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Add Class 2 HTML materials and documentation integration
- Convert Jupyter notebooks to HTML with pre-run cells - Create comprehensive class02/overview.md with learning objectives - Update class02/class02.md with detailed content and links - Integrate Class 2 into documentation structure (docs/make.jl) - Add supporting materials: images, demo script, Project.toml - All materials now accessible through course documentation website
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‎class02/Manifest.toml‎

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# This file is machine-generated - editing it directly is not advised
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julia_version = "1.11.6"
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manifest_format = "2.0"
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project_hash = "da39a3ee5e6b4b0d3255bfef95601890afd80709"
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[deps]

‎class02/Project.toml‎

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[deps]

‎class02/class02.md‎

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Add notes, links, and resources below.
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## Overview
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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.
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## Interactive Materials
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The class is structured around four interactive Jupyter notebooks:
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1. **[Part 1: Minimization via Newton's Method](part1_minimization.html)**
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- Unconstrained optimization fundamentals
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- Newton's method implementation
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- Hessian matrix and regularization
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2. **[Part 1: Root Finding & Backward Euler](part1_root_finding.html)**
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- Root-finding algorithms for implicit integration
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- Fixed-point iteration vs. Newton's method
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- Application to pendulum dynamics
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3. **[Part 2: Equality Constraints](part2_eq_constraints.html)**
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- Lagrange multiplier theory
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- KKT conditions for equality constraints
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- Quadratic programming implementation
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4. **[Part 3: Interior-Point Methods](part3_ipm.html)**
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- Inequality constraint handling
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- Barrier methods and log-barrier functions
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- Comparison with penalty methods
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## Additional Resources
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- **[Lecture Slides (PDF)](ISYE_8803___Lecture_2___Slides.pdf)** - Complete slide deck
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- **[Demo Script](penalty_barrier_demo.py)** - Python demonstration of penalty vs. barrier methods
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- **[LaTeX Source](main.tex)** - Source code for lecture slides
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## Key Learning Outcomes
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- Understand gradient-based optimization methods
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- Implement Newton's method for minimization
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- Apply root-finding techniques for implicit integration
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- Solve equality-constrained optimization problems
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- Compare different constraint handling methods
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- Implement Sequential Quadratic Programming (SQP)
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## Prerequisites
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- Solid understanding of linear algebra and calculus
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- Familiarity with Julia programming
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- Basic knowledge of differential equations
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- Understanding of optimization concepts from Class 1
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## Next Steps
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This class provides the foundation for advanced topics in subsequent classes, including Pontryagin's Maximum Principle, nonlinear trajectory optimization, and stochastic optimal control.
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‎class02/log_barrier.png‎

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‎class02/overview.md‎

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# Class 2 — 08/29/2025
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**Presenter:** Arnaud Deza
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**Topic:** Numerical optimization for control (gradient/SQP/QP); ALM vs. interior-point vs. penalty methods
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---
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## Overview
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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.
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## Learning Objectives
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By the end of this class, students will be able to:
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- Understand the mathematical foundations of gradient-based optimization
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- Implement Newton's method for unconstrained minimization
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- Apply root-finding techniques for implicit integration schemes
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- Solve equality-constrained optimization problems using Lagrange multipliers
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- Compare and contrast different constraint handling methods (ALM, interior-point, penalty)
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- Implement Sequential Quadratic Programming (SQP) for nonlinear optimization
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## Prerequisites
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- Solid understanding of linear algebra and calculus
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- Familiarity with Julia programming
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- Basic knowledge of differential equations
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- Understanding of optimization concepts from Class 1
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## Materials
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### Interactive Notebooks
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The class is structured around four interactive Jupyter notebooks that build upon each other:
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1. **[Part 1: Minimization via Newton's Method](part1_minimization.html)**
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- Unconstrained optimization fundamentals
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- Newton's method for minimization
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- Hessian matrix and positive definiteness
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- Regularization and line search techniques
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- Practical implementation with Julia
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2. **[Part 1: Root Finding & Backward Euler](part1_root_finding.html)**
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- Root-finding algorithms for implicit integration
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- Fixed-point iteration vs. Newton's method
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- Backward Euler implementation for ODEs
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- Convergence analysis and comparison
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- Application to pendulum dynamics
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3. **[Part 2: Equality Constraints](part2_eq_constraints.html)**
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- Lagrange multiplier theory
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- KKT conditions for equality constraints
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- Quadratic programming with equality constraints
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- Visualization of constrained optimization landscapes
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- Practical implementation examples
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4. **[Part 3: Interior-Point Methods](part3_ipm.ipynb)**
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- Inequality constraint handling
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- Barrier methods and log-barrier functions
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- Interior-point algorithm implementation
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- Comparison with penalty methods
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- Convergence properties and practical considerations
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### Additional Resources
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- **[Lecture Slides (PDF)](ISYE_8803___Lecture_2___Slides.pdf)** - Complete slide deck from the presentation
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- **[LaTeX Source Files](main.tex)** - Source code for the lecture slides
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- **[Demo Script](penalty_barrier_demo.py)** - Python demonstration of penalty vs. barrier methods
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## Key Concepts Covered
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### Mathematical Foundations
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- **Gradient and Hessian**: Understanding first and second derivatives in optimization
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- **Newton's Method**: Quadratic convergence and implementation details
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- **KKT Conditions**: Necessary and sufficient conditions for optimality
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- **Duality Theory**: Lagrange multipliers and dual problems
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### Numerical Methods
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- **Root Finding**: Fixed-point iteration, Newton-Raphson method
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- **Implicit Integration**: Backward Euler for stiff ODEs
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- **Sequential Quadratic Programming**: Local quadratic approximations
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- **Interior-Point Methods**: Barrier functions and path-following
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### Constraint Handling
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- **Equality Constraints**: Lagrange multipliers and null-space methods
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- **Inequality Constraints**: Active set methods and interior-point approaches
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- **Penalty Methods**: Quadratic and exact penalty functions
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- **Augmented Lagrangian**: Combining penalty and multiplier methods
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## Practical Applications
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The methods covered in this class are fundamental to:
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- **Optimal Control**: Trajectory optimization and feedback control design
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- **Model Predictive Control**: Real-time optimization with constraints
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- **Robotics**: Motion planning and control with obstacle avoidance
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- **Engineering Design**: Constrained optimization in mechanical systems
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## Further Reading
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- Nocedal, J., & Wright, S. J. (2006). *Numerical Optimization*
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- Boyd, S., & Vandenberghe, L. (2004). *Convex Optimization*
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- Betts, J. T. (2010). *Practical Methods for Optimal Control and Estimation Using Nonlinear Programming*
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## Next Steps
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This class provides the foundation for the advanced topics covered in subsequent classes, including:
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- Pontryagin's Maximum Principle (Class 3)
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- Nonlinear trajectory optimization (Class 5)
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- Stochastic optimal control (Class 7)
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- Physics-Informed Neural Networks (Class 10)
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---
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*For questions or clarifications, please refer to the interactive notebooks or contact the instructor.*

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