docs: promote MOEA/D to the multi/many-objective default
The compare harness shows MOEA/D is the single most consistent performer: top-3 on every multi- and many-objective table (convex, disconnected, spherical and linear fronts; 2 through 10 objectives) and fastest or near-fastest every time. No other algorithm is close to that consistency. This matches the literature view of MOEA/D as a strong, robust, scalable baseline -- with the known caveat that weight-vector spread can leave gaps on highly irregular fronts (the DTLZ/ZDT suite doesn't stress that). But both decision trees buried it: the README filed it under "Want decomposition / weight-vector style" -- a stylistic branch -- and framed it as a speed pick; the book left it out of the TL;DR table entirely. Meanwhile NSGA-II was the listed 2-3-objective default despite losing to MOEA/D on every table and collapsing past ~4 objectives. Both trees now lead the multi- and many-objective branches with MOEA/D, keep NSGA-II as the well-understood alternative and the combinatorial go-to, add MOEA/D to the TL;DR / quick-reference tables, and soften the NSGA-III "strong default" framing to match the data. Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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@@ -121,15 +121,27 @@ picker.
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### Strong default
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[NSGA-II][Nsga2] is the canonical Pareto-based EA. Fast, well-understood,
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maintains diversity via crowding distance. On the harness it lands
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on the Pareto front of every test problem.
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[MOEA/D][Moead] is the most consistent performer on the harness. It
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decomposes the problem into many scalar sub-problems (Tchebycheff or
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weighted sum) and solves them in parallel — fast per generation, and
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robust: it finishes **top-3 on every multi- and many-objective table**
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(convex, disconnected, spherical and linear fronts; 2 through 10
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objectives) and is consistently the fastest or near-fastest. It rarely
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*wins* a table outright — a specialist usually does — but it never lands
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badly. One caveat from the literature: MOEA/D's spread depends on the
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weight-vector distribution and the scalarizing function, so it can leave
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gaps on highly irregular or degenerate fronts; the DTLZ/ZDT suite here
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doesn't stress that.
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NSGA-II is generic over the decision type — drop in
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[`ShuffledPermutation`] + a permutation crossover and it solves
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bi-objective TSP; drop in a binary initializer and [`BitFlipMutation`]
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and it solves bi-objective knapsack. See
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[Multi-objective combinatorial problems](./cookbook/multi-objective-combinatorial.md).
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[NSGA-II][Nsga2] is the other safe default — the canonical Pareto-based
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EA: fast, well-understood, diversity-preserving via crowding distance.
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On the harness it's edged out by MOEA/D on every multi-objective table
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and degrades past ~4 objectives (see the many-objective section), but it
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stays a solid 2–3-objective pick and is the established choice for
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*combinatorial* encodings: drop in [`ShuffledPermutation`] + a
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permutation crossover and it solves bi-objective TSP; drop in a binary
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initializer and [`BitFlipMutation`] and it solves bi-objective knapsack.
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See [Multi-objective combinatorial problems](./cookbook/multi-objective-combinatorial.md).
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### Real-valued, smooth front, want best convergence
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@@ -137,7 +149,7 @@ and it solves bi-objective knapsack. See
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hypervolume outright and converges 100× tighter than the
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dominance-based methods.
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### Better front quality than NSGA-II
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### Better front quality than the default
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[IBEA][Ibea] (indicator-based) is consistently the best of the
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dominance-based methods on the harness — wins ZDT3 hypervolume and
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@@ -152,13 +164,6 @@ in theory; in practice on the harness budgets here it underperforms
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NSGA-II. Worth the higher per-step cost only when exact HV
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contribution is the right discriminator.
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### Decomposition / weight-vector style
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[MOEA/D][Moead] decomposes the multi-objective problem into many scalar
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sub-problems (Tchebycheff or weighted sum) and solves them in
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parallel. Very fast per generation; scales naturally to many
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objectives.
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### Disconnected or non-convex front
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A *disconnected* front (separate arcs, like ZDT3) and a *non-convex but
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@@ -194,18 +199,31 @@ population.
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## Step 2 — many-objective (4+)
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### Strong default
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[MOEA/D][Moead] again. Decomposition sidesteps the *dominance resistance*
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that breaks Pareto-based methods at high objective count — each scalar
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sub-problem still has a clear best, even when almost every pair of
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solutions is mutually non-dominated. On the harness it is **#2 on every
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many-objective table** (DTLZ2 at 4 and 10 objectives, DTLZ1 at 8), and
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fast every time. [NSGA-II][Nsga2] is the cautionary tale: on DTLZ2 at 10
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objectives it finishes *last — behind random search* — because its
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crowding distance has no dominance signal left to refine.
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### Linear / simplex-shaped front (e.g., DTLZ1)
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[GrEA][Grea] — grid coords drive ranking. On DTLZ1 it beats NSGA-III by
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3× and AGE-MOEA by 2.5×.
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[GrEA][Grea] — grid coords drive ranking. On 3-objective DTLZ1 it beats
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NSGA-III by 3× and AGE-MOEA by 2.5×, and it wins the 8-objective DTLZ1
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table outright.
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[MOEA/D][Moead] — decomposition shines on linear fronts; second on DTLZ1
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and among the fastest per generation.
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[MOEA/D][Moead] — also #2 on both DTLZ1 tables.
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### Curved / unknown front geometry
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[NSGA-III][Nsga3] — reference-point niching; canonical many-objective method;
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strong default when the front isn't simplex-shaped.
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[NSGA-III][Nsga3] — reference-point niching; the canonical many-objective
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method by reputation, though on the harness MOEA/D outperforms it on
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every table. Reach for it when you specifically want reference-point
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niching.
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[AGE-MOEA][AgeMoea] — estimates L_p geometry per generation.
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@@ -265,11 +283,12 @@ method on every algorithm in the catalog. See the
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| Multimodal single-objective continuous | [IPOP-CMA-ES][IpopCmaEs] or [Differential Evolution][DifferentialEvolution] |
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| Expensive single-objective | [Bayesian Optimization][BayesianOpt] or [TPE] |
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| Multi-fidelity single-objective | [Hyperband] |
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| 2- or 3-objective default | [NSGA-II][Nsga2] |
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| 2- or 3-objective default | [MOEA/D][Moead] (or [NSGA-II][Nsga2]) |
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| Many-objective default | [MOEA/D][Moead] |
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| 2-objective real-valued smooth front | [MOPSO][Mopso] |
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| Disconnected / non-convex front | [IBEA][Ibea] |
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| Many-objective default (curved front) | [NSGA-III][Nsga3] |
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| Many-objective linear / simplex front | [GrEA][Grea] |
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| Disconnected front | [IBEA][Ibea] |
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| Many-objective, curved front | [NSGA-III][Nsga3] |
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| Many-objective, linear / simplex front | [GrEA][Grea] |
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| Permutation problem (TSP with distance matrix) | [Ant Colony][AntColonyTsp] |
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| Generic permutation problem | [GA][GeneticAlgorithm] + permutation toolkit |
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| Bi-objective combinatorial (TSP / scheduling / knapsack) | [NSGA-II][Nsga2] + matching encoding operators |
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