RVEA, RVEA*, and iRVEA

Author:

Nicolás Rodríguez Uribe

Version:

1.0

Date:

2026-09-09

RVEA in a nutshell

This section is a short, self-contained tutorial on the base RVEA algorithm (Cheng et al., TEVC 2016) for readers who have not met it before. RVEA* and iRVEA (described later in this document) only add mechanisms on top of what follows.

Why reference vectors

RVEA is a decomposition-based algorithm: instead of ranking solutions by Pareto dominance, which quickly becomes uninformative once the number of objectives grows past three or four (almost every solution in a small population is mutually nondominated), it breaks the objective space into a handful of angular subspaces, one per reference vector, and picks a single survivor per subspace. A reference vector is just a unit vector radiating out from the ideal point. jMetal’s default is to generate a uniformly spread set of them with the simplex-lattice construction already used by NSGA-III (ReferencePointGenerator), but that construction can only produce the combinatorial counts C(H + M - 1, M - 1) dictated by its divisions parameter H. Every RVEA builder also accepts a caller-supplied List<double[]> of vectors instead, exactly as MOEA/D does: any set of uniformly distributed weight vectors works, which lets the population size be whatever the vector set’s size is, rather than the nearest simplex-lattice count. jMetal ships several precomputed sets under resources/weightVectorFiles/moead/ (e.g. W3D_100.dat, one vector per line); MOEA/D locates these by their W<M>D_<N>.dat naming convention, and while RVEA has no such convenience wrapper, the same file can be loaded with VectorFileUtils.readVectors(...) and passed straight in as the List<double[]>. These particular files were generated with the Riesz s-Energy method of Blank, Deb, Dhebar, Bandaru, and Seada, TEVC 2021, which — unlike the Das-Dennis simplex lattice — can target an arbitrary number of well-spaced points.

Every generation, RVEA does four things:

  1. Translate. Estimate the ideal point as the componentwise minimum of the current candidate population and subtract it from every objective vector, so the ideal point sits at the origin.

  2. Associate. Assign each translated solution to the reference vector it is closest to in angle. This partitions the population into as many niches as there are reference vectors (some niches may end up empty).

  3. Score. Inside each niche, rank the candidates with the Angle-Penalized Distance (APD, see below) instead of raw distance to the ideal point.

  4. Select. Keep only the best-scoring candidate of each occupied niche. Since empty niches contribute nothing, the surviving population can be smaller than the number of reference vectors.

Angle-Penalized Distance (APD)

Picking the single closest-to-ideal solution in every niche would converge fast but leave the niches uneven, because two very similar, tightly-clustered solutions can both be closer to the ideal point than a lone solution elsewhere in the niche. RVEA countered this with a scalarizing function that penalizes a candidate in proportion to how far it drifts from the centre of its own niche:

APD(f, v, t) = ||f|| * (1 + M * (t / T)^alpha * angle(f, v) / gamma(v))

where, for a translated objective vector f associated with reference vector v:

APD symbols

Symbol

Meaning

||f||

Distance from the (translated) solution to the ideal point: the convergence term. This is the whole score early in the run.

angle(f, v)

Angle between the solution and its reference vector: how far off-centre it sits inside its niche. This is the diversity term.

gamma(v)

The angle from v to its closest neighboring reference vector. It rescales the angular penalty to the local density of vectors, so tightly packed and sparsely packed regions of the objective space are penalized on a comparable scale.

M

Number of objectives.

t, T

Current generation index and the generation horizon (see the Configuration section below for how jMetal derives T from the evaluation budget).

alpha

User parameter controlling how quickly the diversity term is phased in.

The (t / T)^alpha factor is the crux of the design: at generation zero it is 0, so APD reduces to plain distance to the ideal point and the search behaves like a greedy convergence-only method; as t approaches T the factor tends to 1 and off-centre candidates are pushed down the ranking even if they are nominally closer to the ideal point, which spreads the final population evenly along the front. alpha controls how fast that transition happens (2.0 is the value used throughout this document’s examples).

