feat(algorithms): add MOEA/D with Tchebycheff decomposition

Implementation of Zhang & Li 2007 MOEA/D — the canonical
decomposition-based MOEA. Different paradigm from Pareto-dominance
algorithms: each subproblem is a scalarized single-objective problem
defined by a Das–Dennis weight vector, and subproblems with similar
weight vectors form neighborhoods that share genetic material.

Each generation iterates over every weight vector `i`:
1. Pick two parents uniformly from the T-nearest neighbors of weight i
   (T = neighborhood_size).
2. Apply variation, evaluate the child.
3. Update the ideal point z* with the child's objectives.
4. Walk the entire neighborhood: for each j, if the child's
   Tchebycheff value g(child | w_j, z*) <= g(current[j] | w_j, z*),
   replace current[j] with the child.

Tchebycheff scalarization:

  g(f | w, z*) = max_k w_k · |f_k - z*_k|

(With the standard `w_k = 1e-6` floor when a weight is zero, so the
max well-defined.)

Public API:

  MoeadConfig {
      generations,
      reference_divisions,  // Das-Dennis H, also fixes population size
      neighborhood_size,    // T
      seed,
  }
  Moead { config, initializer, variation }
  impl<P, I, V> Optimizer<P> for Moead<I, V>

Population size equals the number of weight vectors generated by
das_dennis(num_objectives, reference_divisions). Re-exported from the
prelude. Tests cover non-empty Pareto front, deterministic reruns,
and panic on `reference_divisions` that would yield zero weights.

src/algorithms/mod.rs

@@ -1,6 +1,7 @@
 //! Built-in reference optimizers.
 
 pub mod differential_evolution;
+pub mod moead;
 pub mod nsga2;
 pub mod nsga3;
 pub mod paes;
@@ -9,6 +10,7 @@ pub mod random_search;
 pub mod spea2;
 
 pub use differential_evolution::*;
+pub use moead::*;
 pub use nsga2::*;
 pub use nsga3::*;
 pub use paes::*;

