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LEC #	TOPICS
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