In 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:

1. 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.

2. 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.

Problem complexity vs. solution complexity

The previous two chapters focus on developing optimal polynomial-time solutions following formal optimization methods from operations research (OR). For some problems, such an approach may not be always effective, and could lead to nonpolynomial-time solutions. For these problems, a customized approach following algorithm design from computer science (CS) could be more effective and lead to a polynomial-time solution.

It is important to distinguish a (solution) algorithm's complexity from the underlying problem's complexity. A problem's complexity determines the potential complexity of any algorithm that is designed to solve this problem. That is, for a problem not in P, unless P = NP, any algorithm that can find an optimal solution to this problem must have nonpolynomial-time complexity. In contrast, if an algorithm (design to solve the problem) has a nonpolynomial-time complexity, we cannot claim that this problem is not in P. Another algorithm designed by someone else may well solve the problem with a polynomial-time complexity.

In this chapter, we illustrate the above approaches and ideas with a case study. This case study is concerned with an optimal relay node assignment problem that arises in cooperative communications (CC) [139]. We first formulate the problem as a mixed-integer linear programming (MILP), following OR's optimization approach.

Review of branch-and-bound framework

In this chapter, we present a general and effective approach to solve NP-hard or NP-complete problems, despite their potential exponential worst-case complexity. Although the complexity of this approach is nonpolynomial, when designed appropriately, this approach may still be a viable approach to solve many problems that arise in practice, particularly when their problem sizes are not terribly large.

The approach that we describe is called branch-and-bound [113]. It is a powerful general-purpose framework to solve nonconvex programming problems. Such a framework aims to obtain a (1 − ∊)-optimal solution for a small given ∊ ≥ 0. Obviously, the smaller ∊ is, the higher is the complexity. Thus, we need to select a suitable ∊ based on the optimality requirement. The efficiency of the algorithm depends on its ability to correctly remove large portions of the solution space during each iterative step within the process of identifying the best solution. As a result, we progressively focus on smaller and smaller portions of the solution space and thus we are able to find a (1 − ∊)-optimal solution much faster than a brute-force exhaustive search.

We now describe the branch-and-bound framework for a minimization problem (the procedure for a maximization problem is similar). Initially, we need to determine the solution space for our problem, including the set of feasible values for each of the optimization variables. The framework consists of two steps, namely, the bounding step and the branching step.

Book overview

The goal of this book is to offer a course to graduate students by demonstrating an important set of mathematical tools (with a focus on optimization techniques) and to show how they can be used to address some challenging problems in wireless networks.

Book organization This book consists of four parts.

1. • Part I, consisting of three chapters, is devoted to optimization and designing algorithms that can offer optimal solutions.

2. • Part II, consisting of five chapters, is devoted to techniques that can offer provably near-optimal solutions.

3. • Part III, consisting of two chapters, is devoted to some highly effective heuristics.

4. • Part IV, consisting of only one chapter, is devoted to some miscellaneous topics in the broader context of wireless network optimizations. This part will be expanded in a future edition.

Structure of each chapter Each chapter starts with a brief overview of a particular optimization technique and subsequently is followed by a comprehensive coverage of a case study in wireless networking. The goal of giving a pointer to the underlying theory at the beginning of each chapter is to offer a direction to students on what they should explore further in formal course work or textbooks in these areas. These pointers are not meant to be comprehensive tutorials, each of which could constitute a book on its own.

Introduction

This chapter is a sequel to the last chapter. In the last chapter, we showed how to design a (1 − ∊)-optimal approximation algorithm for a base station placement problem. The final solution gives a point (or any point within the corresponding fictitious cost point (FCP) subarea) for the base station. In this chapter, we extend the base station placement problem to the next level, where the base station is allowed to move around in the wireless sensor network (WSN).

The benefits of using a mobile base station to prolong sensor network lifetime are easy to understand. Since the base station is the sink node for data collected by all the sensor nodes in the network, the set of sensor nodes near the base station would carry a considerable burden in relaying traffic from other sensor nodes to the base station. By allowing the base station to be mobile, we could alleviate the traffic burden from this fixed set of sensor nodes to other sensor nodes in the network, and thus extend network lifetime. Further, given new advances in unmanned autonomous vehicle (UAV) and customized robotics, having an unmanned vehicle carrying a base station for sensor data collection may not be too far from reality. Although the potential benefit of using a mobile base station to prolong sensor network lifetime is significant, the technical difficulty of this optimization problem is enormous.

A highlight of this book is to include, in each chapter, a comprehensive presentation of a case study that demonstrates how the particular optimization method can be applied in solving a wireless networking problem. These case studies are based on a number of papers written by the authors. The following list acknowledges these publications and their respective journals and publishers on a chapter-by-chapter basis. Portions of these papers have been adapted with permission from the publishers as required. All rights are reserved as stipulated by the various copyright agreements.

Chapter 2

Y.T. Hou, Y. Shi, and H.D. Sherali, “Rate allocation and network lifetime problems for wireless sensor networks,” IEEE/ACM Transactions on Networking, vol. 16, no. 2, pp. 321–334, April 2008. Copyright © 2008 by, and with kind permission from, IEEE.

Chapter 3

J. Liu, Y.T. Hou, Y. Shi, and H.D. Sherali, “Cross-layer optimization for MIMO-based wireless ad hoc networks: routing, power allocation, and bandwidth allocation,” IEEE Journal on Selected Areas in Communications, vol. 26, no. 6, pp. 913–926, August 2008. Copyright © 2008 by, and with kind permission from, IEEE.

Chapter 4

S. Sharma, Y. Shi, Y.T. Hou, and S. Kompella, “An optimal algorithm for relay node assignment in cooperative ad hoc networks,” IEEE/ACM Transactions on Networking, vol. 19, issue 3, pp. 879–892, June 2011. Copyright © 2011 by, and with kind permission from, IEEE.