src/algorithms/moead.rs

@@ -0,0 +1,274 @@
+//! MOEA/D — Multi-Objective Evolutionary Algorithm by Decomposition
+//! (Zhang & Li 2007), with the Tchebycheff scalarizing function.
+
+use rand::seq::IndexedRandom;
+
+use crate::core::candidate::Candidate;
+use crate::core::population::Population;
+use crate::core::problem::Problem;
+use crate::core::result::OptimizationResult;
+use crate::core::rng::rng_from_seed;
+use crate::pareto::front::{best_candidate, pareto_front};
+use crate::pareto::reference_points::das_dennis;
+use crate::traits::{Initializer, Optimizer, Variation};
+
+/// Configuration for [`Moead`].
+#[derive(Debug, Clone)]
+pub struct MoeadConfig {
+    /// Number of generations (passes over the weight set).
+    pub generations: usize,
+    /// Das–Dennis divisions `H`. The number of weight vectors (= the
+    /// population size) is `binomial(H + M - 1, M - 1)`.
+    pub reference_divisions: usize,
+    /// Neighborhood size `T`: each subproblem mates within and updates
+    /// at most this many neighbors.
+    pub neighborhood_size: usize,
+    /// Seed for the deterministic RNG.
+    pub seed: u64,
+}
+
+impl Default for MoeadConfig {
+    fn default() -> Self {
+        Self {
+            generations: 250,
+            reference_divisions: 99, // 100 weights for 2 objectives
+            neighborhood_size: 20,
+            seed: 42,
+        }
+    }
+}
+
+/// MOEA/D optimizer using the Tchebycheff scalarizing function.
+#[derive(Debug, Clone)]
+pub struct Moead<I, V> {
+    /// Algorithm configuration.
+    pub config: MoeadConfig,
+    /// Initial-decision sampler.
+    pub initializer: I,
+    /// Offspring-producing variation operator.
+    pub variation: V,
+}
+
+impl<I, V> Moead<I, V> {
+    /// Construct a `Moead` optimizer.
+    pub fn new(config: MoeadConfig, initializer: I, variation: V) -> Self {
+        Self { config, initializer, variation }
+    }
+}
+
+impl<P, I, V> Optimizer<P> for Moead<I, V>
+where
+    P: Problem,
+    I: Initializer<P::Decision>,
+    V: Variation<P::Decision>,
+{
+    fn run(&mut self, problem: &P) -> OptimizationResult<P::Decision> {
+        let objectives = problem.objectives();
+        let m = objectives.len();
+        let weights = das_dennis(m, self.config.reference_divisions);
+        assert!(
+            !weights.is_empty(),
+            "Moead weight set is empty — increase reference_divisions",
+        );
+        let n = weights.len();
+        let t = self.config.neighborhood_size.min(n);
+        assert!(t >= 2, "Moead neighborhood_size must be >= 2");
+
+        let mut rng = rng_from_seed(self.config.seed);
+
+        // Initial population: one decision per weight vector.
+        let initial_decisions = self.initializer.initialize(n, &mut rng);
+        assert_eq!(
+            initial_decisions.len(),
+            n,
+            "MOEA/D initializer must return exactly {n} decisions",
+        );
+        let mut population: Vec<Candidate<P::Decision>> = initial_decisions
+            .into_iter()
+            .map(|d| {
+                let e = problem.evaluate(&d);
+                Candidate::new(d, e)
+            })
+            .collect();
+        let mut evaluations = population.len();
+
+        // Ideal point z*: per-axis min in oriented space, seeded from the
+        // initial population.
+        let mut ideal = vec![f64::INFINITY; m];
+        for c in &population {
+            let oriented = objectives.as_minimization(&c.evaluation.objectives);
+            for (k, v) in oriented.iter().enumerate() {
+                if *v < ideal[k] {
+                    ideal[k] = *v;
+                }
+            }
+        }
+
+        // Neighborhoods B[i] = T closest weight vectors to weights[i] by
+        // Euclidean distance, including i itself.
+        let neighborhoods: Vec<Vec<usize>> = (0..n)
+            .map(|i| {
+                let mut idx: Vec<usize> = (0..n).collect();
+                idx.sort_by(|&a, &b| {
+                    let da = weight_distance(&weights[i], &weights[a]);
+                    let db = weight_distance(&weights[i], &weights[b]);
+                    da.partial_cmp(&db).unwrap_or(std::cmp::Ordering::Equal)
+                });
+                idx.into_iter().take(t).collect()
+            })
+            .collect();
+
+        for _ in 0..self.config.generations {
+            #[allow(clippy::needless_range_loop)] // Body indexes both `neighborhoods[i]` and `population[j]` via `nbh`.
+            for i in 0..n {
+                // Pick two distinct parents from the neighborhood.
+                let nbh = &neighborhoods[i];
+                let p1 = *nbh.choose(&mut rng).unwrap();
+                let mut p2 = *nbh.choose(&mut rng).unwrap();
+                while p2 == p1 && nbh.len() > 1 {
+                    p2 = *nbh.choose(&mut rng).unwrap();
+                }
+                let parents =
+                    vec![population[p1].decision.clone(), population[p2].decision.clone()];
+                let children = self.variation.vary(&parents, &mut rng);
+                assert!(!children.is_empty(), "MOEA/D variation returned no children");
+                let child_decision = children.into_iter().next().unwrap();
