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docs: promote MOEA/D to the multi/many-objective default

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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>
This commit is contained in:
swaits
2026-05-14 09:06:18 -06:00
co-authored by Claude Opus 4.7
parent 398c82326a
commit 708f1ce9cd
2 changed files with 81 additions and 46 deletions
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README.md
+36 -20
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@@ -426,8 +426,20 @@ START
│
├─ 2 or 3 (multi-objective)
│ │
│ ├─ Strong default, fast, well-understood
│ │ → NSGA-II
│ ├─ Strong default — top-3 on every multi- and
│ │ many-objective table on the harness, fastest or
│ │ near-fastest every time
│ │ → MOEA/D (decomposition into scalar sub-problems;
│ │ robust across convex / disconnected /
│ │ spherical / linear fronts and 2–10
│ │ objectives. Caveat: weight-vector spread
│ │ can leave gaps on highly irregular or
│ │ degenerate fronts)
│ │ → NSGA-II (canonical Pareto EA; well-understood and
│ │ the established choice for combinatorial
│ │ encodings — but edged out by MOEA/D on
│ │ every MO table here, and fades past
│ │ ~4 objectives)
│ │
│ ├─ Real-valued, smooth front, want best convergence
│ │ → MOPSO (multi-objective PSO; on the benches
@@ -435,7 +447,7 @@ START
│ │ convergence by 100× over the
│ │ dominance-based methods)
│ │
│ ├─ Want better front quality than NSGA-II
│ ├─ Want better front quality than the default
│ │ → IBEA (indicator-based; consistently the best
│ │ of the dominance-based methods on these
│ │ benches — wins ZDT3 HV and DTLZ2 mean
@@ -448,9 +460,6 @@ START
│ │ fronts where exact HV-contribution is
│ │ the right discriminator)
│ │
│ ├─ Want decomposition / weight-vector style
│ │ → MOEA/D (very fast per generation, scales well)
│ │
│ ├─ Disconnected front (separate arcs, e.g. ZDT3)
│ │ → IBEA (wins ZDT3 hypervolume on the harness;
│ │ MOEA/D and NSGA-II follow. Geometry-aware
@@ -469,19 +478,26 @@ START
│ → PAES (1+1 ES with a Pareto archive)
│
└─ 4+ (many-objective)
│
├─ Strong default — #2 on every many-objective table on
│ the harness (DTLZ2 at 4 and 10 objectives, DTLZ1 at 8);
│ decomposition sidesteps the dominance collapse that
│ wrecks Pareto-based EAs at high objective count
│ → MOEA/D
│ (NSGA-II is the cautionary tale: on DTLZ2 at 10
│ objectives it finishes last — behind random search)
│
├─ Linear / simplex-shaped front (e.g., DTLZ1)
│ → GrEA (grid coords drive ranking; on DTLZ1
│ here it beats NSGA-III by 3× and
│ AGE-MOEA by 2.5×)
│ → MOEA/D (decomposition shines on linear fronts;
│ second on DTLZ1, also among the
│ fastest per generation)
│ AGE-MOEA by 2.5×, and wins the
│ 8-objective DTLZ1 table outright)
│ → MOEA/D (also #2 on both DTLZ1 tables)
│
├─ Curved / unknown front geometry
│ → NSGA-III (reference-point niching, canonical;
│ a strong default when the front
│ isn't simplex-shaped)
│ → NSGA-III (reference-point niching; canonical by
│ reputation, but MOEA/D outperforms it
│ on every harness table)
│ → AGE-MOEA (estimates L_p geometry per generation)
│ → RVEA (reference vectors with adaptive penalty)
│
@@ -531,21 +547,21 @@ START
| Algorithm | Objectives | Strengths |
|---|---|---|
| **PAES** | 2–3 | 1+1 ES with Pareto archive |
| **NSGA-II** | 2–3 | canonical Pareto-based EA |
| **SPEA2** | 2–3 | strength + density |
| **MOEA/D** | 2+ | decomposition; the most consistent all-rounder — top-3 on every MO/many-objective table here, fastest or near-fastest |
