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.

The slides for this lecture can be found here Lecture Slides (PDF)

The Pluto julia notebook for my final chapter can be found here final chapter

Although the main code for the julia demo's are contained in the Pluto notebook above, the following julia notebooks are the demo's I used in the class recording/presentation.

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

3. Part 2: Equality Constraints

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

4. Part 3: Interior-Point Methods

   • Inequality constraint handling
   • Barrier methods and log-barrier functions
   • Interior-point algorithm implementation

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