Multi-Objective Optimization Lecture

Introduction

• Goal: Optimize multiple objective functions simultaneously
• Context: Falls under the umbrella of optimization

Weighted Sum Method

• Single Objective Function: f = g(x)
• Multiple Objective Functions: Combine g(x) and h(x)
• Weighted Sum: Add functions together with weights
  • Scalars: Use alpha and beta as weights
  • Scaling Values: Use g_0 and h_0 for normalization
• Example: Aircraft design
  • Structural weight (thousands of kg)
  • Coefficient of drag (0.025)
  • Need to scale objectives due to different magnitudes
  • Determine importance with alpha and beta

Pareto Fronts/Frontiers

• Purpose: Show trade-offs between optimal designs for multiple objectives
• Example: Fuel burn vs. zero fuel weight for aircraft design
• Pareto-Optimal Designs: Cannot be improved in one objective without sacrificing the other
• Plot: Green line represents Pareto front
  • Dominated designs: Non-optimal
• Construction Methods:
  • Weighted Sum Method: Sweep through different scalars
  • Epsilon Constraint Method: Minimize one objective, constrain another
    • Preferred method
    • Captures non-convex Pareto fronts accurately
    • Allows for even spacing in a plot

Takeaways

• Multi-objective optimization often requires predefined weightings or constraints
• Pareto Front: Useful for examining trade-offs
• Epsilon Constraint Method: Preferred for constructing Pareto fronts

Activities

• Notebook activities: Construct your own Pareto fronts for a given design or model