SMS-EMOA

Hypervolume contribution provides the environmental selection criterion: candidates compete through the objective-space volume they add relative to a reference point [1]. This steady-state example makes that point explicit.

Goal: run the stable public API in VAMOS 1.0.1, interpret its returned population, and identify which settings control this mechanism.

Capability matrix · Exact configuration reference · Setup and recorded evidence

Run the example

Prerequisite: complete T01 and the coverage setup. The protocol uses ZDT1, 10 real variables, 2 objectives, population 40; 400 evaluations; NumPy; seed 42. Initial evaluations count towards the budget. Settings are illustrative, not tuned.

python smsemoa.py
python smsemoa.py --output smsemoa-local.svg

The second command reruns the same protocol and requires Matplotlib. Use a new filename.

Download the complete script.

Complete runnable script

"""Verified VAMOS 1.0.1 lesson: smsemoa. See coverage-setup for dependencies."""


def execute_example(problem, algorithm, config, budget, seed=42, engine="numpy"):
    import numpy as np

    from vamos import OptimizationResult, optimize

    result = optimize(problem, algorithm=algorithm, algorithm_config=config, max_evaluations=budget, seed=seed, engine=engine)
    G = result.data.get("G")
    if getattr(problem, "n_constraints", 0) and G is None:
        raise ValueError("Declared constraints are missing from the returned result.")
    feasible_rows = np.arange(len(result.F)) if G is None else np.flatnonzero(np.all(G <= 0, axis=1))
    if len(feasible_rows):
        feasible = OptimizationResult({"X": result.X[feasible_rows], "F": result.F[feasible_rows]})
        front, local = feasible.front(return_indices=True)
        front_rows = feasible_rows[local]
    else:
        front, front_rows = np.empty((0, problem.n_obj)), np.empty(0, dtype=int)
    print(f"Algorithm: {algorithm}; encoding: {problem.encoding}; backend: {engine}; seed: {seed}")
    print(f"Returned X: {result.X.shape}; F: {result.F.shape}; result mode: population")
    print(f"Feasible rows: {len(feasible_rows)}/{len(result.F)}; feasible non-dominated rows: {len(front)}")
    print(f"Evaluations: {result.data['evaluations']}")
    return {
        "problem": problem,
        "algorithm": algorithm,
        "config": config,
        "result": result,
        "feasible_rows": feasible_rows,
        "front": front,
        "front_rows": front_rows,
    }


def run_smsemoa(budget=400):
    from vamos.algorithms import SMSEMOAConfig
    from vamos.problems import ZDT1

    problem = ZDT1(n_var=10)
    config = (
        SMSEMOAConfig.builder()
        .pop_size(40)
        .crossover("sbx", prob=1.0, eta=20.0)
        .mutation("pm", prob=1.0 / problem.n_var, eta=20.0)
        .selection("random")
        .reference_point(vector=[1.1, 11.0], adaptive=False)
        .result_mode("population")
        .build()
    )
    return execute_example(problem, "smsemoa", config, budget)