| **NSGA-II** | 2–3 | canonical Pareto-based EA; well-understood, the go-to for combinatorial encodings — but fades past ~4 objectives |
| **MOPSO** | 2–3 | multi-objective PSO; best convergence on smooth real-valued 2-obj fronts |
| **IBEA** | 2+ | indicator-based; consistently best of the dominance-based methods |
| **IBEA** | 2+ | indicator-based; consistently best of the dominance-based methods; wins disconnected fronts |
| **SPEA2** | 2–3 | strength + density |
| **SMS-EMOA** | 2+ | exact HV-contribution selection; high per-step cost, modest gain |
| **HypE** | 2+ | Monte Carlo HV estimation |
| **HypE** | 2+ | Monte Carlo HV estimation; strong on spherical many-objective fronts |
| **ε-MOEA** | 2+ | ε-grid archive; auto-sized |
| **PESA-II** | 2+ | grid-based region selection |
| **AGE-MOEA** | 2+ | adaptive front-geometry estimation |
| **KnEA** | 2+ | knee-point favored survival |
| **MOEA/D** | 2+ | decomposition; fast per-gen |
| **PAES** | 2–3 | 1+1 ES with Pareto archive |
| **NSGA-III** | 4+ | reference-point niching; strong on curved fronts |
| **RVEA** | 4+ | reference vectors with penalty |
| **GrEA** | 4+ | grid coords drive selection; particularly strong on linear/simplex fronts |
| **GrEA** | 4+ | grid coords drive selection; wins linear/simplex fronts at any objective count |
## Current algorithms
docs/book/src/choosing-an-algorithm.md
+45 -26
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@@ -121,15 +121,27 @@ picker.
### Strong default
[NSGA-II][Nsga2] is the canonical Pareto-based EA. Fast, well-understood,
maintains diversity via crowding distance. On the harness it lands
on the Pareto front of every test problem.
[MOEA/D][Moead] is the most consistent performer on the harness. It
decomposes the problem into many scalar sub-problems (Tchebycheff or
weighted sum) and solves them in parallel — fast per generation, and
robust: it finishes **top-3 on every multi- and many-objective table**
(convex, disconnected, spherical and linear fronts; 2 through 10
objectives) and is consistently the fastest or near-fastest. It rarely
*wins* a table outright — a specialist usually does — but it never lands
badly. One caveat from the literature: MOEA/D's spread depends on the
weight-vector distribution and the scalarizing function, so it can leave
gaps on highly irregular or degenerate fronts; the DTLZ/ZDT suite here
doesn't stress that.
NSGA-II is generic over the decision type — drop in
[`ShuffledPermutation`] + a permutation crossover and it solves
bi-objective TSP; drop in a binary initializer and [`BitFlipMutation`]
and it solves bi-objective knapsack. See
[Multi-objective combinatorial problems](./cookbook/multi-objective-combinatorial.md).
[NSGA-II][Nsga2] is the other safe default — the canonical Pareto-based
EA: fast, well-understood, diversity-preserving via crowding distance.
On the harness it's edged out by MOEA/D on every multi-objective table
and degrades past ~4 objectives (see the many-objective section), but it
stays a solid 2–3-objective pick and is the established choice for
*combinatorial* encodings: drop in [`ShuffledPermutation`] + a
permutation crossover and it solves bi-objective TSP; drop in a binary
initializer and [`BitFlipMutation`] and it solves bi-objective knapsack.
See [Multi-objective combinatorial problems](./cookbook/multi-objective-combinatorial.md).
### Real-valued, smooth front, want best convergence
@@ -137,7 +149,7 @@ and it solves bi-objective knapsack. See
hypervolume outright and converges 100× tighter than the
dominance-based methods.
