Convexity and Optimization – Part I

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216 pages

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English

This book contains everything you need to know about convexity in order to get a thorough understanding of linear and convex optimization.

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Description

Content

Convexity plays an important role in many areas of Mathematics, and this book, the first in a series of three on Convexity and Optimization, studies this concept in detail.

The first half of the book is about convex sets. Convex hull, convex cones, separation by hyperplanes, extreme points, faces, and extreme rays are some of the important notions that are considered. Results for the dual cone are interpreted as solvability criteria for systems of linear inequalities. Closed convex sets in general and polyhedra in particular are characterized in terms of extreme points and extreme rays.

The second half is about convex functions. We study, among other things, convexity preserving operations, maxima and minima of convex functions, continuity and differentiability properties, subdifferentials, and conjugate functions.

The book requires knowledge of Linear Algebra and Calculus of Several Variables.

- Preliminaries
- Convex sets
- Affine sets and affine maps
- Convex sets
- Convexity preserving operations
- Convex hull
- Topological properties
- Cones
- The recession cone

- Separation
- Separating hyperplanes
- The dual cone
- Solvability of systems of linear inequalities

- More on convex sets
- Extreme points and faces
- Structure theorems for convex sets

- Polyhedra
- Extreme points and extreme rays
- Polyhedral cones
- The internal structure of polyhedra
- Polyhedron preserving operations
- Separation

- Convex functions
- Basic definitions
- Operations that preserve convexity
- Maximum and minimum
- Some important inequalities
- Solvability of systems of convex inequalities
- Continuity
- The recessive subspace of convex functions
- Closed convex functions
- The support function
- The Minkowski functional

- Smooth convex functions
- Convex functions on R
- Differentiable convex functions
- Strong convexity
- Convex functions with Lipschitz continuous derivatives

- The subdifferential
- The subdifferential
- Closed convex functions
- The conjugate function
- The direction derivative
- Subdifferentiation rules