def plot_example(data, output):
    from pathlib import Path

    import matplotlib

    matplotlib.use("Agg")
    import matplotlib.pyplot as plt
    import numpy as np

    output = Path(output)
    if output.exists():
        raise FileExistsError(f"Choose a new figure filename: {output}")
    if output.suffix.lower() != ".svg":
        raise ValueError("Use an .svg output filename.")
    f = data["result"].F
    with plt.rc_context({"svg.hashsalt": "vamos-coverage", "svg.fonttype": "none"}):
        fig = plt.figure(figsize=(7, 4.5), layout="constrained")
        if f.shape[1] == 3:
            ax = fig.add_subplot(projection="3d")
            ax.scatter(*f.T, c="#007f7a", s=22)
            ax.set(xlabel="f1", ylabel="f2", zlabel="f3")
            ax.set_box_aspect((1, 1, 0.85), zoom=0.85)
        else:
            ax = fig.add_subplot()
            ax.scatter(*f.T, c="#b6bec8", s=25, label="Returned population")
            if len(data["front"]):
                ax.scatter(*data["front"].T, c="#007f7a", s=30, label="Feasible non-dominated rows")
            labels = {
                "WeldedBeamDesignProblem": ("Cost (minimize)", "Deflection proxy (minimize)"),
                "FeatureSelectionProblem": ("Validation error (minimize)", "Feature count (minimize)"),
                "HyperparameterTuningProblem": ("Validation error (minimize)", "Complexity proxy (minimize)"),
            }
            xlabel, ylabel = labels.get(type(data["problem"]).__name__, ("f1 (minimize)", "f2 (minimize)"))
            ax.set(xlabel=xlabel, ylabel=ylabel)
            ax.legend()
        if data["problem"].__class__.__name__ in {"ZDT1", "ZDT1Problem"}:
            x = np.linspace(0, 1, 101)
            ax.plot(x, 1 - np.sqrt(x), "--", c="#7c8795", label="Analytical front")
            ax.legend()
        ax.set_title(f"{data['algorithm'].upper()} · {data['result'].data['evaluations']} evaluations · seed 42")
        fig.savefig(output, metadata={"Date": None}, bbox_inches="tight", pad_inches=0.25)
        plt.close(fig)
    output.write_text("\n".join(line.rstrip() for line in output.read_text(encoding="utf-8").splitlines()) + "\n", encoding="utf-8")


def lesson_cli(function, default_budget=400):
    import argparse

    parser = argparse.ArgumentParser(description="Run a bounded VAMOS example and optionally plot its actual result.")
    parser.add_argument("--budget", type=int, default=default_budget)
    parser.add_argument("--output", help="New SVG filename; requires Matplotlib")
    args = parser.parse_args()
    data = function(budget=args.budget)
    if args.output:
        plot_example(data, args.output)


if __name__ == "__main__":
    lesson_cli(run_smsemoa, default_budget=400)

Recorded output

Algorithm: smsemoa; encoding: real; backend: numpy; seed: 42
Returned X: (40, 10); F: (40, 2); result mode: population
Feasible rows: 40/40; feasible non-dominated rows: 13
Evaluations: 400

Measured with VAMOS 1.0.1; the observations and provenance retain the inputs and results.

Understand the configuration

Setting	Meaning in this example
Fixed reference [1.1, 11]	Both objectives are minimized; this box covers the chosen ZDT1 domain. It is supplied to selection, not inferred from each result.
adaptive=False	Keeps the selection reference fixed throughout this lesson. Adaptive reference behavior is a different experiment.
SBX + polynomial mutation	Real-coded variation; mutation probability is explicitly 1/10.
selection("random")	Parent selection is separate from hypervolume-based environmental selection.

Interpret the result and its limits

Each X row aligns with the same F row. This lesson explicitly requests population mode, then computes the feasible non-dominated subset separately. All objectives are minimized. A non-dominated row is only non-dominated among the available rows; it need not be Pareto optimal. The retained observations contain the actual arrays and configuration.

Hypervolume depends on the reference point and objective scales. The low budget leaves visible distance from the analytical front. Indicator computation also has a cost; this example does not establish suitability for arbitrarily many objectives.

This run has no constraints. Before using a constrained or discrete problem, check the capability matrix and constraint convention. Preserve complete runs through the canonical lifecycle; the teaching JSON is not a replay bundle.

Try one controlled change

Increase the budget to 800 with the same seed and reference point. Compare fronts with that fixed box; then repeat several seeds before drawing a quality conclusion.

Next: use T06 to turn this example into a reproducible comparison.

References

[1] Beume, N., Naujoks, B., and Emmerich, M. (2007). SMS-EMOA: Multiobjective selection based on dominated hypervolume. European Journal of Operational Research, 181(3), 1653–1669.