A small worked example

Take two objectives and three reference vectors at 0°, 45°, and 90° from the first-objective axis, so gamma is 45° for all of them. Consider two translated candidates that both associate with the 45° vector:

Two candidates competing for the same niche

Candidate

Objectives f

||f||

Angle to niche (theta)

Reads as

P

(1, 1.73)

2.00

15°

closer to ideal, off-centre

Q

(2, 2)

2.83

0°

farther from ideal, perfectly centred

With M=2 and alpha=2.0, APD(P, t) = 2 + 1.33 * (t/T)^2 while APD(Q, t) = 2.83 stays constant (its angle penalty is always zero). At t=0 both formulas reduce to plain distance, so P wins (2.00 < 2.83): early on, being closer to the front simply matters more than being centred. Solving for the crossover point shows Q overtakes P once t/T ≈ 0.79, i.e. only in roughly the last fifth of the run does alignment with the reference vector start to outweigh raw convergence. This is exactly the trade-off exercised by the givenCompetingConvergenceAndAngle_whenTimePasses_thenWinnerChanges test in RVEASelectionBehaviorTest.

Keeping the reference vectors useful

A fixed, uniformly spread set of reference vectors only produces a uniformly spread population when every objective is scaled the same way. RVEA periodically (every ceil(fr * T) generations, fr being a user parameter) rescales each predefined vector by the objective ranges observed in the current survivors and renormalizes it, so badly scaled objectives still get evenly covered. The Implemented behavior section below states the exact formulas; RVEA* and iRVEA build directly on this translate/associate/score/select/adapt loop, adding their own strategies for what happens to reference vectors that end up covering no solutions at all.

Repository review

This change extends the RVEA builder, replacement, and environmental selection in jmetal-component with RVEA* and iRVEA. It targets develop at revision d22a7e94f and reuses the existing simplex generator. Other algorithms and the module dependency graph are unchanged.

The existing selection has several behaviors relevant to this work:

  • Empty reference-vector niches borrow the best remaining APD candidate. This enforces a fixed population size, whereas the published selection leaves empty niches empty. It also lets an earlier empty niche take a candidate associated with a later occupied niche.

  • Replacement passes the current parent count to selection. That count cannot represent the configured number of vectors when survivor populations vary.

  • Zero reference vectors are divided by their norm without validation, and nonfinite parameters/objectives are not consistently rejected.

  • The builder’s initial-population closure reads mutable builder state, so changing the builder can affect an algorithm already built from it.

This implementation uses occupied niches, a separate nominal population size, validated directions and parameters, and captured builder components.

Literature and algorithm identity

Literature and algorithm identity

Work

Contribution relevant to this implementation

Cheng et al., TEVC 2016 (author manuscript, RVEA)

RVEA introduces APD and periodic vector scaling. Section VI, Algorithm 4 adds random regeneration of inactive vectors in a second set: RVEA*.

Liu et al., CEC 2019 (author manuscript, iRVEA)

iRVEA replaces at most one inactive adaptive vector per generation and combines APD with regional protection, dominance, and an epsilon archive. This is the iRVEA implemented here.

Tian et al., TEVC 2019

Strengthened dominance relation (SDR), used by iRVEA above six objectives.

Liu et al., RVEA-iGNG, online 2020

Learns front geometry with growing neural gas instead of regenerating individual directions.

Liu et al., CARV-MOEA, 2023

Coordinates vector adaptation and scalarizing functions using local angles and vector age.

Liang et al., TensorRVEA, GECCO 2024 (manuscript)

Tensorizes RVEA for GPU execution; addresses computational throughput.

RVEA-DWC, 2024

Combines local/global vector generation with a modified selection criterion.

WPAEA, 2026

Uses hyperplane projection and adaptive population selection. The publisher’s abstract places it among more recent approaches to irregular fronts.

This focused review was checked on September 9, 2026. RVEA* and iRVEA are established variants; they should not be presented as the latest algorithms or as uniformly superior to newer methods. The newer papers provide context and are not substituted for the two requested algorithms.

