Mathematical Optimization

1. Optimization Methods and Techniques
   1. Exact Algorithms
      1. Branch and Bound
         1. Overview and basic principles
         2. Search tree and bounding criteria
         3. Application examples in integer programming
      2. Cutting Plane Methods
         1. Gomory cuts
         2. Integer polyhedra
         3. Application in optimization problems
      3. Simplex Algorithm
         1. Geometry of the simplex algorithm
         2. Pivot operations
         3. Phase I and Phase II procedures
   2. Heuristic and Metaheuristic Approaches
      1. Genetic Algorithms
         1. Biological inspiration: natural selection and genetics
         2. Encoding solutions and fitness function
         3. Selection, crossover, and mutation operators
         4. Advantages and limitations in problem-solving
      2. Simulated Annealing
         1. Analogy with physical annealing processes
         2. Cooling schedule and temperature parameters
         3. Acceptance criteria for solutions
         4. Use cases and performance considerations
      3. Particle Swarm Optimization
         1. Swarm intelligence principles
         2. Particle dynamics and velocity update rules
         3. Role of global and local best solutions
         4. Convergence characteristics
      4. Tabu Search
         1. Memory structures and tabu list
         2. Short-term vs. long-term memory
         3. Intensification and diversification strategies
         4. Use in large-scale optimization problems
   3. Gradient-Based Methods
      1. Steepest Descent
         1. Concept of steepest descent direction
         2. Line search techniques
         3. Rate of convergence
      2. Newton's Method
         1. Quadratic convergence properties
         2. Application of the Hessian matrix
         3. Linearization and iterative refinement
      3. Quasi-Newton Methods
         1. Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm
         2. Updating the inverse Hessian matrix
         3. Practical implementations and extensions
   4. Derivative-Free Optimization
      1. Direct Search Methods
         1. Nelder-Mead simplex and pattern search
         2. Exploration vs. exploitation balance
      2. Response Surface Methodology
         1. Model-based approaches
         2. Approximation of objective functions
         3. Iterative enhancement
      3. Surrogate Models
         1. Kriging and radial basis functions
         2. Multi-fidelity models
         3. Real-world optimization scenarios
   5. Convex Optimization
      1. Properties of Convex Functions
         1. Definition and examples of convexity
         2. Role in optimization problem formulation
         3. Implications for solution uniqueness
      2. Convex Sets and Convex Problems
         1. Properties of convex sets
         2. Methods for proving convexity
         3. Global vs. local optima
      3. Interior Point Methods
         1. Barrier functions and central path
         2. Primal-dual formulations
         3. Computational efficiency and scalability
   6. Multi-objective Optimization
      1. Concepts in Multi-objective Optimization
         1. Definition of Pareto optimality
         2. Trade-offs between objectives
         3. Visualization techniques: Pareto front
      2. Methods for Solving Multi-objective Problems
         1. Weighted sum approach
         2. Epsilon-constraint method
         3. Evolutionary algorithms for multi-objective optimization
         4. Decision-making processes in multiple criteria environments
   7. Constraint Handling Techniques
      1. Penalty Methods
         1. Interior and exterior penalty functions
         2. Choice of penalty coefficients
         3. Balancing feasibility and optimality
      2. Barrier Methods
         1. Logarithmic barrier functions
         2. Feasibility region approximation
         3. Application in linear and nonlinear problems
      3. Augmented Lagrangian Methods
         1. Combination of penalty and Lagrange multipliers
         2. Improving numerical stability
         3. Implementation challenges and solutions