University of Southern California Dissertations and Theses

Models and algorithms for energy efficient wireless sensor networks

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Models and algorithms for energy efficient wireless sensor networks

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MODELS AND ALGORITHMS FOR ENERGY EFFICIENT WIRELESS SENSOR
NETWORKS

by

Wei Ye

A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(INDUSTRIAL AND SYSTEMS ENGINEERING)

May 2007

Copyright 2007 Wei Ye
Acknowledgements

Thanks to Dr. Fernando Ordonez and Dr. Maged M. Dessouky for their continuous support
and guidance during my PhD study period at USC. They have created a flexible and
pleasant work atmosphere. I have learnt from them so many things in both the scientific
and personal levels. I would like to thank the other member of my advisory committee Dr.
Bhaskar Krishnamachari, Dr. James E. Moore II, Dr. Sheldon M. Ross, Dr. Kurt D. Palmer.
I would like to thank Dr. Richard Waltz for his help with KNITRO 5.1 in solving the
nonlinear programming problems in the robust localization section. I also would like to
thank Dr. Elaine Chew, Dr. F. Stan Settles, Dr. Detlof von Winterfeldt, Dr. Randolph W.
Hall, Dr. Gerald A. Fleischer and Dr. Mohamed I. Dessouky. And I would like to thank all
other faculty and staff members of the Daniel J. Epstein department for such a rich
academic environment. I would like to Ms. Zhihong Sheng, Dr. Worawan Suteewong, Dr.
Luca Quadrifoglio, Mr. Hongzhong Jia, Mr. Ilgaz Sungur and other friends for kind help
for my PhD study.
Thanks to my family for their correspondence and encouragement, it is their love and
support through the years that made this achievement possible
ii
Table of Contents

Acknowledgements ii

List of Figures v

Abstract viii

Chapter One: Introduction 1

Chapter Two: Literature Review 5
2.1 Characteristics of WSNs 5
2.1.1 Deterministic and Randomized Deployment 5
2.1.2 Homogeneous vs. Heterogeneous Nodes 6
2.1.3 Dynamic or Static Network Topology 7
2.1.4 Coverage 7
2.1.5 Data Aggregation 8
2.2 Routing Protocols 8
2.2.1 Network Dependent Routing Protocols 8
2.2.2 Optimization Based Protocols 10
2.3 Important Operational Problems in WSNs 11
2.3.1 Maximal Data Extraction 12
2.3.2 Minimal Energy Consumption 12
2.3.3 Maximal Network Lifetime 13
2.3.4 Localization 13
2.4 Robust Optimization and Its Use in WSN Models 15

Chapter Three: Routing Algorithms in Energy Limited WSNs 19
3.1 Problem Definition 19
3.2 Mathematical Programming Model 21
3.3 Methodology 23
3.3.1 Partial Lagrangian Relaxation 23
3.3.2 Sub-Gradient Projection Method 25
3.4 Centralized Algorithms 27
3.5 Distributed Algorithms 32
3.6 Evaluation Criteria and Simulation Results 35
3.6.1 Evaluation Criteria 35
3.6.2 Simulation Results 37
3.7 Conclusions 42

Chapter Four: Robust Models under Distance Uncertainty 44
4.1 Problem Definition 44
4.1.1 Maximum Data Extraction Problem 44
iii
4.1.2 Minimum Energy Consumption Problem 45
4.1.3 Maximum Lifetime Problem 47
4.2 Methodology and Uncertainty Sets 49
4.2.1 Uncertainty Sets 49
4.2.2 Robust Counterpart Problems 51
4.4. Computational Experiments 60
4.4.1 Experimental Set Up 60
4.4.2 Results for Maximum Data Extraction Problem 63
4.4.3 Result for Minimum Energy Consumption Problem 67
4.4.4 Results for Maximum Lifetime problem 72
4.5 Conclusions 76

Chapter Five: Robust Localization of Sensor Nodes 78
5.1 Problem Description 79
5.2 Methodologies 81
5.2.1 Uncertainty Sets 82
5.2.2 Method-2: Robust Optimization (RO) method 83
5.2.3 Method-3 Minimizing Minimum Mean Square Error (MMMSE) method 87
5.3 Simulations and Results 88
5.3.1 Experimented Set-Up 88
5.3.2 Performance Criteria 89
5.3.3 Experimented Results 89
5.4 Conclusions 98

Chapter Six: Contributions and Extensions 100

References 102

iv

List of Figures

Figure 1.1: A typical sensor network topology 2

Figure 3.1: First order radio model 20

Figure 3.2: A simple WSN topology (n=10) 21

Figure 3.3: Direct Transmission 30

Figure 3.4: One Hop Transmission 30

Figure 3.5: Random 50-node topology for a 0.5 km* 0.5 km network 36

Figure 3.6: Homogeneous experiment for Centralized Algorithms 38

Figure 3.7: Heterogeneous experiment for Centralized Algorithms 39

Figure 3.8: Homogenous experiment for DA, and CA Algorithms 40

Figure 3.9: Heterogeneous experiments for DA, and CA Algorithms 42

Figure 4.1: The comparison of
ac
R and
wc
R in maximum data extraction 64
Problems

Figure 4.2: The mean and standard deviations of objective value for the 65
deterministic and robust solutions in maximum data
extraction problem for different uncertainty level

Figure 4.3: The comparison of
ac
R and
wc
R in maximum data extraction 66
problems as a function of the available energy E of each node

Figure 4.4: The comparison of
ac
R and
wc
R for objective value in 67
minimum energy consumption problems

Figure 4.5: The mean and standard deviations of objective value in minimum 69
energy consumption problem for different uncertainty level

Figure 4.6: The comparison of
ac
R and
wc
R of the objective value in 70
minimum energy consumption problem on different
percentage p of total data to sink node
v

Figure 4.7: The comparison of
ac
R and
wc
R for objective value q (q=1/T) in 71
maximum lifetime problems

