Optimization Theory

  1. Constrained Optimization Theory
    1. Constraint Types and Properties
      1. Equality Constraints
        1. Linear Equality Constraints
          1. Nonlinear Equality Constraints
            1. Constraint Manifolds
            2. Inequality Constraints
              1. Linear Inequality Constraints
                1. Nonlinear Inequality Constraints
                  1. Active and Inactive Constraints
                  2. Feasible Region Geometry
                    1. Constraint Boundaries
                      1. Interior and Boundary Points
                        1. Regular and Singular Points
                      2. Optimality Conditions
                        1. Lagrange Multiplier Theory
                          1. Method of Lagrange Multipliers
                            1. Lagrangian Function
                              1. Interpretation of Multipliers
                              2. Karush-Kuhn-Tucker (KKT) Conditions
                                1. Stationarity Condition
                                  1. Primal Feasibility
                                    1. Dual Feasibility
                                      1. Complementary Slackness
                                        1. First-Order Necessary Conditions
                                        2. Second-Order Conditions
                                          1. Bordered Hessian
                                            1. Sufficient Conditions for Constrained Problems
                                          2. Constraint Qualifications
                                            1. Linear Independence Constraint Qualification (LICQ)
                                              1. Mangasarian-Fromovitz Constraint Qualification (MFCQ)
                                                1. Constant Rank Constraint Qualification (CRCQ)
                                                  1. Slater's Condition for Convex Problems
                                                    1. Abadie's Constraint Qualification
                                                    2. Duality Theory
                                                      1. Lagrangian Duality
                                                        1. Dual Function
                                                          1. Dual Problem Formulation
                                                          2. Weak Duality Theorem
                                                            1. Strong Duality Theorem
                                                              1. Duality Gap
                                                                1. Saddle Point Conditions

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                                                              3. Unconstrained Optimization

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                                                              5. Linear Programming