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AGE-MOEA (2019)

Adaptive GEometry Estimation based Multi-Objective Evolutionary Algorithm — Annibale Panichella, An Adaptive Evolutionary Algorithm based on Non-Euclidean Geometry for Many-objective Optimization, GECCO 2019. DOI: 10.1145/3321707.3321839.

AGE-MOEA is the direct predecessor of AGE-MOEA-2 and the middle link in the lineage NSGA-II → AGE-MOEA → AGE-MOEA-2. Its one big idea: keep NSGA-II's entire evolutionary skeleton, but replace the crowding distance with a survival score that adapts to the shape of the Pareto front.

The idea

NSGA-II's crowding distance implicitly measures spread with rectilinear (Euclidean) geometry, which suits a flat front but misjudges convex or concave ones. Reference-point methods such as NSGA-III place their reference points on a flat hyperplane (Das–Dennis), so they too assume a flat front. AGE-MOEA instead estimates the front's curvature every generation and measures both convergence and diversity in the matching geometry.

The Lₚ manifold. AGE-MOEA models the first non-dominated front as the unit manifold of the Lₚ norm:

(a_1^p + a_2^p + … + a_M^p)^(1/p) = 1

The exponent p is the curvature: p = 1 is a flat hyperplane, p = 2 a hypersphere (concave), and p < 1 a hyperbolic (convex) surface. Fitting the front means finding the p whose unit surface best matches it.

Per-generation recipe (all inside the NSGA-II loop, on the first front F₁):

  1. Normalize F₁ NSGA-III-style — shift the ideal point Z^min to the origin, then divide by the intercepts of the M-dimensional hyperplane through the front's extreme points (falling back to per-axis min–max if that system is singular).
  2. Estimate p cheaply (O(MN)) from a single representative point — the front's central point C (the one nearest the diagonal bisector) — via the closed form p = log(M) / (log(M) − log(Σ Cᵢ)).
  3. Score = diversity / proximity, both in the Lₚ norm with that p:
    • proximity = ‖f(S)‖ₚ — distance to the ideal point (convergence);
    • diversity — a greedy maximin: pre-seed the extreme solutions with a score of +∞, then repeatedly add the point whose summed distance to its two nearest already-selected neighbors is largest, recording that sum as its score.
    • Non-first fronts get score = 1 / proximity (proximity only).

Environmental selection then fills front by front and splits the last front by descending survival score — the NSGA-II structure, with this score in place of crowding distance.

In OptunaSharp

AGE-MOEA is validated internally as the base of the AGE-MOEA-2 port: AgeMoea2Survival derives from AgeMoeaSurvival (the 2019 survival, with FindCornerSolutions, normalization, and the greedy SurvivalScore), and the byte-exact Layer-1 driver AgeMoea2Optimizer accepts either survival strategy. Passing the 2019 base recovers AGE-MOEA-1 for free — the reference implementation runs it by supplying the default geometry/distance kernels to the same code:

// (internal Layer-1 optimizer, used by the parity harness)
var optimizer = new AgeMoea2Optimizer(problem, popSize, nGen, seed, survival: new AgeMoeaSurvival());

AGE-MOEA has no public Optuna-idiomatic sampler of its own; the public, ask/tell entry point is AgeMoea2Sampler (the improved 2022 variant). Like AGE-MOEA-2, AGE-MOEA is absent from Optuna core, so it lives in OptunaSharp as a beyond-core algorithm validated against its reference implementation (pymoo), not against Optuna.

Relationship to the lineage

  • From NSGA-II: inherits the whole (μ+λ) skeleton — Rₜ = Pₜ ∪ Qₜ, fast non-dominated sort, front-by-front filling, binary tournament + SBX + polynomial mutation. Only the crowding distance is replaced.
  • To AGE-MOEA-2: the 2022 version keeps everything here and overrides exactly two kernels — the curvature estimate (a single-point heuristic → Newton–Raphson) and the pairwise distance (raw Lₚ → geodesic).

See also

AGE-MOEA-2 (2022) · NSGA-II (2002) · Samplers · Multi-objective Optimization · the API reference.

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