Multi-Objective Optimization¶
How ASCICat handles multiple objectives.
The Challenge¶
Catalyst selection involves competing objectives:
- Maximize activity
- Maximize stability
- Minimize cost
No single catalyst optimizes all simultaneously.
Approaches¶
1. Weighted Sum (ASCICat)¶
\[\phi = \sum_i w_i \cdot S_i\]
Advantages:
- Single interpretable metric
- Reproducible rankings
- Explicit trade-offs
Limitations:
- Non-convex Pareto regions missed
- Weight selection required
2. Pareto Optimization¶
Find non-dominated solutions where no other solution is better on all objectives.
Advantages:
- No weight specification needed
- Complete trade-off information
Limitations:
- Returns set, not ranking
- Requires subjective final selection
3. Goal Programming¶
Minimize deviation from targets:
\[\min \sum_i |S_i - S_i^{target}|\]
Advantages:
- Aspiration-based
- Handles constraints
Limitations:
- Targets often arbitrary
Why Weighted Sum?¶
ASCICat uses weighted sum because:
- Interpretability - Weights directly reflect priorities
- Reproducibility - Same weights = same ranking
- Comparability - Enables cross-study comparison
- Simplicity - Easy to understand and implement
Weight Selection¶
Default: Equal Weights¶
\[w_a = w_s = w_c \approx 0.33\]
Unbiased starting point for exploration.
Application-Specific¶
| Application | Suggested Weights |
|---|---|
| Fundamental research | (0.5, 0.3, 0.2) |
| Industrial catalyst | (0.35, 0.40, 0.25) |
| Large-scale deployment | (0.30, 0.25, 0.45) |
Sensitivity Analysis¶
Don't trust single weight choice - explore the weight space!
Mathematical Properties¶
Convex Combination¶
For \(w_i \geq 0\) and \(\sum w_i = 1\):
- \(\phi \in [\min S_i, \max S_i]\)
- Linear in scores
- Continuous in weights
Pareto Optimality¶
Weighted sum solutions are Pareto-optimal when:
- All weights positive
- Pareto frontier is convex
Trade-off Analysis¶
For two catalysts A and B:
\[\Delta \phi = w_a(S_a^A - S_a^B) + w_s(S_s^A - S_s^B) + w_c(S_c^A - S_c^B)\]
A is preferred when \(\Delta \phi > 0\).
References¶
- Marler, R. T. & Arora, J. S. Struct. Multidiscip. Optim. 26, 369 (2004)
- Hwang, C.-L. & Masud, A. S. M. Multiple Objective Decision Making (Springer, 1979)