Convex Optimization Theory
Table of Contents:
Basic Concepts of Convex Analysis
- Convex Sets and Functions
- Convex Functions
- Closedness and Semicontinuity
- Operations with Convex Functions
- Characterizations of Differentiable Convex Functions
- Convex and Affine Hulls
- Relative Interior and Closure
- Calculus of Relative Interiors and Closures
- Continuity of Convex Functions
- Closures of Functions
- Recession Cones
- Directions of Recession of a Convex Function
- Nonemptiness of Intersections of Closed Sets
- Closedness Under Linear Transformations
- Hyperplanes
- Hyperplane Separation
- Proper Hyperplane Separation
- Nonvertical Hyperplane Separation
- Conjugate Functions
- Summary
Basic Concepts of Polyhedral Convexity
- Extreme Points
- Polar Cones
- Polyhedral Sets and Functions
- Polyhedral Cones and Farkas' Lemma
- Structure of Polyhedral Sets
- Polyhedral Functions
- Polyhedral Aspects of Optimization
Basic Concepts of Convex Optimization
- Constrained Optimization
- Existence of Optimal Solutions
- Partial Minimization of Convex Functions
- Saddle Point and Minimax Theory
Geometric Duality Framework
- Min Common/Max Crossing Duality
- Some Special Cases
- Connection to Conjugate Convex Functions
- General Optimization Duality
- Optimization with Inequality Constraints
- Augmented Lagrangian Duality
- Minimax Problems
- Strong Duality Theorem
- Existence of Dual Optimal Solutions
- Duality and Polyhedral Convexity
- Summary
Duality and Optimization
- Nonlinear Farkas' Lemma
- Linear Programming Duality
- Convex Programming Duality
- Strong Duality Theorem - Inequality Constraints
- Optimality Conditions
- Partially Polyhedral Constraints
- Duality and Existence of Optimal Primal Solutions
- Fenchel Duality
- Conic Duality
- Subgradients and Optimality Conditions
- Subgradients of Conjugate Functions
- Subdifferential Calculus
- Optimality Conditions
- Directional Derivatives
- Minimax Theory
- Minimax Duality Theorems
- Saddle Point Theorems
- Theorems of the Alternative
- Nonconvex Problems
- Duality Gap in Separable Problems
- Duality Gap in Minimax Problems
References
Index
[Return to Athena Scientific Homepage]
info@athenasc.com