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Class 2 — 08/29/2025

Presenter: Arnaud Deza

Topic: Numerical optimization for control (gradient/SQP/QP); ALM vs. interior-point vs. penalty methods

Overview

This class covers the fundamental numerical optimization techniques essential for optimal control problems. We explore gradient-based methods, Sequential Quadratic Programming (SQP), and various approaches to handling constraints including Augmented Lagrangian Methods (ALM), interior-point methods, and penalty methods.

Interactive Materials

The class is structured around 1 slide deck and four interactive Jupyter notebooks:

  1. Part 1a: Root Finding & Backward Euler

    • Root-finding algorithms for implicit integration
    • Fixed-point iteration vs. Newton's method
    • Application to pendulum dynamics

  2. Part 1b: Minimization via Newton's Method

    • Unconstrained optimization fundamentals
    • Newton's method implementation
    • Globalization strategies: Hessian matrix and regularization

  3. Part 2: Equality Constraints

    • Lagrange multiplier theory
    • KKT conditions for equality constraints
    • Quadratic programming implementation

  4. Part 3: Interior-Point Methods

    • Inequality constraint handling
    • Barrier methods and log-barrier functions
    • Comparison with penalty methods

Additional Resources

Key Learning Outcomes

  • Understand gradient-based optimization methods
  • Implement Newton's method for minimization
  • Apply root-finding techniques for implicit integration
  • Solve equality-constrained optimization problems
  • Compare different constraint handling methods
  • Implement Sequential Quadratic Programming (SQP)

Next Steps

This class provides the foundation for advanced topics in subsequent classes, including Pontryagin's Maximum Principle, nonlinear trajectory optimization, and stochastic optimal control.