### Better front quality than NSGA-II
### Better front quality than the default
[IBEA][Ibea] (indicator-based) is consistently the best of the
dominance-based methods on the harness — wins ZDT3 hypervolume and
@@ -152,13 +164,6 @@ in theory; in practice on the harness budgets here it underperforms
NSGA-II. Worth the higher per-step cost only when exact HV
contribution is the right discriminator.
### Decomposition / weight-vector style
[MOEA/D][Moead] decomposes the multi-objective problem into many scalar
sub-problems (Tchebycheff or weighted sum) and solves them in
parallel. Very fast per generation; scales naturally to many
objectives.
### Disconnected or non-convex front
A *disconnected* front (separate arcs, like ZDT3) and a *non-convex but
@@ -194,18 +199,31 @@ population.
## Step 2 — many-objective (4+)
### Strong default
[MOEA/D][Moead] again. Decomposition sidesteps the *dominance resistance*
that breaks Pareto-based methods at high objective count — each scalar
sub-problem still has a clear best, even when almost every pair of
solutions is mutually non-dominated. On the harness it is **#2 on every
many-objective table** (DTLZ2 at 4 and 10 objectives, DTLZ1 at 8), and
fast every time. [NSGA-II][Nsga2] is the cautionary tale: on DTLZ2 at 10
objectives it finishes *last — behind random search* — because its
crowding distance has no dominance signal left to refine.
### Linear / simplex-shaped front (e.g., DTLZ1)
[GrEA][Grea] — grid coords drive ranking. On DTLZ1 it beats NSGA-III by
3× and AGE-MOEA by 2.5×.
[GrEA][Grea] — grid coords drive ranking. On 3-objective DTLZ1 it beats
NSGA-III by 3× and AGE-MOEA by 2.5×, and it wins the 8-objective DTLZ1
table outright.
[MOEA/D][Moead] — decomposition shines on linear fronts; second on DTLZ1
and among the fastest per generation.
[MOEA/D][Moead] — also #2 on both DTLZ1 tables.
### Curved / unknown front geometry
[NSGA-III][Nsga3] — reference-point niching; canonical many-objective method;
strong default when the front isn't simplex-shaped.
[NSGA-III][Nsga3] — reference-point niching; the canonical many-objective
method by reputation, though on the harness MOEA/D outperforms it on
every table. Reach for it when you specifically want reference-point
niching.
[AGE-MOEA][AgeMoea] — estimates L_p geometry per generation.
@@ -265,11 +283,12 @@ method on every algorithm in the catalog. See the
| Multimodal single-objective continuous | [IPOP-CMA-ES][IpopCmaEs] or [Differential Evolution][DifferentialEvolution] |
| Expensive single-objective | [Bayesian Optimization][BayesianOpt] or [TPE] |
| Multi-fidelity single-objective | [Hyperband] |
| 2- or 3-objective default | [NSGA-II][Nsga2] |
| 2- or 3-objective default | [MOEA/D][Moead] (or [NSGA-II][Nsga2]) |
| Many-objective default | [MOEA/D][Moead] |
| 2-objective real-valued smooth front | [MOPSO][Mopso] |
| Disconnected / non-convex front | [IBEA][Ibea] |
| Many-objective default (curved front) | [NSGA-III][Nsga3] |
| Many-objective linear / simplex front | [GrEA][Grea] |
| Disconnected front | [IBEA][Ibea] |
| Many-objective, curved front | [NSGA-III][Nsga3] |
| Many-objective, linear / simplex front | [GrEA][Grea] |
| Permutation problem (TSP with distance matrix) | [Ant Colony][AntColonyTsp] |
| Generic permutation problem | [GA][GeneticAlgorithm] + permutation toolkit |
| Bi-objective combinatorial (TSP / scheduling / knapsack) | [NSGA-II][Nsga2] + matching encoding operators |