Figure 4.8: The mean and standard deviations of objective value for the 72
deterministic and robust solutions in minimum energy consumption
problem for different percentage P of total data that must be sent
to the sink
max
250000, 10000, 1 0.1
ii
ED τ == =

Figure 4.9: The comparison of
ac
R and
wc
R for objective value in minimum 73
energy consumption problems in different location of the sink
node (p1 = 90%, p2 = 50%, = 250000, = 10000,
i
E
max
i
D
R =0.1, τ = 0.5)

Figure 4.10: The mean and standard deviations for the deterministic and robust 74
solutions of objective value T in maximum lifetime problem for
different uncertainty level

Figure 4.11: The comparison of
ac
R and
wc
R of the objective value q (q=1/T) 75
for maximum lifetime problem for different percentage p of total
data to sink node

Figure 5.1: The topology description of malicious distance by attackers 80

Figure 5.2: Experiments for different sensors n=6, 7, 8, 9, and 10 when K=1, 90
error =10%

Figure 5.3: M1, M2 and M3 for different uncertainty β when Error=0%, 92
K=1 and n=10

Figure 5.4: M1, M2 and M3 for different uncertainty β when Error=10%, 92
K=1 and n=10

Figure 5.5: M1, M2 and M3 for different uncertainty β when Error=20%, 93
K=1 and n=10

Figure 5.6: M1, M2 and M3 for different uncertainty β when Error=0%, 94
and n=10

Figure 5.7: M1, M2 and M3 for different uncertainty β when Error=10% 95
and n=10
vi
Figure 5.8: M1, M2 and M3 for different uncertainty β when Error=20%, 96
K=1 and n=10

Figure 5.9: Solution time of M1, M2 and M3 for different bad information K 97
when Error=0% and n=10

Figure 5.10: M1, M2 and M3 for different β when Error=0%, n=10, K=5 98
vii

Abstract

Wireless Sensor Networks (WSNs) is an area of active research in industry and academia.
WSNs can be used in a wide array of applications such as, battlefield surveillance,
aerospace exploration, environmental monitoring, products tracking and supply chain
management, homeland security applications, and so on. In this dissertation, we study
algorithms that address two challenges faced by WSNs in applications: the need to operate
distributedly and to take into account uncertain conditions. We first work
on constructing efficient distributed routing algorithms for maximal data extraction
problem, the second part of this thesis focuses on the effect of considering uncertain
conditions for routing in WSN. The last part of the thesis introduces methodologies that
address uncertainty to achieve secure localization in hostile environments.

We develop efficient distributed routing algorithms for data extraction by using the
Lagrangian relaxation method and the sub-gradient projection method on a maximal data
extraction formulation of the routing problem. We show through computational
experiments that, for the problem considered, both centralized and distributed versions of
the algorithm arrive at routing solutions that are on average better than 10% from optimal
after only a few iterations.

viii
We use robust optimization to address distance uncertainty in WSN routing. We develop
models that incorporate uncertainty for three important problems in WSN operations:
Maximum data extraction, Minimum energy consumption, and Maximum network lifetime
problems. Our computational experiments show that as the uncertainty increases a robust
solution for these problems provides a significant improvement in worst case performance
at the expense of a small loss in optimality when compared to the optimal solution of a
fixed scenario.

Finally, we consider the robust analysis in secure localization for energy limited wireless
sensor networks under malicious attackers in hostile environments. We present three
methods: Minimum Mean Square Error (MMSE), Robust Optimization (RO) and
Minimizing Minimum Mean Square Error (MMMSE) to make the estimated location
attack-tolerant. Simulations show that the robust method obtains accurate location
estimates in the presences of an unknown number of tampered distance data and large
uncertainty set.

ix
Chapter One: Introduction
Wireless Sensor Networks (WSNs) is an area of active research in the world. WSNs are
made up of inexpensive sensors that can be quickly deployed in large numbers with low
set-up costs. Each node operates autonomously and is equipped with data processing,
memory and capable of short range wireless communications. But they can work
cooperatively and cover a large area. These sensors can be used to detect temperature,
noise, pressure, in unfriendly environments. They can be used to detect the presence of
noxious, poisonous or other dangerous materials, providing early detection and
identification before serious damage can result. WSNs are critical to achieve today's
missions at home and abroad. WSNs can be placed in high density locations, such as, big
airports, train stations, shopping malls, to monitor environmental applications. It is
possible that in the future WSNs will become integral to our everyday lives in ways that
are difficult to imagine today, for a description see Elson and Estrin (2004).
WSNs are made up of two parts as shown in Figure 1.1. One part is called sensor network,
which consists of inexpensive sensors with sensing, processing, and communication
function that can be deployed in large quantities without close human supervision,
possibly into unfriendly territory, for various information gathering tasks. Each node has
a limited energy supply and can collect the data that needs to be transmitted to the sink
node. Within its energy limits, WSNs must sense, process, and transmit information to the
sink node, where a remote end-user can access it. The other part is called a centralized
control unit. The positions of sensor nodes need not be engineered or predetermined. For
1
example in battlefield or environmental monitoring of remote geographical regions,
sensors can be randomly deployed by simply throwing from the aircraft, once deployed,
the nodes can communicate with each other and build connected networks. Sensor
networks can detect information of interest such as moisture, temperature and pressure.
Then they get this data from the environment and send it into the control unit. Remote
users can use the Internet to connect to the control unit, to learn the conditions of the
sensor network.

