How does multi-objective optimization work in MatCraft?

Multi-objective optimization (MOO) is the process of simultaneously optimizing two or more conflicting objectives. Most real-world materials problems are inherently multi-objective — you want a material that is strong and lightweight, or conductive and cheap.

MatCraft's Approach

MatCraft handles multi-objective optimization through a combination of surrogate models, scalarization, and Pareto analysis:

1. Per-Objective Surrogates

A separate surrogate model (or output head) is trained for each objective. This means the model independently predicts water flux, salt rejection, and cost for a given composition. Each surrogate has its own uncertainty estimate.

2. Scalarization for CMA-ES

CMA-ES natively optimizes a single scalar value. MatCraft converts multiple objectives into a scalar using one of these strategies:

• Weighted sum: score = w1 obj1_normalized + w2 obj2_normalized. Simple but can miss concave regions of the Pareto front.
• Chebyshev scalarization (default): Minimizes the worst-case weighted deviation from the ideal point. This can find solutions in concave regions and produces a more uniform Pareto front.
• Random weight sampling: On each iteration, random weight vectors are sampled. Over many iterations, this covers the full Pareto front.

objectives:
  - name: hardness
    direction: maximize
    weight: 0.6    # Optional; used for weighted scalarization
  - name: cost
    direction: minimize
    weight: 0.4

3. Hypervolume-Based Acquisition

For the active learning loop, the acquisition function is extended to multi-objective settings using the Expected Hypervolume Improvement (EHVI). This measures how much a new candidate would expand the dominated volume of the Pareto front, naturally balancing all objectives.

4. Pareto Front Construction

After the campaign completes (or at any intermediate point), MatCraft extracts the Pareto front from all evaluated candidates. The front is available as a DataFrame, a plot, or an interactive visualization.

results = campaign.run()

# Get Pareto-optimal candidates
pareto = results.pareto_front()
print(pareto[["hardness", "cost", "iron", "chromium", "nickel"]])

# Find the "knee" point -- the solution with the best balance
knee = results.pareto_knee()
print(f"Balanced solution: {knee}")

Number of Objectives

MatCraft supports 2-6 objectives efficiently. Beyond 6 objectives, the Pareto front becomes very large (most solutions are non-dominated in high dimensions), and the hypervolume computation becomes expensive. For many-objective problems (>6), consider grouping related objectives or using preference-based scalarization.