Numerical Methods

1. Nonlinear Systems and Optimization
   1. Basic Concepts
      1. Optimization Goals
         1. Minimize Cost Functions
         2. Maximize Efficiency or Profit
         3. Achieve Specific Performance Metrics
         4. Balance Trade-offs in Multi-objective Problems
      2. Types of Nonlinear Systems
         1. Systems with Polynomial Equations
         2. Systems with Trigonometric Functions
         3. Systems with Exponential Functions
         4. Hybrid Systems (mix of different types)
      3. Constraints Handling
         1. Equality Constraints
         2. Inequality Constraints
         3. No-Constraints and Unconstrained Optimization
         4. Lagrange Multipliers for Constrained Problems
   2. Methods
      1. Direct Methods
         1. Evaluation of Objective Functions
         2. Penalty Methods for Constraints
         3. Barrier Methods for Constraint Handling
      2. Iterative Optimization Techniques
         1. Gradient Descent
            1. Steepest Descent
            2. Learning Rate Adjustments
            3. Batch vs. Stochastic Gradient Descent
         2. Newton Method for Systems
            1. Advantages of Quadratic Convergence
            2. Hessian Matrix Utilization
            3. Damping and Line Search Strategies
         3. Conjugate Gradient Method
            1. Applications in Quadratic Problems
            2. Comparison with Gradient Descent
            3. Preconditioning in Conjugate Gradient
         4. Other Iterative Approaches
            1. Quasi-Newton Methods
               1. Broyden-Fletcher-Goldfarb-Shanno (BFGS)
               2. Limited-memory BFGS (L-BFGS)
            2. Trust Region Methods
            3. Sequential Quadratic Programming (SQP)
   3. Advanced Topics
      1. Global Optimization Techniques
         1. Genetic Algorithms
         2. Simulated Annealing
         3. Particle Swarm Optimization
         4. Application of Metaheuristic Algorithms
      2. Convex and Non-Convex Optimization
         1. Identifying Convex vs. Non-convex Problems
         2. Challenges in Non-convex Optimization
         3. Local vs. Global Minima Finding
      3. Duality Theory
         1. Relationship between Primal and Dual Problems
         2. Strong and Weak Duality Concepts
      4. Multi-objective Optimization
         1. Pareto Efficiency and Fronts
         2. Weighting Methods for Trade-offs
         3. Interactive and Non-interactive Methods
   4. Application in Machine Learning and Operations Research
      1. Training Machine Learning Models
         1. Optimization of Loss Functions
         2. Scale and Dimensionality Considerations
      2. Operations Research Applications
         1. Supply Chain Optimization
         2. Resource Allocation and Scheduling
      3. Real-world Challenges and Case Studies
         1. Scalability in Large-scale Systems
         2. Dealing with Data Uncertainty and Variability
   5. Software Tools and Frameworks
      1. Optimization Solvers
         1. Commercial Tools: CPLEX, Gurobi
         2. Open-source Options: IPOPT, COIN-OR
      2. Machine Learning Libraries
         1. TensorFlow and PyTorch Integration
         2. Optimization Algorithms in Scikit-learn
      3. Integration with Cloud Platforms
         1. Leveraging Cloud Computing Resources
         2. Distributed Optimization and Parallel Algorithms