Fig1.1: A typical sensor network topology

The use of WSNs aims at revolutionizing the availability and quality of information in a
wide array of applications, such as, industrial control and monitoring, marine
environmental monitoring, forest wildfire surveillance, battlefield surveillance and
Control unit
Sink
Node
Sensor Networks Field End Users
Sensors
Processors
Radio Tx/Rx
Internet
Memory

Energy Resource/ Battery

2
military sensing, air quality monitoring and radiation sensing, bridge structural integrity
monitoring, products tracking and supply chain management, intelligent agriculture,
homeland security applications, see for instance Elson and Estrin (2004), Estrin et al.
(1999), Akyildiz et al.(2002), Harter et al. (1999).
Although there are many possible applications of WSNs, the unique features and
application requirement of sensor networks, such as limited power, dense deployment and
random network topology, make inefficient many protocols and algorithms that exist for
traditional wireless networking, and force a different operation of the network, see Estrin
et al. (1999) and Akyildiz et al.(2002).
In this dissertation, we study algorithms that address two challenges faced by WSNs in
applications: the need to operate distributedly and to take into account uncertain
conditions. We first work on constructing efficient distributed routing algorithms for
maximal data extraction problem, the second part of this thesis focuses on the effect of
considering uncertain conditions for routing in WSN. The last part of the thesis
introduces methodologies that address uncertainty to achieve secure localization in
hostile environments.
The dissertation is organized as follows. We first present a literature review of related
works in Section 2. In Section 3, we design the routing models and algorithms for energy
limited WSNs. Centralized Algorithms and Distributed Algorithms based on the
Lagrangian relaxation method and sub-gradient project method are discussed. Robust
routing problems under distance of uncertainty are introduced in Section 4. Some results
based on robust optimization method are given. In Section 5, we consider securing
3
location under a malicious attack in unfriendly environment. In Section 6, we present the
main conclusions of the dissertation.

