1 |
Convex and Nonconvex Optimization Problems
Why is Convexity Important in Optimization
Lagrange Multipliers and Duality
Min Common / Max Crossing Duality |

2 |
Convex Sets and Functions
Epigraphs
Closed Convex Functions
Recognizing Convex Functions |

3 |
Differentiable Convex Functions
Convex and Affine Bulls
Caratheodory's Theorem
Closure, Relative Interior, Continuity |

4 |
Review of Relative Interior
Algebra of Relative Interiors and Closures
Continuity of Convex Functions
Recession Cones |

5 |
Global and Local Minima
Weierstrass' Theorem
The Projection Theorem
Recession Cones of Convex Functions
Existence of Optimal Solutions |

6 |
Nonemptiness of Closed Set Intersections
Existence of Optimal Solutions
Special Cases: Linear and Quadric Programs
Preservation of Closure under Linear Transformation and Partial Minimization |

7 |
Preservation of Closure under Partial Minimization
Hyperplanes
Hyperplane Separation
Nonvertical Hyperplanes
Min Common and Max Crossing Problems |

8 |
Min Common / Max Crossing Problems
Weak Duality
Strong Duality
Existence of Optimal Solutions
Minimax Problems |

9 |
Min-Max Problems
Saddle Points
Min Common / Max Crossing for Min-Max |

10 |
Polar Cones and Polar Cone Theorem
Polyhedral and Finitely Generated Cones
Farkas Lemma, Minkowski-Weyl Theorem
Polyhedral Sets and Functions |

11 |
Extreme Points
Extreme Points of Polyhedral Sets
Extreme Points and Linear / Integer Programming |

12 |
Polyhedral Aspects of Duality
Hyperplane Proper Polyhedral Separation
Min Common / Max Crossing Theorem under Polyhedral Assumptions
Nonlinear Farkas Lemma
Application to Convex Programming |

13 |
Directional Derivatives of One-Dimensional Convex Functions
Directional Derivatives of Multi-Dimensional Convex Functions
Subgradients and Subdifferentials
Properties of Subgradients |

14 |
Conical Approximations
Cone of Feasible Directions
Tangent and Normal Cones
Conditions for Optimality |

15 |
Introduction to Lagrange Multipliers
Enhanced Fritz John Theory |

16 |
Enhanced Fritz John Conditions
Pseudonormality
Constraint Qualifications |

17 |
Sensitivity Issues
Exact Penalty Functions
Extended Representations |

18 |
Convexity, Geometric Multipliers, and Duality
Relation of Geometric and Lagrange Multipliers
The Dual Function and the Dual Problem
Weak and Strong Duality
Duality and Geometric Multipliers |

19 |
Linear and Quadric Programming Duality
Conditions for Existence of Geometric Multipliers
Conditions for Strong Duality |

20 |
The Primal Function
Conditions for Strong Duality
Sensitivity
Fritz John Conditions for Convex Programming |

21 |
Fenchel Duality
Conjugate Convex Functions
Relation of Primal and Dual Functions
Fenchel Duality Theorems |

22 |
Fenchel Duality
Fenchel Duality Theorems
Cone Programming
Semidefinite Programming |

23 |
Overview of Dual Methods
Nondifferentiable Optimization |

24 |
Subgradient Methods
Stepsize Rules and Convergence Analysis |

25 |
Incremental Subgradient Methods
Convergence Rate Analysis and Randomized Methods |

26 |
Additional Dual Methods
Cutting Plane Methods
Decomposition |