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
- Part II Separation Theorem, Extreme Points, Recessive Directions, and Geometry of Polyhedral Sets
- 6 Separation Theorem and Geometry of Convex Sets
- 7 Geometry of Polyhedral Sets
- 8 Exercises for Part II
- Part III Convex Functions
- Part IV Convex Programming, Lagrange Duality, Saddle Points
- Book part
- References
- Index
6 - Separation Theorem and Geometry of Convex Sets
from Part II - Separation Theorem, Extreme Points, Recessive Directions, and Geometry of Polyhedral Sets
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
- Part II Separation Theorem, Extreme Points, Recessive Directions, and Geometry of Polyhedral Sets
- 6 Separation Theorem and Geometry of Convex Sets
- 7 Geometry of Polyhedral Sets
- 8 Exercises for Part II
- Part III Convex Functions
- Part IV Convex Programming, Lagrange Duality, Saddle Points
- Book part
- References
- Index
Summary
In this chapter, we prove the Separation Theorem for convex sets and extract from it basic results on the geometry of closed convex sets, specifically (a) supporting hyperplanes, extreme points, and the (finite-dimensional) Krein--Milman Theorem, (b) recessive directions and recessive cone of a convex set, (c) the definition and basic properties of the dual cone, (d) the (finite-dimensional) Dubovitski--Milutin Lemma, (e) existence of bases and extreme rays of nontrivial closed pointed cones and their relation to extreme points of the cone’s base, (f)the (finite-dimensional) Krein--Milman Theorem in conic form, and (g) polarity.
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- Essential Mathematics for Convex Optimization , pp. 83 - 117Publisher: Cambridge University PressPrint publication year: 2025
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