4

Chapter Two: Literature Review

There is a significant body of work in the literature on WSNs. We give the brief review of
prior work related to this dissertation. This review section first describes the main
characteristics present in a WSN, we then present a literature review on routing protocols,
and on other important optimization problems for WSN. We finish with a short
description of robust optimization literature, which is the method we use to address
uncertainty.
2.1 Characteristics of WSNs
We first describe the main features in a WSN: Sensor networks are application-specific.
Traditional wireless networks are deployed to adapt to a wide variety of applications. But
it is reasonable to design a sensor network to address specific problems and thus the
design of the network can be tailored to the sensing task at hand. Depending on the
specific application, WSNs can have different characteristics, such as the type of node
deployment, the type of nodes, dynamic or static network topology, the type of coverage,
and data manipulation, see Estrin et al. (1999), Sadagopan and Krishnamachari (2004),
and Al-Karaki and Kammal (2004).
2.1.1 Deterministic and Randomized Deployment
There are two possible types of sensor deployments: deterministic and randomized. In the
deterministic deployment, the sensors are manually put and the location of each node is
5
known exactly. However, in the random case, the sensor nodes are placed randomly, with
possibly unknown exact locations, and use ad-hoc paths. Node deployments are
application dependent and influence the performance of the routing protocols, see Al-
Karaki and Kammal (2004). In our work, we consider both cases.
2.1.2 Homogeneous vs. Heterogeneous Nodes
In many papers, all sensor nodes are presumed to be homogeneous, for example, all
nodes have the same energy and same memory capacity. However, depending on the
specific application different sensor nodes may have different role or capability. For
example, some applications might need a mixture of sensors for monitoring light,
temperature, and humidity of the surrounding environment, detecting motion via acoustic
ways, and capturing the image or video tracking of moving objects, see Al-Karaki and
Kammal (2004), and Svaizer et al. (1997). For some applications, different sensor nodes
have different initial energy. Data receiving and transmission can be generated from
different sensors at different rates, subject to diverse quality of service constraints, and
can follow multiple data transmission paths. For example, hierarchical protocols specify a
cluster-head node that is different from the normal sensors, which can be selected from
the deployed sensors or can be more powerful than other sensor nodes in terms of energy,
bandwidth, and memory. Hence, the main burden of transmission to the sink node is
handled by the set of cluster-heads. The existence of heterogeneous sensors improves a
large amount of technical issues related to data routing, see Heinzelman et al. (2000),
6
Lindsey et al. (2001), and Intagagonwiwat et al. (2000). In our work, we consider
homogeneous and heterogeneous nodes.
2.1.3 Dynamic or Static Network Topology
Sensor nodes are assumed to be stationary in most of the sensor network architectures.
However, the network topology can be dynamic for some applications, allowing the sink
node and sensor nodes to move. For example, the network topology is dynamic in a target
detection/tracking application, while it is static in environmental monitoring applications.
Routing data from or to moving nodes is more challenging, see Al-Karaki et al. (2004),
Sichitiu et al. (2003) and Patwari et al. (2003). Dynamic events in most applications
require periodic transmission and consequently generate significant track to be routed to
the sink node, see Al-Karaki et al. (2004). In our work, we consider static topology.
2.1.4 Coverage
In many papers, it is assumed that sensors can detect all the targets in a fixed area.
However in some WSNs applications, each sensor obtains the view of a certain range of
the environment since each sensor has limited sensing range. A given sensor's view of the
environment is limited both in range and in accuracy; it can only cover a limited physical
area of the environment see Abrams et al. (2004). For example, video sensors and
acoustic sensors can detect any target within a given area, see Savarese et al. (2002).
Hence, the coverage area is also an important design parameter in WSNs.
7
2.1.5 Data Aggregation
For some applications, sensor nodes may get lots of redundant data. Similar information
from multiple sensors can be aggregated or redundant data can be removed to reduce the
amount of data transmitted leading to energy savings. Data aggregation is the
combination of data from different nodes in term of a certain function of aggregation, e.g.,
duplicate suppression, minima, maxima and average. This technique has been used to
achieve energy efficiency and optimize data transfer transportation in some routing
protocols. Signal processing methods can also be used for data aggregation, which it is
referred to as data fusion, where the incoming signals are combined by using some
techniques such as beam forming to produce a more accurate output signal and reduce the
noise in these signals, see Heinzelman et al (2000), and Al-Karaki et al. (2004).
2.2 Routing Protocols
There exist a number of papers about routing protocols. Here we classify recent routing
protocols for sensor networks in two broad classes, which we discuss below in more
detail: network dependent protocols and optimization based protocols.
2.2.1 Network Dependent Routing Protocols
Network dependent routing protocols exploit various network properties for efficiency.
For example LEACH proposed by Heinzelman et al. (2000), PEGASIS proposed by
Lindsey et al. (2001), Directed Diffusion proposed by Intagagonwiwat et al. (2000), and
SPIN proposed by Heinzelman et al. (1999) exploit the possibility of data aggregation at
the nodes to achieve important energy savings. Low Energy Adaptive Clustering
8
Hierarchy (LEACH) is a hierarchical clustering algorithm for sensor networks. In
LEACH, some sensor nodes are randomly selected as cluster-heads and the cluster-head
nodes aggregate data arriving from nodes that belong to the respective cluster, and
transmit a compressed packet to the sink node in order to reduce the amount of data that
must be sent to the sink node. An improvement for LEACH protocol, called Power-
Efficient GAthering in Sensor Information Systems (PEGASIS), was presented by
Lindsey et al. (2001) and is the near optimal chain based protocol for a data-gathering
problem in sensor networks. The basic principal of the protocol is that nodes need only
communicate with their closest neighbors and they take turns in connection with the base
station to extend the network lifetime. Directed Diffusion has described one way of
disseminating data, where the sink nodes send out interest, the description of tasks, to all
sensors. After the interest has been propagated through the entire network, gradient paths
from each source node back to the sink node are built. Then sources nodes send the data
to the sink node along the interest’s gradient paths. Interest and data propagation and
aggregation are dealt with locally. An adaptive Protocol named Sensor Protocols for
Information via Negotiation (SPIN) disseminate the data at each sensor, based on the
assumption that all sensors in the network are potential base-stations, enabling a user to
query any node and receive the required information immediately. Nodes running the
SPIN protocol assign a high-level name to finish describing their collected data (meta-
data), and perform meta-data negotiations before any data is transmitted.
The algorithms proposed by Braginsky and Estrin (2002), Tian and Georganas (2003)
rely on high-energy agents in the network to create directed paths between the source of
9
information and the sink node. Thus the algorithms can save energy typically used for
flooding queries through the network. In general terms, since network dependent
protocols exploit particular features of the network, they could conceivably perform
poorly in a network without these properties, in addition it is difficult to obtain
performance bounds for these routing heuristics as they do not compute or approximate
an optimal solution.
A number of protocols use node clustering to break down the problem. For instance,
Ghiasi et al. (2002), Li et al. (2001) and Srivastava et al. (2002). Ghiasi et al. (2002)
provide an algorithm for clustering the sensor nodes such that each cluster (which has a
master) is balanced and the total distance between sensor nodes and master nodes is
minimized. This clustering maintains a certain degree of service quality and a reasonable
system lifetime. Li et al. (2001) proposed a hierarchical power-aware routing protocol to
extend the lifetime of the network. This protocol divides the network into group of
sensors in geographic clusters. Srivastava et al. (2002) optimally solve the problem of
scheduling the nodes to be switched on such that there exists at-least one path from one
side to an other at every time during intruder detection and minimizing the overall power
dissipation.
2.2.2 Optimization Based Protocols
Optimization based protocols implement an iterative optimization algorithm on some
problems over the sensor network. Examples of such protocols for sensor networks
include: sub-gradient algorithms for the maximum lifetime problem, i.e. maximizing the
10
time until the first node runs out of energy, see Sankar and Liu (2004), Madan and
Lall(2004) and Hua and Yum (2005), and approximate solutions to maximum data
extraction problem, see Sadagopan et al. (2004). The prior work on sub-gradient based
methods considers additional assumptions, such as the use of potential functions, to
ensure efficient performance. The sub-gradient optimization method is classic in non-
linear optimization, see for instance Bertsekas (1999), and has been used to develop
distributed algorithms for network flow, see Tseng et al. (1990) and flow control in
networks without energy constraints but with fixed capacity, see Low and Lapsley (1999).
Sadagopan et al. (2004) propose an approximate algorithm that uses network topology
and current energy information to derive a metric with which to route the information.
Optimization based protocols are both general and can provide performance bounds
based on the optimization problem over the network.
2.3 Important Operational Problems in WSNs
There are many open issues in WSNs. In this dissertation, we try to find energy efficient
distributed protocols in practice for data extraction that arise in many WSN applications
and these algorithms are insensitive to the uncertain conditions under which they have to
operate in unfriendly environments. We first consider maximizing data extraction
problem in WSNs and present its robust model under distance uncertainty to show the
robust solution is insensitive to the uncertain conditions. We also present robust models
for other two important problems in WSNs Operations: minimizing energy consumption,
and maximizing the lifetime of network. Finally we also consider robust analysis in
11
securing localization under malicious attackers in hostile environments. Here, we give
brief description related to these problems.
2.3.1 Maximal Data Extraction
Within its energy limits, the WSN must sense, process, and transmit information to a base
station or sink node, where a remote end-user can access it. Since communication is often
the most expensive operation for a sensor node, an efficient algorithm to route the data
gathered is crucial to efficiently use the limited energy. For instance, if every node
transmits its data directly to the sink node, nodes with little data to send will be left with
unused energy while data can be stranded in a node that depleted its energy. Thereby the
problem of maximizing data extraction in energy-limited WSN is also interesting
problem, see Ye and Ordonez (2005), Sadagopan and Krishnamachari (2004), and
Ordonez and Krishnamachari (2004).
2.3.2 Minimal Energy Consumption
Minimizing energy consumption problem is related to the problem of maximizing data
extraction in energy limited WSNs. The objective of the problem is to minimize the
amount of energy consumed to extract certain information from sensors to the sink node.
Sensor nodes can use up their limited supply of energy performing computations and
transmitting information. As such, energy-conserving forms of communication and
computation are essential. In a multi-hop WSN, each node plays a dual role as data
sender and data router. The malfunctioning of some sensor nodes due to power failure
can cause significant topological changes and might require rerouting of packets and
12
reorganization of the network, see Heinzelman et al. (2000), Lindsey et al. (2001) and
Ordonez and Krishnamachari (2004).
2.3.3 Maximal Network Lifetime
Every node consumes energy while transmitting and receiving data, limiting the life of
each node. The lifetime of each node is defined as the expected time for the energy of this
node to be exhausted. We define the lifetime T of the system as the time that the first
sensor is drained of its energy. Alternatively, the system lifetime of a sensor network is
the minimum lifetime of all nodes in the network. An interesting problem is how to
maximize the network lifetime for WSN with limited energy, and has been the subject of
a number of works; see Sankar et al. (2004), Madan and Lall (2004), Hua and Yum
(2005), Sankar and Liu (2004), Zhang and Hou (2005), and Giridhar and Kumar (2005).
2.3.4 Localization
Sensor’s location estimation plays an important role in many applications of sensor
networks, see Varshney (1996). In general, two main categories of localization techniques
are used in sensor networks localization: range-based and range-free schemes.
The range-based schemes includes the protocol based on Time of Arrival (TOA)
proposed by Wellenhoff et al. (1997), Time Difference of Arrive (TDOA) proposed by
Harter et al. (1999) and Savvides et al. (2001), and Direction of Arrival (DOA) proposed
by Nicelescu and Nath (2003) that calculates the relative angle between two nodes and
further is used to calculate the distance. It also includes a localization scheme based on
connectivity constraints and relative signal angles between neighbors see Doherty et al.
13
(2001). DOA method and TDOA are used to estimate the acoustic source location.
Svaizer et al. (1997), Sheng et al. (2005), Rabbat et al. (2004) and Liu et al. (2005)
describe that received signal energy is used to estimate the source location. The ML
(Maximum likelihood) estimation algorithm and the EM (Expectation Maximization)
algorithm are proposed by Svaizer et al. (1997) and Liu et al. (2005) to estimate source
location. Since the transmission signal from the target is attenuated with the square
distances, the noise greatly influences the sensors far apart from the targets. The signals
thus do not help a lot to improve the estimation accuracy, see Rabbat et al. (2004).
The range-free schemes do not require the measurement of physical distance-related
properties and are used for the location applications with less accurate requirements, see
Bulusu et al. (2000), He et al. (2003), Nagpal et al. (2003), and Niculescu and Nath (2003
b). For example, one uses the centroid of all locations in the received beacon signals to
calculate a sensor’s location described by Bulusu et al. (2000). One uses the minimum
hop count and the average hop size to estimate the distance between nodes and then
determine sensor nodes’ locations, see Niculescu and Nath (2003b). However none of
these schemes will work properly under hostile conditions as most existing localization
methods are vulnerable to malicious attacks.
Few methods have been proposed for secure distance and location verification. Liu et al.
(2005) have proposed an attack-resistant Minimum Mean Square Error-based location
estimation and a voting-based location estimation methods to cope with attack in location
scenarios. Li et al. (2005) have presented a Least Median of Squares position estimator.
14
Both methods apply minimum square error-based methods to realize a secure localization.
Capkun and Hubaux (2004) have proposed a method for secure positioning in sensor
networks based on Verifiable Multilateration in the presence of adversaries.
Sastry et al. (2003) have pointed out a novel distance bounding protocol based on
ultrasound and radio wireless technologies. Both proposals pay more attention to distance
verification and neighbor authorization. Biswas and Ye (2004) discuss one Semidefinite
Programming (SDP) relaxation based method for the position estimation problem and the
optimization problem is set up so as to minimize the error in sensor positions to fit
distance measures, and it show that very few anchor nodes are required to accurately
estimate the position of all the unknown nodes in a network.
2.4 Robust Optimization and Its Use in WSN Models