The authors’ PlatEMO RVEAa implementation was consulted to cross-check RVEA* ordering and final truncation. The new Java selection code is independently implemented from the mathematical description.

Implemented behavior

All objectives are minimized. The builders reject constrained problems. RVEABuilder, RVEAStarBuilder, and IRVEABuilder return the existing EvolutionaryAlgorithm<S> and use its evaluation, variation, replacement, termination, and observer components.

For a translated objective vector f, assigned direction v, and smallest neighbor angle gamma(v), selection minimizes

APD(f, v, t) = ||f|| * (1 + M * (t / T)^alpha * angle(f, v) / gamma(v))

The candidate population’s current minimum supplies the translation; there is no historical ideal point. Predefined directions are periodically rescaled from their original values using survivor objective ranges, then normalized.

RVEA family behavior

Variant

Survivor selection

Vector updates

Population size

RVEA

Minimum APD in each occupied niche

Periodic range scaling

At most N

RVEA*

Nondominated candidates, then APD over both sets

Range scaling of predefined vectors; random regeneration of inactive additional vectors

Up to 2N during evolution; at most N after final angular truncation

iRVEA

APD, regional protection, Pareto/SDR secondary selection, clustering, late epsilon selection

Range scaling of predefined vectors; at most one inactive adaptive vector replaced using a least-covered candidate direction

At most N; may be smaller when insufficient candidates survive

An offspring batch retains its configured size even when the parent population shrinks. Random mating samples with replacement, so a single surviving parent is supported. iRVEA’s archive holds at most N solution copies and is updated using additive epsilon fitness with kappa = 0.05.

Explicit implementation choices

The iRVEA manuscript leaves some details open and its pseudocode contains inconsistent counts. The following choices make the implementation reviewable:

  • “Least similarity” means maximizing the minimum angle to active vectors. Activity is measured over both sets using translated objectives, consistently with APD. No replacement is needed if every candidate direction is covered.

  • Coarse regions use nearest-direction association in range-normalized space. The default target is 40; a single-layer simplex lattice chooses the largest available count no greater than the target (36 for three objectives), with the objective-axis lattice as the minimum. The target is capped at N.

  • A dominated representative selected to protect an otherwise unrepresented region is reserved through subsequent dominance and truncation steps. This follows the stated region-preservation purpose; applying the manuscript’s later global Pareto filter literally would discard that representative again.

  • Hierarchical reduction uses average linkage on angles and retains the member nearest the translated ideal point. Crowding reduction compares the sorted angular neighbor distances; equal crowding favors the shorter norm, then stable input order. The manuscript does not specify linkage or ties.

  • SDR uses normalized objective sums and the adaptive angular threshold of the authors’ SDR reference implementation. Epsilon fitness also uses current objective ranges, with a separate scale for each target column, held constant during iterative deletion.

  • After t > 0.8*T, epsilon selection keeps up to N - |P_APD| remaining candidates. This resolves the inconsistent deletion count in Algorithm 3. Solutions are never invented or cloned just to fill an empty slot.

  • A zero translated objective vector has zero APD and associates with the first direction. Angular denominators are floored at 1e-12. A scaling or regeneration step yielding the zero vector retains the previous direction. Duplicate directions and constant objective ranges therefore remain finite.

These are documented interpretations of the papers, not a claim of bitwise equivalence to an unavailable original iRVEA program or reproduction of its published benchmark tables.

Configuration

The existing RVEA constructors and component setters remain available. The builders now reject constrained problems, invalid numerical directions, and budgets smaller than the initial population. Callers must also allow a survivor population smaller than N instead of assuming empty niches are filled.

