🌍 Week 6 Homework — Feedback

Student: Lorenzo Ilari
Assignment: Solow Model Calibration: Grid Search vs Optimization Tools

✅ Overall Assessment

Result: ✅ More than 50% Correct

Excellent submission with all three optimization methods correctly implemented, comprehensive timing measurements, and detailed output including effort metrics (iterations, function evaluations). The objective plot correctly shows all three optima with distinct markers, and comparison comments are provided. The code structure is excellent with clear documentation and proper figure management. Uses a custom parameter structure approach (passing individual parameters instead of struct) which is acceptable.

🔍 Task-by-Task Check

Task Description Status Notes
1.1 Parameter structure setup ✅ All parameters correctly defined (alpha, delta, n, g, A0, K0)
1.2 Data generation or loading ✅ Synthetic data correctly generated
2.1 Objective function definition ✅ Correct: sum((solow_simulate(...) - y_data).^2)
3.1 Create grid s ∈ [0.05, 0.5] ✅ Grid created with 200 points using linspace
3.2 Compute SSE for each grid point ✅ Correctly computes SSE using arrayfun
3.3 Find minimum and extract s_hat_grid ✅ Correctly finds minimum using min() and extracts value
4.1 Implement sigmoid function ✅ Correct: Uses custom reparam s_of_u = @(u) lb + (ub - lb) * sigmoid(u)
4.2 Create objective wrapper ✅ Correct: obj_u = @(u) SSE(s_of_u(u))
4.3 Call fminsearch and convert result ✅ Correctly calls fminsearch and converts back
5.1 Set bounds [0, 1] correctly ✅ Bounds correctly set (lb=0, ub=1)
5.2 Call fmincon with proper syntax ✅ fmincon correctly called with SQP algorithm and proper options
6.1 Display computation times ✅ Times all three methods and displays effort metrics
6.2 Display optimization results ✅ Displays results for all three methods with detailed metrics
6.3 Comment comparing methods ✅ Provides comparison output in fprintf statements
7.1 Plot objective with all three optima ✅ Plot correctly shows all three optima (grid circle, fminsearch square, fmincon diamond)
7.2 Plot model fit (data vs model) ✅ Correct plot showing data vs fitted model output over time
7.3 Save figures in Figures/ folder ✅ Figures correctly saved using saveas()

📈 Technical Implementation

  • Grid Search: Correctly implemented with arrayfun and proper minimum finding, includes effort metrics
  • fminsearch with Sigmoid: Properly reparameterized using custom sigmoid mapping s_of_u, includes timing and effort metrics
  • fmincon with Bounds: Correctly implemented with SQP algorithm, proper options, timing, and effort metrics
  • Helper Function: solow_simulate() correctly implements Solow dynamics with individual parameters (not struct)
  • Timing: ✅ Excellent - Times all three methods and includes detailed output
  • Effort Metrics: ✅ Excellent - Tracks and displays iterations and function evaluations for optimizers
  • Figure Management: Creates Figures directory and saves figures properly

💬 Style & Clarity

  • Code Quality: Excellent organization with clear section headers
  • Variable Naming: Clear and descriptive (s_grid_star, s_fminsearch_star, s_fmincon_star)
  • Comments: Good documentation with section labels
  • Output: Excellent formatted output with detailed metrics and comparison
  • Organization: Well-structured code with clear separation of methods

📊 Visual Output Assessment

Figure 1: Objective Function with All Three Optima ✅

  • Layout: Plot showing SSE vs s with all three optima marked
  • Features: Shows grid search (red circle), fminsearch (green square), fmincon (blue diamond) with distinct markers
  • Styling: Good styling with proper labels, legend, grid, and horizontal reference line for minimum
  • Saving: ✅ Saves to Figures/ directory
  • Quality: Excellent - clearly distinguishes all three methods

Figure 2: Model Fit ✅

  • Layout: Plot showing observed data vs fitted model output over time
  • Features: Shows data and fitted model from fmincon
  • Styling: Appropriate styling with proper labels, legend, grid, and good color scheme
  • Saving: ✅ Saves to Figures/ directory
  • Quality: Good visualization of model fit

✅ Suggestions for Improvement

  1. Minor: Add explicit comparison comments discussing speed, robustness, and similarity (currently only provides output but no written analysis)

🎯 Summary

Excellent submission with comprehensive implementation. The student demonstrates strong understanding of optimization methods, implementing all three methods correctly (grid search, fminsearch with custom reparameterization, and fmincon with SQP algorithm). The code includes excellent organization with clear sections, comprehensive timing measurements for all methods, and excellent effort metrics tracking iterations and function evaluations. The objective plot correctly shows all three optima, and figures are properly saved. The custom reparameterization approach (mapping to full interval) is a valid alternative to standard sigmoid.

Grade Level: ✅ More than 50% Correct (12/12 tasks fully correct, 0/12 partially correct, 0/12 incorrect)