Sensor Networks are deployed in different type of regions. And it is reasonable to assume
that there is uncertainty in distance information for the quality of radio channel in real
WSNs. This uncertainty in distance is important because sensors transmitting data
consume an amount of energy proportional to approximately the square of the distance.
We present robust models under distance uncertainty for these problems in energy limited
WSNs, see Biswas and Ye (2004), Patwari et al.(2003), and Whitehouse et al. (2005).
Our work involves the use of robust optimization methods to problems in the application
of WSNs. Robust convex optimization problems for different uncertainty sets of data and
the robust methodology that we follow have been discussed: Ben-Tal and Nemirovski
(1998) for convex optimization, Ben-Tal et al (2000) for semidefinite programing,
15
Bertsimas and Sim (2003) for integer and network flow optimization. In a number of
applications this methodology has shown that it can provide solutions which exhibit an
important improvement in the worst case performance at a small additional cost for the
expected case, see Goldfarb and Iyengar (2003) and Bertismas and Sim (2004).

The robust solution for an optimization problem under uncertainty, as defined by Ben-Tal
and Nemirovski (1998), is the solution that has the best objective value in its worst case
uncertainty scenario. Attractive features of a robust solution are that while it is only close
to optimal for any specific scenario, it behaves well over all likely uncertainty outcomes.
To introduce the robust optimization methodology, we consider the following
optimization problem under uncertainty:
min ( , )
s.t. (, ) 0
fxu
gxu ≤

where the uncertainty parameter u belongs to a closed bounded and convex uncertainty
set u . The robust solution is obtained by solving the following Robust Counterpart
problem (RC):
∈ ∪
min max ( , )
s.t. ( , ) 0,for all
x u
fxu
gxu u ≤ ∈ ∪
（2.1）
or
,
min
s.t. ( , ) 0, for all
(, ) , for all
x
gxu u
fxu u
γ
γ
γ

≤ ∈
≤ ∈
∪
∪
（2.2）
16
Note that the constraints in (2.1) and (2.2) need to be satisfied for every u , thus for
continuous uncertainty sets these robust counterpart problems consider infinitely many
constraints. For many problems finding the robust solution is no harder than solving the
deterministic problem. The complexity of solving problem (RC) has been shown to be the
same as the complexity of solving the deterministic problem (fixed u ) for various
problems and uncertainty sets . For example, the robust counterpart of an LP is
equivalent to an LP when ∪ is a polyhedron and to a quadratically constrained convex
program when ∪ is a bounded ellipsoidal set; in addition, the size of the resulting RC
problem is bounded by a polynomial of the deterministic problem’s dimensions. For
example, in the case where the objective function is linear in u, say
∈ ∪
∈ ∪
∪
( , ) ( )
T
f xu c x u = , and
the uncertainty set u is a polyhedron, say
{| } UuMu q = ≤ ,
then the constraint
(, ) , fxu u U γ ≤ ∀ ∈
is equivalent to
max ( )
T
cx u
Mu q
γ ≤
≤

