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DTLZ9 with 3 objectives

The problem

Deb, Thiele, Laumanns and Zitzler's report (2001) builds its scalable test problems three ways; the last, the constraint surface approach, gives DTLZ8 and DTLZ9: each objective is a function of its own block of variables, so that the objectives can take values independently, and constraints on the objectives cut the front out of their space. (The report's DTLZ8 is the 2002 conference paper's DTLZ7, and its DTLZ9 isn't in the paper; genoxide follows the report.)

DTLZ9 (section 8.9, eq. 29), with 3 objectives and 30 variables in [0, 1]:

minimize   fⱼ = Σ xᵢ^0.1 over the j-th block of 10 variables,  j = 1, 2, 3
subject to f₃² + f₁² − 1 ≥ 0
           f₃² + f₂² − 1 ≥ 0

The blocks are DTLZ8's (see its page). Unlike DTLZ8's, the objectives are sums, from 0 to 10.

The front is the curve f₁ = f₂ = cos θ, f₃ = sin θ, θ in [0, π/2], where both constraints meet: for a given f₃ in [0, 1], each constraint asks fⱼ ≥ √(1 − f₃²) alone. With f₃ and either other objective it's a quarter of the unit circle. Its ideal point is (0, 0, 0) and its nadir point (1, 1, 1). The report describes the front but doesn't write it out; genoxide derives it from the constraints, and its tests check it against random feasible solutions.

What makes it hard

The power 0.1 first: on the front each block sums to at most 1 over 10 variables, so each variable is about (fⱼ/10)¹⁰, below 10⁻¹⁰, or exactly 0, and a random solution has its objectives near the top of their range: "the density of solutions gets thinner towards the Pareto-optimal region". Then the curve: as on DTLZ8's line, a solution with f₁ ≠ f₂ next to it is only weakly dominated, and a population spreads over the surfaces around the curve (the report's figures 30 and 31).

Representation

A Real genome of 30 genes in [0, 1], the report's n = 10M. The problem is genoxide's Dtlz9::<3>, whose fitness is the three objectives and the total constraint violation, 0 when it is feasible; in Python, gx.problems.Dtlz9(). 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, with a population of 100, simulated binary crossover with η = 20 and polynomial mutation with η = 20 at a rate of 1/30 per gene:

  • NSGA-II (Deb, Pratap, Agarwal and Meyarivan, 2002, IEEE Transactions on Evolutionary Computation 6(2): 182-197), crossover at a rate of 0.9, for 500 generations, as long as the report's runs (its figures 30 and 31);
  • SMS-EMOA (Beume, Naujoks and Emmerich, 2007, European Journal of Operational Research 181(3): 1653-1669), which keeps the solutions that add the most hypervolume, with as many children a generation as the population, for 20,000 generations, 2,000,000 evaluations. A solution that only a point of the front weakly dominates adds almost no hypervolume next to it, and is the first to go.

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 100 points of the front, about what a front of 100 solutions spread like it has.

NSGA-II's run with seed 1 ends with 100 solutions, 100 feasible, IGD+ 3.4675, hypervolume 0.0000, 0.0% of the sample's; SMS-EMOA's with 100 solutions, 100 feasible, IGD+ 0.0024, hypervolume 0.2661, 99.9% of the sample's. Over seeds 1 to 20, NSGA-II ends with IGD+ from 3.3579 to 4.8193 and 0.0% to 0.0% of the sample's hypervolume, no run reaching the target, and SMS-EMOA with IGD+ from 0.0024 to 0.0026 and 99.8% to 99.9% of the sample's hypervolume, every run reaching the target. SMS-EMOA's IGD+ is the sample's own: its 100 solutions lie on the curve, spread along it. The report's NSGA-II and SPEA2, after 500 generations, "could not cover the entire range of the circle", with many solutions away from it.