The constructors accept either simplex divisions H or a caller-supplied List<double[]>. The initial population size must equal the vector count: N = C(H + M - 1, M - 1). For three objectives, H=12 gives N=91. Two-layer vectors can be supplied through ReferencePointGenerator.generateTwoLayers; for example (8, 3, 2) gives 156 directions. Predefined vectors are copied and normalized at construction.

var problem = new DTLZ7(22, 3);
var algorithm = new IRVEABuilder<>(
    problem, 91, 25000,
    new SBXCrossover(1.0, 20.0),
    new PolynomialMutation(1.0 / problem.numberOfVariables(), 20.0),
    2.0, 0.1, 12)
    .setNumberOfSubregions(40)
    .build();
algorithm.run();
var population = algorithm.result();

Set JMetalRandom.getInstance().setSeed(...) before building to reproduce the initial random adaptive vectors as well as the search. Build a fresh algorithm for each run. Default mating, crossover, and mutation use jMetal’s shared random generator; concurrent runs need appropriate random isolation.

setTermination, setEvaluation, setCreateInitialPopulation, setSelection, and setVariation are supported. Custom components must obey their existing jMetal contracts. Termination must be TerminationByEvaluations. The initial population consumes N evaluations and the generation horizon is T = max(1, ceil((maxEvaluations - N) / offspringSize)). Generation indices run from zero; adaptation occurs at zero and every ceil(fr*T) selections. The last complete offspring batch may overshoot the budget by fewer than offspringSize evaluations. A 25,000 budget with batches of 91 uses 25,025.

Running the examples

Build from the repository root with Maven and Java 21 or later.

mvn -DskipTests=false clean package
java -cp jmetal-auto/target/jmetal-auto-7.6-SNAPSHOT-jar-with-dependencies.jar org.uma.jmetal.component.examples.multiobjective.rvea.RVEAStarDTLZ2Example
java -cp jmetal-auto/target/jmetal-auto-7.6-SNAPSHOT-jar-with-dependencies.jar org.uma.jmetal.component.examples.multiobjective.rvea.IRVEADTLZ7Example
java -cp jmetal-auto/target/jmetal-auto-7.6-SNAPSHOT-jar-with-dependencies.jar org.uma.jmetal.component.examples.multiobjective.rvea.RVEAVariantsExample DTLZ5 25000 42

The existing jmetal-auto distribution JAR includes jmetal-component and its dependencies. Alternatively, run these main classes directly in an IDE.

RVEAVariantsExample runs all three algorithms and accepts DTLZ2 (regular), DTLZ5 (degenerate), DTLZ7 (disconnected), or ZDT1/2/3, followed by a budget and seed. The ZDT problems use two objectives, 30 decision variables, 100 initial solutions, and 100 predefined vectors (H=99). Their reference front filenames do not include the .3D suffix used by the DTLZ examples. The two minimal examples use seed 42 and 25,000 evaluations. Each run writes its nondominated subset to FUN.<algorithm>.<problem>.csv and the corresponding decision variables to VAR.<algorithm>.<problem>.csv. It reports evaluations, survivors, nondominated count, runtime, and raw IGD+ against the repository’s reference front. Smaller IGD+ is better; compare values within the same problem. A single example run is not a statistical comparison of algorithms.

Verification

Tests cover APD convergence/diversity tradeoffs, translation and scale changes, adaptation scheduling, empty niches, regeneration, the one-vector replacement limit, protected dominated regions, Pareto/SDR switching, late supplementation, epsilon deletion, clustering, archive isolation, numerical degeneracy, builder validation, odd offspring batches, and repeatability across fresh builds. Runs on DTLZ2/5/7, MaF1, and seven-objective DTLZ2 exercise complete pipelines; a seeded DTLZ2 test checks progress toward the unit-sphere front.

mvn -pl jmetal-component -am "-Dtest=*RVEA*Test,ReferencePointGeneratorTest" "-Dsurefire.failIfNoSpecifiedTests=false" test
mvn validate

develop does not bind Checkstyle to validate; this command checks the Maven reactor. New Java files are formatted with Google Java Format. Coverage is measured with a JaCoCo agent supporting the locally installed JDK.

