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Convex Analysis and Optimization >> Content Detail



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LEC #TOPICS
1Convex and Nonconvex Optimization Problems

Why is Convexity Important in Optimization

Lagrange Multipliers and Duality

Min Common / Max Crossing Duality
2Convex Sets and Functions

Epigraphs

Closed Convex Functions

Recognizing Convex Functions
3Differentiable Convex Functions

Convex and Affine Bulls

Caratheodory's Theorem

Closure, Relative Interior, Continuity
4Review of Relative Interior

Algebra of Relative Interiors and Closures

Continuity of Convex Functions

Recession Cones
5Global and Local Minima

Weierstrass' Theorem

The Projection Theorem

Recession Cones of Convex Functions

Existence of Optimal Solutions
6Nonemptiness of Closed Set Intersections

Existence of Optimal Solutions

Special Cases: Linear and Quadric Programs

Preservation of Closure under Linear Transformation and Partial Minimization
7Preservation of Closure under Partial Minimization

Hyperplanes

Hyperplane Separation

Nonvertical Hyperplanes

Min Common and Max Crossing Problems
8Min Common / Max Crossing Problems

Weak Duality

Strong Duality

Existence of Optimal Solutions

Minimax Problems
9Min-Max Problems

Saddle Points

Min Common / Max Crossing for Min-Max
10Polar Cones and Polar Cone Theorem

Polyhedral and Finitely Generated Cones

Farkas Lemma, Minkowski-Weyl Theorem

Polyhedral Sets and Functions
11Extreme Points

Extreme Points of Polyhedral Sets

Extreme Points and Linear / Integer Programming
12Polyhedral Aspects of Duality

Hyperplane Proper Polyhedral Separation

Min Common / Max Crossing Theorem under Polyhedral Assumptions

Nonlinear Farkas Lemma

Application to Convex Programming
13Directional Derivatives of One-Dimensional Convex Functions

Directional Derivatives of Multi-Dimensional Convex Functions

Subgradients and Subdifferentials

Properties of Subgradients
14Conical Approximations

Cone of Feasible Directions

Tangent and Normal Cones

Conditions for Optimality
15Introduction to Lagrange Multipliers

Enhanced Fritz John Theory
16Enhanced Fritz John Conditions

Pseudonormality

Constraint Qualifications
17Sensitivity Issues

Exact Penalty Functions

Extended Representations
18Convexity, 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
19Linear and Quadric Programming Duality

Conditions for Existence of Geometric Multipliers

Conditions for Strong Duality
20The Primal Function

Conditions for Strong Duality

Sensitivity

Fritz John Conditions for Convex Programming
21Fenchel Duality

Conjugate Convex Functions

Relation of Primal and Dual Functions

Fenchel Duality Theorems
22Fenchel Duality

Fenchel Duality Theorems

Cone Programming

Semidefinite Programming
23Overview of Dual Methods

Nondifferentiable Optimization
24Subgradient Methods

Stepsize Rules and Convergence Analysis
25Incremental Subgradient Methods

Convergence Rate Analysis and Randomized Methods
26Additional Dual Methods

Cutting Plane Methods

Decomposition

 








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