Convex Analysis and Optimization
Table of Contents:
Basic Convexity Concepts
- Linear Algebra and Real Analysis
- Vectors and Matrices
- Topological Properties
- Square Matrices
- Derivatives
- Convex Sets and Functions
- Convex and Affine Hulls
- Relative Interior, Closure, and Continuity
- Recession Cones
- Nonemptiness of Intersections of Closed Sets
- Closedness Under Linear Transformations
- Notes, Sources, and Exercises
Convexity and Optimization
- Global and Local Minima
- The Projection Theorem
- Directions of Recession and Existence of Optimal Solutions
- Existence of Solutions of Convex Programs
- Unbounded Optimal Solution Sets
- Partial Minimization of Convex Functions
- Hyperplanes
- An Elementary Form of Duality
- Nonvertical Hyperplanes
- Min Common/Max Crossing Duality
- Saddle Point and Minimax Theory
- Min Common/Max Crossing Framework for Minimax
- Minimax Theorems
- Saddle Point Theorems
- Notes, Sources, and Exercises
Polyhedral Convexity
- Polar Cones
- Polyhedral Cones and Polyhedral Sets
- Farkas' Lemma and Minkowski-Weyl Theorem
- Polyhedral Sets
- Polyhedral Functions
- Extreme Points
- Extreme Points of Polyhedral Sets
- Polyhedral Aspects of Optimization
- Linear Programming
- Integer Programming
- Polyhedral Aspects of Duality
- Polyhedral Proper Separation
- Min Common/Max Crossing Duality
- Minimax Theory Under Polyhedral Assumptions
- A Nonlinear Version of Farkas' Lemma
- Convex Programming
- Notes, Sources, and Exercises
Subgradients and Constrained
Optimization
- Directional Derivatives
- Subgradients and Subdifferentials
- Epsilon-Subgradients
- Subgradients of Extended Real-Valued Functions
- Directional Derivative of the Max Function
- Conical Approximations
- Optimality Conditions
- Notes, Sources, and Exercises
Lagrange Multipliers
- Introduction to Lagrange Multipliers
- Enhanced Fritz John Optimality Conditions
- Informative Lagrange Multipliers
- Sensitivity
- Alternative Lagrange Multipliers
- Pseudonormality and Constraint Qualifications
- Exact Penalty Functions
- Using the Extended Representation
- Extensions Under Convexity Assumptions
- Notes, Sources, and Exercises
Lagrangian Duality
- Geometric Multipliers
- Duality Theory
- Linear and Quadratic Programming Duality
- Existence of Geometric Multipliers
- Convex Cost -- Linear Constraints
- Convex Cost -- Convex Constraints
- Strong Duality and the Primal Function
- Duality Gap and the Primal Function
- Conditions for No Duality Gap
- Subgradients of the Primal Function
- Sensitivity Analysis
- Fritz John Conditions when there is no Optimal Solution
- Enhanced Fritz John Conditions
- Informative Geometric Multipliers
- Notes, Sources, and Exercises
Conjugate Duality
- Conjugate Functions
- Fenchel Duality Theorems
- Connection of Fenchel Duality and Minimax Theory
- Conic Duality
- Exact Penalty Functions
- Notes, Sources, and Exercises
Dual Computational Methods
- Dual Derivatives and Subgradients
- Subgradient Methods
- Analysis of Subgradient Methods
- Subgradient Methods with Randomization
- Cutting Plane Methods
- Ascent Methods
- Notes, Sources, and Exercises
References
Index
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