This bound on the maximization linear programming problem is equivalent to
guaranteeing the existence of a dual solution whose objective function value satisfies the
bound.
In other words, we can represent these infinitely many constraints with the system
17
,(),
TT
qy M y c x y γ 0 ≤ = ≥
Repeating this procedure for each of the robust constraints shows that the robust
counterpart of a linear programming problem with polyhedral uncertainty sets is an LP
that is of size polynomial in the size of the original deterministic problem.
18

Chapter Three: Routing Algorithms in Energy Limited WSNs
Since communication is one of the most expensive operations for a sensor node, an
efficient algorithm to route the data gathered is crucial to using the limited energy
efficiently. For instance, if every node transmits its data directly to the sink node, nodes
with little data to send will be left with unused energy while data can be stranded in a
node that has depleted its energy. In addition, since the limited energy in effect limits the
life of the network, the WSN can not spend much time coordinating a routing policy and
must reach an efficient data gathering mechanism rapidly. If not it would spend a
significant percentage of its lifetime operating inefficiently. Each wireless node is
assumed to have the capability to relay packets and be able to dynamically adjust its
transmission energy depending on the distance over which it transmits a bit of data.
3.1 Problem Definition
We consider a WSN with n fixed sensor nodes that gather data to be sent to a sink node,
denoted as the node . Let be the total amount of the data (bytes) collected by
the sensor i , and
1 n +
max
i
D
max
i
E be the total energy of the node i . Let be the Euclidean
distance between nodes i and . We denote by N the set of sensor nodes, and A the set of
directed arcs ( , ), in the complete graph
ij
d
j
i j ,{ iN j N 1} n ∈ ∈+ ∪ .
Sadagopan et al (2004), Lindsey et al. (2001), Intagagonwiwat et al. (2000) show that the
energy consumed in transmitting data from one sensor to an other depends on the
distance between them according to the following radio model: We consider that a radio
19
dissipates
elec
ε =400nJ/byte to run the transmitter or receiver circuitry and
amp
ε =800pJ/byte/m
2
for the transmitter amplifier.

Figure 3.1: First Order Radio Model
The equations used to calculate transmission energy costs and receiving energy costs for
a k-byte message and the distance (As is shown in Figure 3.1) are d
2
Transmitting: (,) * **
Receiving : ( ) *
Tx elec amp
Rx elec
E kd k k d
Ek k
εε
ε
=+
=

It is also assumed that the radio channel is symmetric so that the energy required to
transmit a byte of information from the node i to the node j is the same as energy
required to transmit a byte of information from the node j to the node i for a given
signal to noise ratio (SNR).
K-byte
d
Transmit
Electroni
Transmit
Electronics
Tx
Amplifier
E
Tx
(d)
ε
elec
*K ε
amp
E
Rx

K-byte
Receive Electronics
ε
elec
*K
20

3.2 Mathematical Programming Model

As is shown in Figure 3.2, the problem of maximizing data output given limited energy at
the nodes and the above energy expenditure in transmissions and receptions can be
written as a linear programming problem, see Sadagopan et al. (2004), Braginskey and
Estrin (2002).
Let
ij
f be the amount of the data transmitted from the node i to the node j , we formulate
the maximal data extraction problem as follows:
2
4
7
8
6
Sink
9
3
1
10
5
E
10
, D
10
f
10, n+1
f
4,n+1
f
54
f
9,n+1
Figure 3.2: A Simple WSN Topology (n=10)
f
24
f
13
f
67
f
79
f
39
f
34
f
89
E
4
, D
4
E
9
, D
9
E
5
, D
5
E
8
, D
8
E
7
, D
7
E
2
, D
2
E
3
, D
3
E
6
, D
6
E
1
D
1
21
{, 1|( , ) }
2
{|( , ) } {|( , ) }
max
{|( , ) } {|( , ) }
{|( , ) } {|( , ) }
max
s.t.
(1 )
ij
ij n i j A
i
ij ij ji
ji j A j ji A
i
ij ji
ji j A j ji A
ij ji
ji j A j ji A
f
f dfEiN
f fD i N
ff
β
=+ ∈
∈∈
∈∈
∈∈
+ + ≤ ∀ ∈
− ≤ ∀ ∈
− ≥
∑
∑∑
∑∑
∑∑
0
0(,)
ij
iN
fijA
∀ ∈
≥ ∀ ∈

(3.1)
(3.2)
(3.3)
(3.4)
Where

amp
max
,
i
i
elec elec
E
E
ε
β
ε ε
==

The constraints in this problem include:
(1) Energy constraint (3.1): the amount of the data transmitted and received by a sensor is
limited by the energy of the sensor,
(2) Conservation of flow constraints (3.2) and (3.3): the difference of the amount of data
transmitted and received must be less than or equal to the amount of the data collected by
the sensor itself, and also must be greater than or equal to 0.
(3) Non-negativity of flow constraints (3.4).

