Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgments
- Copyright Permissions
- 1 Introduction
- Part I Methods for Optimal Solutions
- 2 Linear programming and applications
- 3 Convex programming and applications
- 4 Design of polynomial-time exact algorithm
- Part II Methods for Near-optimal and Approximation Solutions
- Part III Methods for Efficient Heuristic Solutions
- Part IV Other Topics
- References
- Index
3 - Convex programming and applications
from Part I - Methods for Optimal Solutions
Published online by Cambridge University Press: 05 May 2014
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgments
- Copyright Permissions
- 1 Introduction
- Part I Methods for Optimal Solutions
- 2 Linear programming and applications
- 3 Convex programming and applications
- 4 Design of polynomial-time exact algorithm
- Part II Methods for Near-optimal and Approximation Solutions
- Part III Methods for Efficient Heuristic Solutions
- Part IV Other Topics
- References
- Index
Summary
They can because they think they can.
VirgilIn the previous chapter, we described LP and illustrated its application to solve some interesting problems in wireless networks. In this chapter, we describe convex programming [11], which is a popular tool to solve a wide range of problems in wireless networks. In terms of problem space, LP can be viewed as a special case of convex optimization. To facilitate our description, we define the following terms:
Convex set: A set is convex if for any two of its elements z1 and z2 and for any λ ∈ [0, 1], λz1 + (1 − λ)z2 is also an element of this set. For example, the set {(x, y): x2 + y2 ≤ 1} is a convex set but the set {(x, y) : 1 ≤ x2 + y2 ≤ 2} is not a convex set.
Convex and concave functions: A function f (x) is a convex function if for any x1 and x2 and any λ ∈ [0, 1], f (λx1 + (1 − λ)x2) ≤ λf (x1) + (1 − λ)f (x2), where x can be a single variable or a vector of variables.
- Type
- Chapter
- Information
- Applied Optimization Methods for Wireless Networks , pp. 38 - 60Publisher: Cambridge University PressPrint publication year: 2014