NSGA-III's value over NSGA-II shows up at 3+ objectives, where
crowding distance loses its diversity signal. Adds a third comparison
section to `examples/compare.rs`:
DTLZ2 (3-objective, 12-D, the textbook benchmark for many-objective
algorithms): unit-sphere-octant Pareto front. Compares RandomSearch,
NSGA-II, SPEA2, and NSGA-III on:
- mean distance from front points to the unit sphere
(closed-form: |1 - sqrt(f1² + f2² + f3²)|),
- spacing,
- front size,
- wall-clock ms.
NSGA-III config: H=12 reference divisions (91 reference points,
matching the canonical setup from Deb & Jain 2014).
Also wires NSGA-III into the existing ZDT1 (2-objective) section even
though it's not its sweet spot — useful as a regression check that the
algorithm at least keeps up with NSGA-II on bi-objective problems.
A comparison example that runs every applicable optimizer on ZDT1 and
Rastrigin across N seeds and reports mean ± stddev for each quality
metric. Designed so a new algorithm slots in by adding a single runner
function — no harness changes needed.
ZDT1 (multi-objective, dim=30):
Reports hypervolume_2d (against ref point [1.1, 1.1]), spacing, mean
L2 distance to the analytical Pareto front, front size, and wall-clock
ms. RandomSearch, PAES, and NSGA-II all use bounds-aware operators
(RealBounds, BoundedGaussianMutation, SBX+PolyMut) so the Problem
itself stays unclamped — apples-to-apples.
Rastrigin (single-objective, dim=5):
Reports mean ± stddev best objective and ms. RandomSearch, PAES,
NSGA-II (degenerate single-obj case), and DE.
Default budget: 10 seeds × 25,000 evaluations on ZDT1, × 50,000 on
Rastrigin. Run with:
cargo run --release --example compare
Replace the v0.1 `GaussianMutation` + clamp-inside-`evaluate` setup
with the canonical NSGA-II operator pair: SBX (η_c=15, per-var prob 0.5)
followed by PolynomialMutation (η_m=20, per-var prob 1/dim), composed
via `CompositeVariation`. Both are bounds-aware on their own, so the
in-evaluate clamping is dropped.
Result on ZDT1 (dim=30, pop=100, gens=1000, seed=42): mean L2 distance
to the analytical Pareto front is 0.00152 — comfortably within the
published NSGA-II range for this benchmark.
Note on the previous number: the v0.1 setup reported 0.00072 at 40k
evals, but that was an artifact of clamping inside `evaluate`. Out-of-
bounds Gaussian mutations on `x[0]` were snapping to 0, which
coincides with the ZDT1 Pareto-front extreme (f1=0). The new operator
pair has no such free lunch — it runs the actual NSGA-II algorithm —
and the new measurement is what honest convergence on ZDT1 actually
looks like.
Generations bumped from 400 to 1000 (40k → 100k evaluations) to give
the operators headroom; matches the budget DE uses for Rastrigin so
the example feels balanced.
Two canonical optimization benchmarks in a single runnable example:
- ZDT1 (Zitzler-Deb-Thiele 1): 30-D, two minimization objectives,
closed-form Pareto front \\(f_2 = 1 - \\sqrt{f_1}\\) for
\\(f_1 \\in [0, 1]\\). Solved with NSGA-II.
- Rastrigin: highly multimodal single-objective, global minimum
\\(f = 0\\) at the origin. Solved with DE.
Both are public-domain mathematical formulas. Implemented as Problem
impls in examples/benchmarks.rs; main() runs each, prints front /
best, and (for ZDT1) reports the mean L2 distance from the known
analytical Pareto front so the example doubles as a sanity check on
solution quality.
The three runnable examples called out in spec §18.5 / §19. All open
with `use heuropt::prelude::*;` so they double as a check that the
prelude is sufficient on its own:
- toy_nsga2.rs: Schaffer N.1 solved with NSGA-II.
- random_search.rs: 2D sphere solved with RandomSearch.
- custom_optimizer.rs: a minimal hill-climber implementing
`Optimizer<P>` directly, demonstrating spec §2.3.
swaits