
Overview Lecture: A New Look at Convex Analysis and Optimization (PDF) 
1 
Cover Page of Lecture Notes (PDF)
Convex and Nonconvex Optimization Problems (PDF)
Why is Convexity Important in Optimization
Lagrange Multipliers and Duality
Min Common/Max Crossing Duality 
2 
Convex Sets and Functions (PDF)
Epigraphs
Closed Convex Functions
Recognizing Convex Functions 
3 
Differentiable Convex Functions (PDF)
Convex and Affine Bulls
Caratheodory's Theorem
Closure, Relative Interior, Continuity 
4 
Review of Relative Interior (PDF)
Algebra of Relative Interiors and Closures
Continuity of Convex Functions
Recession Cones 
5 
Global and Local Minima (PDF)
Weierstrass' Theorem
The Projection Theorem
Recession Cones of Convex Functions
Existence of Optimal Solutions 
6 
Nonemptiness of Closed Set Intersections (PDF)
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 (PDF)
Hyperplanes
Hyperplane Separation
Nonvertical Hyperplanes
Min Common and Max Crossing Problems 
8 
Min Common / Max Crossing Problems (PDF)
Weak Duality
Strong Duality
Existence of Optimal Solutions
Minimax Problems 
9 
MinMax Problems (PDF)
Saddle Points
Min Common / Max Crossing for MinMax 
10 
Polar Cones and Polar Cone Theorem (PDF)
Polyhedral and Finitely Generated Cones
Farkas Lemma, MinkowskiWeyl Theorem
Polyhedral Sets and Functions 
11 
Extreme Points (PDF)
Extreme Points of Polyhedral Sets
Extreme Points and Linear / Integer Programming 
12 
Polyhedral Aspects of Duality (PDF)
Hyperplane Proper Polyhedral Separation
Min Common / Max Crossing Theorem under Polyhedral Assumptions
Nonlinear Farkas Lemma
Application to Convex Programming 
13 
Directional Derivatives of OneDimensional Convex Functions (PDF)
Directional Derivatives of MultiDimensional Convex Functions
Subgradients and Subdifferentials
Properties of Subgradients 
14 
Conical Approximations (PDF)
Cone of Feasible Directions
Tangent and Normal Cones
Conditions for Optimality 
15 
Introduction to Lagrange Multipliers (PDF)
Enhanced Fritz John Theory 
16 
Enhanced Fritz John Conditions (PDF)
Pseudonormality
Constraint Qualifications 
17 
Sensitivity Issues (PDF)
Exact Penalty Functions
Extended Representations 
18 
Convexity, Geometric Multipliers, and Duality (PDF)
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 (PDF)
Conditions for Existence of Geometric Multipliers
Conditions for Strong Duality 
20 
The Primal Function (PDF)
Conditions for Strong Duality
Sensitivity
Fritz John Conditions for Convex Programming 
21 
Fenchel Duality (PDF)
Conjugate Convex Functions
Relation of Primal and Dual Functions
Fenchel Duality Theorems 
22 
Fenchel Duality (PDF)
Fenchel Duality Theorems
Cone Programming
Semidefinite Programming 
23 
Overview of Dual Methods (PDF)
Nondifferentiable Optimization 
24 
Subgradient Methods (PDF)
Stepsize Rules and Convergence Analysis 
25 
Incremental Subgradient Methods (PDF)
Convergence Rate Analysis and Randomized Methods 
26 
Additional Dual Methods (PDF)
Cutting Plane Methods
Decomposition 