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.

Learning Objectives

By the end of this class, students will be able to:

• Understand the mathematical foundations of gradient-based optimization
• Implement Newton's method for unconstrained minimization
• Apply root-finding techniques for implicit integration schemes
• Solve equality-constrained optimization problems using Lagrange multipliers
• Compare and contrast different constraint handling methods (ALM, interior-point, penalty)
• Implement Sequential Quadratic Programming (SQP) for nonlinear optimization

Prerequisites

• Solid understanding of linear algebra and calculus
• Familiarity with Julia programming
• Basic knowledge of differential equations
• Understanding of optimization concepts from Class 1

Materials

Interactive Notebooks

The class is structured around four interactive Jupyter notebooks that build upon each other:

1. Part 1a: Root Finding & Backward Euler

   • Root-finding algorithms for implicit integration
   • Fixed-point iteration vs. Newton's method
   • Backward Euler implementation for ODEs
   • Convergence analysis and comparison
   • Application to pendulum dynamics

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

   • Unconstrained optimization fundamentals
   • Newton's method for minimization
   • Hessian matrix and positive definiteness
   • Regularization and line search techniques
   • Practical implementation with Julia

3. Part 2: Equality Constraints

   • Lagrange multiplier theory
   • KKT conditions for equality constraints
   • Quadratic programming with equality constraints
   • Visualization of constrained optimization landscapes
   • Practical implementation examples

4. Part 3: Interior-Point Methods

   • Inequality constraint handling
   • Barrier methods and log-barrier functions
   • Interior-point algorithm implementation
   • Comparison with penalty methods
   • Convergence properties and practical considerations

Additional Resources

• Lecture Slides (PDF) - Complete slide deck from the presentation
• LaTeX Source Files - Source code for the lecture slides
• Demo Script - Python demonstration of penalty vs. barrier methods

Key Concepts Covered

Mathematical Foundations

• Gradient and Hessian: Understanding first and second derivatives in optimization
• Newton's Method: Quadratic convergence and implementation details
• KKT Conditions: Necessary and sufficient conditions for optimality
• Duality Theory: Lagrange multipliers and dual problems

Numerical Methods

• Root Finding: Fixed-point iteration, Newton-Raphson method
• Implicit Integration: Backward Euler for stiff ODEs
• Sequential Quadratic Programming: Local quadratic approximations
• Interior-Point Methods: Barrier functions and path-following

Constraint Handling

• Equality Constraints: Lagrange multipliers and null-space methods
• Inequality Constraints: Active set methods and interior-point approaches
• Penalty Methods: Quadratic and exact penalty functions
• Augmented Lagrangian: Combining penalty and multiplier methods

Practical Applications

The methods covered in this class are fundamental to:

• Optimal Control: Trajectory optimization and feedback control design
• Model Predictive Control: Real-time optimization with constraints
• Robotics: Motion planning and control with obstacle avoidance
• Engineering Design: Constrained optimization in mechanical systems

Further Reading

Next Steps

This class provides the foundation for the advanced topics covered in subsequent classes, including:

• Pontryagin's Maximum Principle (Class 3)
• Nonlinear trajectory optimization (Class 5)
• Stochastic optimal control (Class 7)
• Physics-Informed Neural Networks (Class 10)

For questions or clarifications, please reach out to Arnaud Deza at adeza3@gatech.edu