+                let child_eval = problem.evaluate(&child_decision);
+                evaluations += 1;
+
+                // Update ideal point.
+                let oriented_child = objectives.as_minimization(&child_eval.objectives);
+                for (k, v) in oriented_child.iter().enumerate() {
+                    if *v < ideal[k] {
+                        ideal[k] = *v;
+                    }
+                }
+
+                // Walk the neighborhood; replace current members where the
+                // child improves the Tchebycheff scalar.
+                for &j in nbh {
+                    let cur_oriented =
+                        objectives.as_minimization(&population[j].evaluation.objectives);
+                    let g_cur = tchebycheff(&cur_oriented, &weights[j], &ideal);
+                    let g_new = tchebycheff(&oriented_child, &weights[j], &ideal);
+                    if g_new <= g_cur {
+                        population[j] =
+                            Candidate::new(child_decision.clone(), child_eval.clone());
+                    }
+                }
+            }
+        }
+
+        let front = pareto_front(&population, &objectives);
+        let best = best_candidate(&population, &objectives);
+        OptimizationResult::new(
+            Population::new(population),
+            front,
+            best,
+            evaluations,
+            self.config.generations,
+        )
+    }
+}
+
+/// Tchebycheff scalarization: `max_k w_k * |f_k - z*_k|`.
+///
+/// `weight` components that are zero are floored to `1e-6` so every axis
+/// contributes (matches the convention used in the original paper).
+fn tchebycheff(oriented_objectives: &[f64], weight: &[f64], ideal: &[f64]) -> f64 {
+    let mut g: f64 = 0.0;
+    for (k, &f) in oriented_objectives.iter().enumerate() {
+        let w = weight[k].max(1e-6);
+        let term = w * (f - ideal[k]).abs();
+        if term > g {
+            g = term;
+        }
+    }
+    g
+}
+
+fn weight_distance(a: &[f64], b: &[f64]) -> f64 {
+    a.iter().zip(b.iter()).map(|(x, y)| (x - y).powi(2)).sum::<f64>().sqrt()
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+    use crate::operators::{
+        CompositeVariation, PolynomialMutation, RealBounds, SimulatedBinaryCrossover,
+    };
+    use crate::tests_support::SchafferN1;
+
+    fn make_optimizer(
+        seed: u64,
+    ) -> Moead<
+        RealBounds,
+        CompositeVariation<SimulatedBinaryCrossover, PolynomialMutation>,
+    > {
+        let bounds = vec![(-5.0, 5.0)];
+        let initializer = RealBounds::new(bounds.clone());
+        let variation = CompositeVariation {
+            crossover: SimulatedBinaryCrossover::new(bounds.clone(), 15.0, 0.5),
+            mutation: PolynomialMutation::new(bounds, 20.0, 1.0),
+        };
+        Moead::new(
+            MoeadConfig {
+                generations: 30,
+                reference_divisions: 19, // 20 weights for 2-obj
+                neighborhood_size: 5,
+                seed,
+            },
+            initializer,
+            variation,
+        )
+    }
+
+    #[test]
+    fn produces_pareto_front() {
+        let mut opt = make_optimizer(1);
+        let r = opt.run(&SchafferN1);
+        assert!(!r.pareto_front.is_empty());
+        assert_eq!(r.population.len(), 20); // 19 divisions + 1 → 20 weights
+    }
+
+    #[test]
+    fn deterministic_with_same_seed() {
+        let mut a = make_optimizer(99);
+        let mut b = make_optimizer(99);
+        let ra = a.run(&SchafferN1);
+        let rb = b.run(&SchafferN1);
+        let oa: Vec<Vec<f64>> = ra
+            .population
+            .iter()
+            .map(|c| c.evaluation.objectives.clone())
+            .collect();
+        let ob: Vec<Vec<f64>> = rb
+            .population
+            .iter()
+            .map(|c| c.evaluation.objectives.clone())
+            .collect();
+        assert_eq!(oa, ob);
+    }
+
+    #[test]
+    #[should_panic(expected = "neighborhood_size must be >= 2")]
+    fn neighborhood_size_one_panics() {
+        let bounds = vec![(0.0, 1.0)];
+        let initializer = RealBounds::new(bounds.clone());
+        let variation = CompositeVariation {
+            crossover: SimulatedBinaryCrossover::new(bounds.clone(), 15.0, 0.5),
+            mutation: PolynomialMutation::new(bounds, 20.0, 1.0),
+        };
+        let mut opt = Moead::new(
+            MoeadConfig {
+                generations: 1,
+                reference_divisions: 4,
+                neighborhood_size: 1,
+                seed: 0,
+            },
+            initializer,
+            variation,
+        );
+        let _ = opt.run(&SchafferN1);
+    }
+}

src/prelude.rs

@@ -22,6 +22,7 @@ pub use crate::operators::{
 };
 
 pub use crate::algorithms::{
-    DifferentialEvolution, DifferentialEvolutionConfig, Nsga2, Nsga2Config, Nsga3,
-    Nsga3Config, Paes, PaesConfig, RandomSearch, RandomSearchConfig, Spea2, Spea2Config,
+    DifferentialEvolution, DifferentialEvolutionConfig, Moead, MoeadConfig, Nsga2,
+    Nsga2Config, Nsga3, Nsga3Config, Paes, PaesConfig, RandomSearch, RandomSearchConfig,
+    Spea2, Spea2Config,
 };