To simplify notation, we have normalized the energy in terms of receptions, that is to say,
each reception consumes a unit of energy, while each transmission from the node i to the
22
node j consumes . The amount of data that can be received and transmitted by
a node is limited by the energy of the node. In addition each node considers a maximal
amount of data to be transmitted. If there are no energy limit constraints (3.1), then
we can extract all data available in the network, and the objective value of
2
(1 )
ij
d β +
max
i
D
{, 1|( , ) }
max
ij
ij n i j A
f
=+ ∈
∑
is equal to
max
i
D
∑

3.3 Methodology

Our method is approximately divided into two steps:

(1) We construct the Lagrangian dual problem D: by relaxing some
constraints of the primal problem.
0
min ( )
p
Dp
≥
(2) Then we solve the dual problem to obtain the optimal objective function value using
the sub-gradient projection method, and in the process provide a routing solution that
achieves this value,
3.3.1 Partial Lagrangian Relaxation

We now consider the problem obtained by the Lagrangian relaxation of the energy
constraints. This is obtained by including these constraints in the objective, with a
multiplier, or price
i
p . This leads to the problem:
23

2
{
22
{, 1|( , } { , 1|( , ) }
{} } , 1|( , ) |( , ) {|( , ) }
L( , )
=((1)
(1) ( )
i
ii i i j i
ij ij ij ij
ij n i j A ij n i j A i N
i
ij j
i
ji
iN ij n i j A j ji A
ij
ji j A
fP
fp d f E)
i
f pd p f pd p p p E
f β
ββ
=+ ∈ ≠ + ∈ ∈
∈ =+ ∈ ∈ ∈
−++ −
= − −+ + − −− +
∑∑ ∑ ∑
∑∑ ∑

max
{|( , ) } {|( ,) }
2
and use X denote the set of feasible flows below, that is:
0
0, (,)
1
ij
i
ij ji
ji j A j ji A
i
ij ij ij
ij
fX
ffD iN
X
E
fbb ij A
d β
∈∈
∈
⎧
≤ − ≤ ∀ ∈
⎪
⎪
=
⎨
⎪
≤ ≤ = ∀ ∈
⎪
+
⎩
∑∑

Notice that the first term in L( is separable for variables f,P)
ij
f , hence the objective
function of the dual problem is:
2
{( , ) }
D( )= maxL( , ) ( ) ( ),
where ( ) max[ ( )]
ij
ii
iN
ii j
ij ij
ij A
ij
fX
fX
pfpBp pE
Bpfpdp β
∈
∈
∈
∈
=+
=−−p−
∑
∑

(3.5)
and we define .
1
1
n
p
+
=−
(3.6) The dual problem is the .
0
:min ( )
p
DD
≥
p
p
Since the original problem is a linear program (LP), then its Lagrangian dual,
, is also an LP. Also, since both are feasible, the primal and dual attain
the same finite optimal objective function value, see [Bertsekas (1999)]. Therefore, we
solve the dual problem to obtain the optimal objective function value, and in the process
provide a routing solution that achieves this value.
0
:min ( )
p
D D
≥
24

3.3.2 Sub-Gradient Projection Method

The sub-gradient optimization method can be used to solve . We now
describe in general the sub-gradient projection method, see [Low and Lapsey(1999), and
Bertsekas (1999)]. It is an iterative method with iterations defined by
, and [ , where
0
min ( )
p
Dp
≥
(1) [ () ()]
t
pt pt g t α
+
+= − ] max{ ,0} z
+
= z ( 1) pt + is price of the t +1
iteration, is price of the t iteration, g(t) is the sub-gradient of at , and ( ) pt ( ) Dp ( ) pt
0
t
α > is the step-size of t iteration. Thus at each iteration of the sub-gradient method,
we take a step in the direction of a negative sub-gradient. A sub-gradient of at
is defined as any vector g that satisfies the inequality
( ) Dp ( ) pt
( ()) () (())
T
g p pt Dp Dpt −≤ − for
any p .

For a given price of iteration t, we denote () pt
*
()
ij
f t , the solution at t iteration that
maximizes the Lagrangian ((), ())
ij
L ft t p over ()
ij
ftX ∈ .
*
p is the optimal solution of the
price that minimizes dual problem . ( ) Dp

Proposition 3.1: For any sub-gradient g(t), we have
25
(1) [() ()]
ii
t
i
tppt g t α
+
+= − , for all step size such that
*
(()) ( )
0
(())^2
t i
i
Dpt D p
gt
α
−
<=

∑
, then
sub-gradient projection method for the dual problem converges and its
price converges to
0
:min ( )
p
D D
≥
p
( ) pt
*
p
Proof 3.1: Refer to Bertsekas (1999), pp.610-612 and Exercise 6.3.1 in pp.629.

Proposition 3.2:
The sub-gradients of at are given by (()) Dpt ( ) pt

*2 *
{|( , ) } {|( ,) }
(())
()( ()(1 ) ()
i
iij ij ji
ji j A j ji A
Dpt
ftd ftE
p
β
∈∈
∂
=− + + − ) ∑∑
∂
. (3.7)
Proof of Proposition 3.2:

*2 *
{} {}
*
|( , ) |( , )
()
let () ( ()(1 ) () ),
() ( ( ), ())
We know
argmax
ji
ii
ji j A j ji A
ij ij
ft
gt f t d f t E
ft Lft pt
β
∈∈
=− + + −
=
∑∑
}
*
{
*2
*2
**
{
*
{|( ,) {|( , ) }
{|( , ) } |( , ) }
,1|(,)} {|(,)}
()
(())
() )
()
()( ()(1
()
[ ( ) ( )( ()(1
()
)
(1 ) ( )
T
i i
ji
jji A iN j i j A
ii
ji
iN j i j A jji A
ij
iN
ij ij
ij ij
i
ij
ij n ijA jijA
f
gppt
tE
pt d f E
pft d
t
ft p f t
t
f
t β
β
∈ ∈∈
∈∈ ∈
∈ =+ ∈ ∈
+
− −
+−
=++
=− +
−
+
∑∑
∑
∑
∑∑
∑∑ ∑
*2
{} {}
*
2
*
{ , 1|( , ) } |( , ) |( , )
*
{|( ,) }
()( )]
)()
[() ()(1) ()
ij ij
i
i
iN
ij
i
ij
ij n i j A j i j A j ji A
ji
jji A
ji
fptf d f E
dft
tt t β
β
∈ =+ ∈ ∈ ∈
∈
−+
)]E
−
+ −
−+
∑∑ ∑ ∑
∑

