc2Dtlz1::<3>::default();
// with GENOXIDE_TRACE=<file>, a trace of the runs for the plot on the example's page
let mut trace = trace::Trace::from_env();
// the sample: at least as many points of the optimal front as the population has, about what
// a front of that many solutions can be
let sample = normalized(&problem, &problem.optimal_front(92).expect("known"));
let rate = 1.0 / problem.variables() as f64;
// NSGA-III with the settings of the C-TAEA paper's C-NSGA-III, constraint dominance
let algorithm = Nsga3::builder(
problem.representation(),
[Minimize; 3],
multi::das_dennis::<3>(12),
)
.population_size(92)
.crossover(SimulatedBinaryCrossover::new(30.0)?)
.crossover_rate(1.0)
.mutate(PolynomialMutation::per_gene(rate, 20.0)?)
.seed(1)
.build()?;
run(
&problem, "NSGA-III", algorithm, problem, 1_000, &sample, &mut trace,
)?;
// the same on DTLZ1, without the constraints, whose solutions the problem then scores
let algorithm = Nsga3::builder(
problem.representation(),
[Minimize; 3],
multi::das_dennis::<3>(12),
)
.population_size(92)
.crossover(SimulatedBinaryCrossover::new(30.0)?)
.crossover_rate(1.0)
.mutate(PolynomialMutation::per_gene(rate, 20.0)?)
.seed(1)
.build()?;
run(
&problem,
"NSGA-III, unconstrained",
algorithm,
Dtlz1::<3>::new(problem.variables()),
1_000,
&sample,
&mut trace,
)?;
let whole = normalized(&problem, &problem.optimal_front(3_000).expect("known"));
println!(
"the whole front: hypervolume {:.4}; a sample of {} of its points: {:.4}",
hypervolume(&whole, &REFERENCE, &[Minimize; 3]),
sample.len(),
hypervolume(&sample, &REFERENCE, &[Minimize; 3])
);
trace.write(&problem);
Ok(())
}
// runs `algorithm` on `fitness` for `generations`, and prints its final front's size, how many of
// its solutions the problem finds feasible, their IGD+ to 2,000 points of the optimal front and
// their hypervolume, with normalized objectives, as a share of the sample's
fn run<A, F>(
problem: &Dc2Dtlz1<3>,
name: &'static str,
algorithm: A,
fitness: F,
generations: u64,
sample: &[[f64; 3]],
trace: &mut trace::Trace,
) -> Result<()>
where
A: MultiObjectiveAlgorithm<3, Genome = Reals>,
F: MultiFitnessFunction<Reals, 3> + Sync,
{
let mut record = trace.front(name, *problem);
let outcome = MultiEngine::new(algorithm, fitness)
.stop_when(Stop::generations(generations))
.on_generation(|snapshot| record(snapshot))
.run()?;
let scores: Vec<([f64; 3], f64)> = outcome
.front()
.iter()
.map(|x| problem.evaluate(x.genome()))
.collect();
let feasible: Vec<[f64; 3]> = scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect();
let noun = if scores.len() == 1 {
"solution"
} else {
"solutions"
};
print!(
"{name}, {generations} generations: {} {noun}, ",
scores.len()
);
if feasible.is_empty() {
let least = scores.iter().map(|s| s.1).fold(f64::INFINITY, f64::min);
println!("none feasible, the least violation {least:.4}");
return Ok(());
}
let found = normalized(problem, &feasible);
let optimal = normalized(problem, &problem.optimal_front(2_000).expect("known"));
let distance = igd_plus(&found, &optimal, &[Minimize; 3]);
let volume = hypervolume(&found, &REFERENCE, &[Minimize; 3]);
let percent = 100.0 * volume / hypervolume(sample, &REFERENCE, &[Minimize; 3]);
println!(
"{} feasible, IGD+ {distance:.4}, hypervolume {volume:.4}, {percent:.1}% of the \
sample's",
feasible.len()
);
Ok(())
}
// the objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in each
pub fn normalized(problem: &Dc2Dtlz1<3>, points: &[[f64; 3]]) -> Vec<[f64; 3]> {
let ideal = problem.ideal_point().expect("known");
let nadir = problem.nadir_point().expect("known");
let scale = |p: &[f64; 3]| std::array::from_fn(|j| (p[j] - ideal[j]) / (nadir[j] - ideal[j]));
points.iter().map(scale).collect()
}
python examples/dc2_dtlz1_3obj/main.py
"""DC2-DTLZ1 with 3 objectives: minimize three objectives of DTLZ1 subject to two constraints on its
distance function, which leave almost nothing but the front feasible, with NSGA-III.
