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
- Part III Convex Functions
- Part IV Convex Programming, Lagrange Duality, Saddle Points
- 16 Convex Programming Problems and Convex Theorem of the Alternative
- 17 Lagrange Function and Lagrange Duality
- 18 ★ Convex Programming in Cone-Constrained Form
- 19 Optimality Conditions in Convex Programming
- 20 ★ Cone-Convex Functions: Elementary Calculus and Examples
- 21 ★ Mathematical Programming Optimality Conditions
- 22 Saddle Points
- 23 Exercises for Part IV
- Book part
- References
- Index
16 - Convex Programming Problems and Convex Theorem of the Alternative
from Part IV - Convex Programming, Lagrange Duality, Saddle Points
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
- Part III Convex Functions
- Part IV Convex Programming, Lagrange Duality, Saddle Points
- 16 Convex Programming Problems and Convex Theorem of the Alternative
- 17 Lagrange Function and Lagrange Duality
- 18 ★ Convex Programming in Cone-Constrained Form
- 19 Optimality Conditions in Convex Programming
- 20 ★ Cone-Convex Functions: Elementary Calculus and Examples
- 21 ★ Mathematical Programming Optimality Conditions
- 22 Saddle Points
- 23 Exercises for Part IV
- Book part
- References
- Index
Summary
In this chapter, we (a) outline the subject and the terminology of mathematical and convex programming, (b) introduce the Slater and relaxed Slater conditions and formulate the Convex Theorem on the Alternative -- the basis of Lagrange duality theory in convex programming, (c) introduce the notions of cone-convexity and of the convex programming problem in cone-constrained form, thus extending the standard mathematical programming setup of convex optimization, and (d) formulate and prove the Convex Theorem on the Alternative in cone-constrained form, justifying, as a byproduct, the standard Convex Theorem on the Alternative.
Keywords
mathematical programming problemconvex programming probemoptimal value of an optimization problemSlater conditionrelaxed Slater conditionstrictly feasible and essentially strictly feasible solutionsConvex Theorem on the Alternativecone-convexityconic inequalityconvex problem in cone-constrained formSlater condition in cone-constrained formConvex Theorem on the Alternative in cone-constrained form
Information
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- Essential Mathematics for Convex Optimization , pp. 235 - 248Publisher: Cambridge University PressPrint publication year: 2025
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