mvn -pl jmetal-component -am "-Dtest=*RVEA*Test,ReferencePointGeneratorTest" "-Dsurefire.failIfNoSpecifiedTests=false" org.jacoco:jacoco-maven-plugin:0.8.13:prepare-agent test org.jacoco:jacoco-maven-plugin:0.8.13:report

Verified in this workspace: the complete clean package reactor reported 2,006 tests with no failures or errors; the 13 skipped tests are pre-existing. All 66 new test cases passed. JaCoCo reports 497/509 covered lines (97.6%) across the added and modified RVEA production classes, excluding the examples and the unchanged reference-point generator. Google Java Format’s dry run reported no changes, and mvn validate passed.

The example executions below used seed 42 and 25,025 actual evaluations. These are smoke-run observations, not multi-seed experimental conclusions.

Three-objective smoke-run results

Algorithm

Problem

Final survivors

Exported nondominated solutions

IGD+

RVEA*

DTLZ2

91

91

0.023796

iRVEA

DTLZ7

91

90

0.039189

RVEA

DTLZ5

50

10

0.036877

RVEA*

DTLZ5

85

85

0.003019

iRVEA

DTLZ5

91

90

0.002880

Additional two-objective runs

ZDT1, ZDT2, and ZDT3 provide convex, concave, and disconnected reference fronts, respectively. All nine executions below use the default 30 decision variables, seed 42, population 100, and exactly 25,000 evaluations. Crossover, mutation, alpha, and adaptation frequency are the same as in the preceding examples.

Two-objective smoke-run results

Problem

Algorithm

Nondominated solutions

IGD+

ZDT1

RVEA

68

0.038176

ZDT1

RVEA*

100

0.003824

ZDT1

iRVEA

100

0.003330

ZDT2

RVEA

44

0.043022

ZDT2

RVEA*

100

0.003357

ZDT2

iRVEA

100

0.003046

ZDT3

RVEA

41

0.054089

ZDT3

RVEA*

100

0.003358

ZDT3

iRVEA

93

0.004571

These single runs favor iRVEA on ZDT1/2 and RVEA* on ZDT3 in IGD+. They do not establish statistical superiority. The plots show all exported nondominated solutions, use identical axis limits within each problem, and preserve the gaps between ZDT3 reference-front segments.

After rebuilding the distribution JAR, run from the repository root:

java -cp jmetal-auto/target/jmetal-auto-7.6-SNAPSHOT-jar-with-dependencies.jar org.uma.jmetal.component.examples.multiobjective.rvea.RVEAVariantsExample ZDT1 25000 42
java -cp jmetal-auto/target/jmetal-auto-7.6-SNAPSHOT-jar-with-dependencies.jar org.uma.jmetal.component.examples.multiobjective.rvea.RVEAVariantsExample ZDT2 25000 42
java -cp jmetal-auto/target/jmetal-auto-7.6-SNAPSHOT-jar-with-dependencies.jar org.uma.jmetal.component.examples.multiobjective.rvea.RVEAVariantsExample ZDT3 25000 42
python results/rvea/zdt/plot_results.py --input-dir .

The plotting script verifies the CSV dimensions, variable bounds, nondominance, and IGD+ against the recorded executions in results/rvea/zdt/runs.log. It writes input hashes and full-precision metrics to plot_data.json, a nine-panel overview, individual problem PNGs, and a four-page PDF.

The recorded FUN and VAR files are also included in results/rvea/data. Both plotting scripts use these archived inputs by default, so the figures can be regenerated from a fresh checkout with Python, NumPy, and Matplotlib:

python results/rvea/plots/plot_results.py
python results/rvea/zdt/plot_results.py

Use --input-dir . to plot freshly executed examples from the repository root. The DTLZ overview and PDF contain the five three-objective runs above. These archived single-seed runs were first recorded on the earlier 6.9.4 development baseline. All 14 runs were repeated with the 7.6 distribution after integrating the current develop base; every FUN and VAR file matched the archived inputs byte for byte. The new execution output is in reproduction.log.