Book contents
- Essential Mathematics for Convex Optimization
- Reviews
- Essential Mathematics for Convex Optimization
- Copyright page
- Contents
- Preface
- Main Notational Conventions
- Part I Convex sets in Rn: From First Acquaintance to Linear Programming Duality
- 1 First Acquaintance with Convex Sets
- 2 Theorems of Carathéodory, Radon, and Helly
- 3 Polyhedral Representations and Fourier–Motzkin Elimination
- 4 General Theorem of the Alternative and Linear Programming Duality
- 5 Exercises for Part I
- Part II Separation Theorem, Extreme Points, Recessive Directions, and Geometry of Polyhedral Sets
- Part III Convex Functions
- Part IV Convex Programming, Lagrange Duality, Saddle Points
- Book part
- References
- Index
1 - First Acquaintance with Convex Sets
from Part I - Convex sets in Rn: From First Acquaintance to Linear Programming Duality
Published online by Cambridge University Press: 22 October 2025
Book contents
- Essential Mathematics for Convex Optimization
- Reviews
- Essential Mathematics for Convex Optimization
- Copyright page
- Contents
- Preface
- Main Notational Conventions
- Part I Convex sets in Rn: From First Acquaintance to Linear Programming Duality
- 1 First Acquaintance with Convex Sets
- 2 Theorems of Carathéodory, Radon, and Helly
- 3 Polyhedral Representations and Fourier–Motzkin Elimination
- 4 General Theorem of the Alternative and Linear Programming Duality
- 5 Exercises for Part I
- Part II Separation Theorem, Extreme Points, Recessive Directions, and Geometry of Polyhedral Sets
- Part III Convex Functions
- Part IV Convex Programming, Lagrange Duality, Saddle Points
- Book part
- References
- Index
Summary
In this chapter, we (a) present the definition and basic examples (linear and affine subspace, polyhedron, norm ball, simplex, cone) of convex sets, (b) introduce and study the notions of convex combination and convex hull, (c) outline basic convexity-preserving operations with sets, (d) establish nonemptiness of the relative interior of a nonempty convex set and the fact that the relative interior of a convex sets is dense in its closure, and (e) introduce the notions of conic and perspective transforms of a convex set.
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- Essential Mathematics for Convex Optimization , pp. 3 - 27Publisher: Cambridge University PressPrint publication year: 2025
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