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    Lecture Notes

    LEC # TOPICS
      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

    Min-Max Problems (PDF)

    Saddle Points

    Min Common / Max Crossing for Min-Max
    10

    Polar Cones and Polar Cone Theorem (PDF)

    Polyhedral and Finitely Generated Cones

    Farkas Lemma, Minkowski-Weyl 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 One-Dimensional Convex Functions (PDF)

    Directional Derivatives of Multi-Dimensional 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