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
- 9 First Acquaintance with Convex Functions
- 10 How to Detect Convexity
- 11 Minima and Maxima of Convex Functions
- 12 Subgradients
- 13 ★ Legendre Transform
- 14 ★ Functions of Eigenvalues of Symmetric Matrices
- 15 Exercises for Part III
- Part IV Convex Programming, Lagrange Duality, Saddle Points
- Book part
- References
- Index
11 - Minima and Maxima of Convex Functions
from Part III - Convex Functions
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
- 9 First Acquaintance with Convex Functions
- 10 How to Detect Convexity
- 11 Minima and Maxima of Convex Functions
- 12 Subgradients
- 13 ★ Legendre Transform
- 14 ★ Functions of Eigenvalues of Symmetric Matrices
- 15 Exercises for Part III
- Part IV Convex Programming, Lagrange Duality, Saddle Points
- Book part
- References
- Index
Summary
In this chapter, we (a) demonstrate that every local minimizer of a convex function is its global minimizer, (b) show that the Fermat rule provides a necessary and sufficient condition for an interior point of the domain of a convex function to be a global minimizer of the function, provided the function is differentiable at the point, (c) introduce radial and normal cones and express in terms of these cones the necessary and sufficient condition for a point to be the minimizer of a convex function whenever the function is differentiable at the point, (d) introduce the symmetry principle and (e) provide basic information on maximizers of convex functions.
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- Essential Mathematics for Convex Optimization , pp. 175 - 183Publisher: Cambridge University PressPrint publication year: 2025
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