Optimization Theory

  1. Linear Programming
    1. Problem Formulation
      1. Standard Form
        1. Canonical Representation
          1. Slack and Surplus Variables
          2. Converting to Standard Form
            1. Inequality Constraints
              1. Unrestricted Variables
                1. Minimization to Maximization
              2. Geometric Foundations
                1. Polyhedra and Polytopes
                  1. Vertices and Extreme Points
                    1. Edges and Faces
                      1. Convex Hull Representation
                      2. Basic Feasible Solutions
                        1. Relationship to Extreme Points
                          1. Degeneracy
                            1. Adjacent Basic Solutions
                            2. Fundamental Theorem of Linear Programming
                            3. Simplex Method
                              1. Simplex Algorithm
                                1. Initial Basic Feasible Solution
                                  1. Optimality Test
                                    1. Pivot Selection Rules
                                      1. Tableau Operations
                                      2. Two-Phase Simplex Method
                                        1. Phase I: Finding Initial Solution
                                          1. Phase II: Optimization
                                            1. Artificial Variables
                                            2. Degeneracy and Cycling
                                              1. Bland's Pivoting Rule
                                                1. Lexicographic Rule
                                                2. Computational Complexity
                                                3. Duality in Linear Programming
                                                  1. Primal-Dual Relationships
                                                    1. Dual Problem Construction
                                                      1. Symmetric and Asymmetric Forms
                                                      2. Fundamental Theorems of Duality
                                                        1. Weak Duality
                                                          1. Strong Duality
                                                            1. Complementary Slackness
                                                            2. Economic Interpretation
                                                              1. Shadow Prices
                                                                1. Marginal Values
                                                              2. Sensitivity Analysis
                                                                1. Parametric Programming
                                                                  1. Right-Hand Side Changes
                                                                    1. Objective Function Coefficient Changes
                                                                      1. Adding New Variables
                                                                        1. Adding New Constraints
                                                                        2. Interior Point Methods
                                                                          1. Barrier Methods
                                                                            1. Logarithmic Barrier Function
                                                                              1. Central Path
                                                                              2. Primal-Dual Interior Point Methods
                                                                                1. Newton's Method for KKT System
                                                                                  1. Path-Following Algorithms
                                                                                  2. Polynomial-Time Complexity

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                                                                                4. Constrained Optimization Theory

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                                                                                6. Nonlinear Programming