constrained optimization

**Constrained Optimization** in semiconductor manufacturing is the **optimization of process objectives (yield, CD, uniformity) subject to explicit constraints on process parameters and output specifications** — finding the best solution within the feasible operating region defined by equipment limits and quality requirements. **Types of Constraints** - **Equipment Limits**: Temperature range, pressure range, gas flow capacity, power limits. - **Quality Specs**: CD ± tolerance, thickness ± tolerance, defect density < maximum. - **Process Windows**: Combinations that must be avoided (e.g., high power + low pressure causes arcing). - **Cost Constraints**: Material usage limits, maximum number of process steps. **Why It Matters** - **Feasibility**: The true optimum may be infeasible — constrained optimization finds the best achievable solution. - **Robustness**: Constraints on spec limits ensure the optimized recipe actually works in production. - **Methods**: Lagrange multipliers, penalty methods, interior point, and SQP handle different constraint types. **Constrained Optimization** is **optimizing within reality** — finding the best process conditions while respecting every equipment limit and quality specification.