Mathematical Optimization

1. Types of Optimization Problems
   1. Linear Optimization
      1. Linear Programming (LP)
         1. Formulation of LP problems
            1. Defining decision variables
            2. Establishing objective functions
            3. Representing constraints as linear inequalities or equations
         2. Graphical method
            1. Geometric interpretation with two-variable problems
            2. Feasible region and corner-point evaluation
         3. Simplex method
            1. Step-by-step procedure using tableau format
            2. Pivot operations and optimality criteria
            3. Handling degeneracy and unbounded solutions
         4. Duality in LP
            1. Definition and interpretation of the dual problem
            2. Weak duality and strong duality theorems
            3. Economic interpretation of dual variables
         5. Sensitivity analysis
            1. Impact of changes in coefficients of objective function
            2. Variations in constraint right-hand side values
            3. Shadow prices and reduced costs analysis
   2. Nonlinear Optimization
      1. Nonlinear Programming (NLP)
         1. Unconstrained optimization
            1. Objective functions and local/global minima
            2. First-order and second-order conditions for optimality
         2. Constrained optimization
            1. Formulating constrained problems with equality and inequality constraints
            2. Methods of converting constrained problems to unconstrained
         3. Gradient descent methods
            1. Basic concepts and convergence criteria
            2. Variants: Stochastic Gradient Descent, Adaptive methods
         4. Newton's method
            1. Second-order optimization techniques
            2. Convergence properties and limitations
         5. Lagrange multipliers
            1. Method for incorporating equality constraints
            2. Lagrangian function formulation and solving
         6. Karush-Kuhn-Tucker (KKT) conditions
            1. Necessary conditions for optimality with inequality constraints
            2. Complementarity, dual feasibility, and primal feasibility
   3. Integer Optimization
      1. Integer Linear Programming (ILP)
         1. Difference from continuous LP, binary and integer variables
         2. Branch and bound method
            1. Tree representation of solution space
            2. Branching rules and bounding criteria
         3. Cutting plane methods
            1. Gomory cuts and other cutting plane techniques
            2. Separation of integer feasible solutions from fractional solutions
      2. Mixed-Integer Programming (MIP)
         1. Problem formulation with both integer and continuous variables
         2. Applications in industrial and logistics problems
         3. General algorithms and software for MIP
   4. Dynamic Optimization
      1. Dynamic Programming (DP)
         1. Bellman equation
            1. Recursive decomposition of problems
            2. Value function and state transition functions
         2. Principle of optimality
            1. Use in breaking down multi-stage decision processes
         3. Applications in decision making
            1. Resource allocation, equipment replacement, and shortest path problems
      2. Stochastic Dynamic Programming
         1. Handling uncertainty in decision making over time
         2. Markov Decision Processes (MDP)
         3. Risk-sensitive and robust dynamic optimization
   5. Combinatorial Optimization
      1. Traveling Salesman Problem (TSP)
         1. Exact algorithms: branch and cut, dynamic programming approaches
         2. Heuristic methods: nearest neighbor, genetic algorithms, simulated annealing
         3. NP-hard nature and approximation algorithms
      2. Knapsack problem
         1. Variants: 0/1 knapsack, bounded, unbounded knapsack
         2. Greedy method and dynamic programming solutions
         3. Applications in resource allocation and budgeting
      3. Graph algorithms
         1. Minimum spanning tree algorithms: Kruskal, Prim
         2. Shortest path algorithms: Dijkstra, Bellman-Ford
         3. Network flow problems and solutions: maximum flow, minimum cost flow