From genoxide's problems.Dc2Dtlz1; run evaluates it in Rust. Runs NSGA-III with the settings of the
C-TAEA paper's C-NSGA-III, a population of 92 for 1,000 generations, twice: with constraint
dominance, and on DTLZ1 without the constraints. Prints each final front's size, how many of its
solutions are feasible, their IGD+ to 2,000 points of the optimal front and their hypervolume, with
the objectives normalized by the front's ideal and nadir points, as a share of that of a sample of
the front with at least as many points as the population; then the hypervolumes of the whole front
and of the sample.
With ``GENOXIDE_TRACE=<file>``, it also writes a trace of its runs for the plot on the example's
page, with trace.py.
python examples/dc2_dtlz1_3obj/main.py
"""
import genoxide as gx
from trace import Trace
# the reference point of the hypervolume, with the objectives normalized by the front's ideal and
# nadir points
REFERENCE = [1.1, 1.1, 1.1]
problem = gx.problems.Dc2Dtlz1()
ideal, nadir = problem.ideal_point, problem.nadir_point
rate = 1 / problem.dimensions
def normalized(points):
"""The objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in
each."""
return (points - ideal) / (nadir - ideal)
# with GENOXIDE_TRACE=<file>, a trace of the runs for the plot on the example's page
trace = Trace(problem, normalized, REFERENCE)
# the sample: at least as many points of the optimal front as the population has, about what a
# front of that many solutions can be
sample = normalized(problem.optimal_front(92))
def run(name, algorithm, fitness, generations):
"""Runs ``algorithm`` on ``fitness`` for ``generations``, and prints its final front's size,
how many of its solutions the problem finds feasible, their IGD+ to 2,000 points of the
optimal front and their hypervolume, with normalized objectives, as a share of the
sample's."""
result = algorithm.run(fitness, generations=generations, on_generation=trace.front(name))
objectives, violations = problem.evaluate(result.front_genomes)
feasible = objectives[violations == 0]
noun = "solution" if len(objectives) == 1 else "solutions"
start = f"{name}, {generations} generations: {len(objectives)} {noun}, "
if len(feasible) == 0:
print(start + f"none feasible, the least violation {violations.min():.4f}")
return
optimal = normalized(problem.optimal_front(2_000))
distance = gx.indicators.igd_plus(normalized(feasible), optimal)
volume = gx.indicators.hypervolume(normalized(feasible), REFERENCE)
percent = 100 * volume / gx.indicators.hypervolume(sample, REFERENCE)
print(
start + f"{len(feasible)} feasible, IGD+ {distance:.4f}, hypervolume {volume:.4f}, "
f"{percent:.1f}% of the sample's"
)
# NSGA-III with the settings of the C-TAEA paper's C-NSGA-III, constraint dominance
algorithm = gx.Nsga3(
problem.genome,
objectives=problem.objectives,
reference_directions=gx.das_dennis(3, 12),
population_size=92,
crossover=gx.SimulatedBinaryCrossover(30),
crossover_rate=1.0,
mutation=gx.PolynomialMutation(20, rate=rate),
seed=1,
)
run("NSGA-III", algorithm, problem, 1_000)
# the same on DTLZ1, without the constraints, whose solutions the problem then scores
algorithm = gx.Nsga3(
problem.genome,
objectives=problem.objectives,
reference_directions=gx.das_dennis(3, 12),
population_size=92,
crossover=gx.SimulatedBinaryCrossover(30),
crossover_rate=1.0,
mutation=gx.PolynomialMutation(20, rate=rate),
seed=1,
)
run("NSGA-III, unconstrained", algorithm, gx.problems.Dtlz1(variables=problem.dimensions), 1_000)
whole = normalized(problem.optimal_front(3_000))
print(
f"the whole front: hypervolume {gx.indicators.hypervolume(whole, REFERENCE):.4f}; "
f"a sample of {len(sample)} of its points: "
f"{gx.indicators.hypervolume(sample, REFERENCE):.4f}"
)
trace.write()
What it prints, from a seeded run:
NSGA-III, 1000 generations: 1 solution, none feasible, the least violation 0.8975
NSGA-III, unconstrained, 1000 generations: 92 solutions, 92 feasible, IGD+ 0.0289, hypervolume 1.1203, 99.7% of the sample's
the whole front: hypervolume 1.1577; a sample of 105 of its points: 1.1239