Reference: Deb, K., Thiele, L., Laumanns, M. and Zitzler, E. (2001). Scalable Test Problems for Evolutionary Multi-Objective Optimization. TIK-Report 112, Computer Engineering and Networks Laboratory, ETH Zürich.

Known optimum: the curve f₁ = f₂ = cos θ, f₃ = sin θ, θ in [0, π/2]; ideal point (0, 0, 0), nadir point (1, 1, 1); hypervolume 0.2697 (normalized objectives, reference point (1.1, 1.1, 1.1))

Source: examples/dtlz9_3obj

Interactive run: tachsin.gr/projects/genoxide/examples/dtlz9-3obj

cargo run --release --example dtlz9_3obj

//! DTLZ9 with 3 objectives: minimize three objectives over 30 variables subject to 2 constraints,
//! whose front is a curve, with NSGA-II and SMS-EMOA.
//!
//! From genoxide's `multi::problems::Dtlz9`. Runs NSGA-II for the report's 500 generations and
//! SMS-EMOA for 20,000, with a population of 100. 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 dtlz9_3obj
//! ```

mod trace;

use genoxide::Objective::Minimize;
use genoxide::multi::MultiFitnessFunction;
use genoxide::multi::indicator::{hypervolume, igd_plus};
use genoxide::multi::problems::{Dtlz9, 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 = Dtlz9::<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(100).expect("known"));
    let rate = 1.0 / problem.variables() as f64;
    // NSGA-II for the report's 500 generations
    let algorithm = Nsga2::builder(problem.representation(), [Minimize; 3])
        .population_size(100)
        .crossover(SimulatedBinaryCrossover::new(20.0)?)
        .mutate(PolynomialMutation::per_gene(rate, 20.0)?)
        .seed(1)
        .build()?;
    run(
        &problem, "NSGA-II", algorithm, problem, 500, &sample, &mut trace,
    )?;
    // SMS-EMOA, which keeps the solutions that add the most hypervolume, for 20,000 generations
    let algorithm = SmsEmoa::builder(problem.representation(), [Minimize; 3])
        .population_size(100)
        .crossover(SimulatedBinaryCrossover::new(20.0)?)
        .mutate(PolynomialMutation::per_gene(rate, 20.0)?)
        .seed(1)
        .build()?;
    run(
        &problem, "SMS-EMOA", algorithm, problem, 20_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: &Dtlz9<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: &Dtlz9<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/dtlz9_3obj/main.py

"""DTLZ9 with 3 objectives: minimize three objectives over 30 variables subject to 2 constraints,
whose front is a curve, with NSGA-II and SMS-EMOA.

From genoxide's problems.Dtlz9; run evaluates it in Rust. Runs NSGA-II for the report's 500
generations and SMS-EMOA for 20,000, with a population of 100. 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/dtlz9_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.Dtlz9()
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(100))


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-II for the report's 500 generations
algorithm = gx.Nsga2(
    problem.genome,
    objectives=problem.objectives,
    population_size=100,
    crossover=gx.SimulatedBinaryCrossover(20),
    mutation=gx.PolynomialMutation(20, rate=rate),
    seed=1,
)
run("NSGA-II", algorithm, problem, 500)
# SMS-EMOA, which keeps the solutions that add the most hypervolume, for 20,000 generations
algorithm = gx.SmsEmoa(
    problem.genome,
    objectives=problem.objectives,
    population_size=100,
    crossover=gx.SimulatedBinaryCrossover(20),
    mutation=gx.PolynomialMutation(20, rate=rate),
    seed=1,
)
run("SMS-EMOA", algorithm, problem, 20_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-II, 500 generations: 100 solutions, 100 feasible, IGD+ 3.4675, hypervolume 0.0000, 0.0% of the sample's
SMS-EMOA, 20000 generations: 100 solutions, 100 feasible, IGD+ 0.0024, hypervolume 0.2661, 99.9% of the sample's
the whole front: hypervolume 0.2697; a sample of 100 of its points: 0.2664