Multi-Objective Optimization

Multi-Objective Optimization

Most real-world materials optimization problems involve multiple conflicting objectives. Increasing a battery's energy density often reduces its cycle life. Making a membrane more permeable typically lowers its selectivity. Multi-objective optimization (MOO) finds the best possible trade-offs among these competing goals.

Why Not Just Combine Objectives?

A common approach is to combine objectives into a single score using weights: score = w1 obj1 + w2 obj2. This has significant drawbacks:

• Choosing weights is arbitrary: The "right" weights depend on the application context, which may not be known during optimization.
• Non-convex Pareto fronts: Weighted sums can only find solutions on the convex hull of the Pareto front, missing potentially valuable solutions in concave regions.
• Scale sensitivity: Objectives with larger numerical values dominate unless carefully normalized.

MatCraft instead computes the full Pareto front, letting you choose trade-offs after seeing all options.

Expected Hypervolume Improvement (EHVI)

For multi-objective acquisition, MatCraft uses Expected Hypervolume Improvement (EHVI). This is the multi-objective generalization of Expected Improvement:

EHVI(x) = E[HV(P ∪ {f(x)}) - HV(P)]

Where P is the current Pareto front and f(x) is the predicted objective vector for candidate x. EHVI naturally balances:

• Convergence: Candidates that extend the Pareto front outward (better objective values).
• Diversity: Candidates that fill gaps between existing Pareto solutions.

Number of Objectives

MatCraft supports up to 5 simultaneous objectives, but performance and interpretability vary:

| Objectives | Pareto Front | Recommendation | |—————-|———————|————————| | 2 | A curve in 2D space | Ideal. Easy to visualize and interpret. | | 3 | A surface in 3D space | Good. 3D interactive plots available. | | 4 | A hypersurface | Challenging. Use parallel coordinates for visualization. | | 5 | High-dimensional surface | Difficult. Consider reducing to essential objectives. |

Defining Multiple Objectives

objectives:
  - name: energy_density
    direction: maximize
    unit: Wh/kg

  - name: cycle_life
    direction: maximize
    unit: cycles

  - name: cost
    direction: minimize
    unit: USD/kWh

Objectives can mix maximize and minimize directions. Internally, MatCraft negates minimize objectives so that all optimization is performed as maximization.

Objective Normalization

Before computing hypervolume or EHVI, objectives are normalized to [0, 1] based on the observed range. This ensures that objectives with different scales contribute equally:

normalized_obj = (obj - obj_min) / (obj_max - obj_min)

Normalization is updated at each iteration as new evaluations expand the observed range.

Constraint Handling in MOO

Constraints interact with multi-objective optimization in two ways:

1. Feasible-first ranking: Feasible solutions always rank above infeasible ones, regardless of objective values.
2. Constraint violation degree: Among infeasible solutions, those with smaller total constraint violation are preferred.

This ensures that the optimizer first finds the feasible region, then optimizes within it.

Decomposition Strategies

For problems with many objectives (4+), MatCraft can optionally decompose the multi-objective problem into multiple single-objective subproblems using reference direction methods:

optimizer:
  active_learning:
    decomposition: true
    n_reference_directions: 20

Each reference direction defines a weighted scalarization. The overall Pareto front is assembled from all subproblem solutions.

Practical Tips

1. Start with 2 objectives: Validate your setup with two objectives, then add more if needed.
2. Check for redundancy: If two objectives are strongly correlated (r > 0.9), one may be redundant. Use a scatter plot to check.
3. Set reference points: Explicit reference points make hypervolume comparisons consistent across campaigns.
4. Use constraints instead of objectives: If you have a hard requirement (e.g., strength > 100 MPa), express it as a constraint rather than an objective.

Post-Optimization Decision Making

After obtaining the Pareto front, use one of these strategies to select a final design:

• Knee-point selection: Choose the solution with maximum curvature on the front.
• Preference articulation: Apply weights or thresholds based on application requirements.
• Robust selection: Among Pareto solutions, pick the one with smallest uncertainty (most reliable prediction).

See Pareto Analysis for implementation details.