Given p let
f
ij
=
argmax , ) (
ij
p L f . From the definition of ( ) Dp , we have
26
2
{ {|(, ) } ,1|(,)} {|(,)}
**2
{, 1|(, ) } { |(, ) } { |( , ) }
() ()( )]
(1 ) [ ( ) ( )( () () )]
(1 )
i
i
i
iN ji j A
ij jij ji
ij n i j A j ji A
ii
ij ij ij ji
ij n ijA iN jijA j ji A
Dp f p f d f E
*
ftp ft d ftE
β
β
∈ ∈ =+ ∈ ∈
=+ ∈ ∈ ∈ ∈
=− + −
+ ≥− +
+
∑∑ ∑ ∑
∑∑ ∑ ∑
−

Replacing this inequality in the previous equation and substituting the definition
of (()) D pt yields ( ()) () (()) ()
T
gppt Dp Dpt t ≤− − ,
which proves that is the sub-
gradient of the
*2 *
{} {} |( , ) |( , )
() ( ( )(1 ) () )
ji
ii
ji j A j ji A
ij ij
gt f t d f t E β
∈∈
=− + + −
∑ ∑
(()) D pt at . ( ) pt

So, according to proposition 3.1 and 3.2, we can compute the iteration price
*2 *
{|( , ) } {|( , ) }
(1) [() ()]
[() ( (1 ) )]
ii
tij ij ji
ji j A j ji A
ii i
t
tppt gt
pt f d f E
α
αβ
+
+
∈∈
+ = −
= + + + −
∑∑

and its price will converge to () pt
*
p .
Although sub-gradient type algorithms converge to the optimal solution, this convergence
can be very slow, see [Bertsekas (1999)]. We are interested in studying whether, for the
problem in question, this convergence is sufficiently fast in the first iterations so that the
routing solutions are already reasonably close to the optimal to be considered efficient
routing heuristics.

3.4 Centralized Algorithms
We now describe the implementation detail for the sub-gradient projection method for the
maximal data extraction problem. The following implementation uses a routing solution
27
for the problem without energy constraints and decides a unique step length at each
iterate. Hence these are centralized algorithms. Although these are not realistic
algorithms from an implementation point of view, we seek to determine whether these
algorithms can achieve fast enough convergence to an efficient solution as to justify
pursuing this approach in a distributed setting.
Our first centralized algorithm from Lagrangian relaxation of energy constraints uses the
optimal value of the dual problem to calculate the step-size. It is therefore a purely
theoretical algorithm which will be used for comparison purposes. The algorithm
considers a fixed integer value m to control the rate of decrease of the diminishing step
size, an iteration limit ITLIM, and an optimal tolerance value of TOL.
*
D(p )

2
{( , ) }
Centralized Algorithm-1
1: At =0, set ( ) 0, , or other initial values.
2: While t ITLIM and |D( ( ))-D( ( ))|>TOL do
3: Solve problem B( ( ))= max[
Algorithm 3.1:
(
i
i
ij ij
fX
ij A
tpt iN
pt pt
pt f p d β
∈
∈
=∀∈
≤
−− ∑
*
*
2
*2 *
{|( , ) }
)],
let ( ) be the optimal solution.
4: Set D( ( ))=B( ( )) + ( )
(() ( ))
5: Set
(())
6: Set ( ) ( (1 )
ij
ii
iN
t i
i
i
ij ij ji
ji j A
pp
ft
pt p t p t E
mDpt Dp
mt g t
gt f d f
α
β
∈
∈
−
∑
−
=
+ ∑
=− + + ∑
{|( , ) }
)
7: Compute a new price: ( 1) [ ( ) ( )]
8: Set 1
i
jji A
ii i
t
E
pt pt g t
tt
α
∈
+
− ∑
+= −
= +

28
In the Algorithm 3.1, the formula
m
mt +
, satisfies the usual condition for the diminishing
the step size
t=0
0, and , see [Bertsekas (1999)]
t
mm
mt mt
∞
→∞
→=∞
++
∑

Our next two centralized methods consider variations of the Algorithm 3.1, which use a
lower bound obtained from feasible flow instead of the optimal value .
*
() Dp

A. lower bound 1: _ LB δ
We update algorithm 3.1 simply by using a lower bound _ LB δ instead of the optimal
solution value of when computing the step size.
*
( ) Dp
The t-th iteration lower bound _ () LBt δ is obtained by scaling down the flow
computed at each iteration to obtain a feasible flow. This guarantees that it provides a
lower bound, where
( ) ft
( ) _ LB t δ is the value of the flow at each iteration, and is given by
*
{, 1|( , ) }
*2
{|( , ) } {|( , ) }
i
{g ( ) 0, }
()_()
where =
min
(1 )
ij
ij n i j A
i
ij ij ji
ji j A j ji A
ti
LB t t
E
*
f df
f δ
δ
β
δ
=+ ∈
∈∈
>∀
=
++
∑
∑∑

The best lower bound is denoted by
0,1,2,...
__
max t
LB LB t() δ δ
=
=

29

B. Lower bound2: OneHop_LB
At the beginning of the centralized algorithm, we also can use direct communication (see
Figure 3.3) from each node to sink node to get the lower bound
max 2
()
_min{ ,
1
i
i
ij
Ei
DT LB D
d β
= ∑
+
}

to instead of Dp . But we can get better lower bound Onehop_LB (As is shown in
Figure 3.4) than to replace to compute a new price.
*
( )
_ DT LB
*
( ) Dp
In Figure 3.4, we consider some nodes that cannot send all their collected data if they
send the data directly to the sink node since they have limited energy and larger distance.
Let these nodes be set K and let nodes that have sent all their information and have
residual energy be the set L. The Onehop_LB initial solutions allows a node k K ∈ to
send part of its information to the sink through some node l L ∈ , if it is beneficial to the
network.

Figure 3.3: Direct Transmit Figure 3.4: One Hop Transmit
30
We get the improved lower bound Onehop_LB using the Procedure 3.4.1 below
Note that the lower bound2: Onehop_LB is fixed, independent of the current iterate of
the sub-gradient algorithm and can be determined a priori; whereas lower bound1
_ LB δ is updated from the current optimal solution
*
ij
f (t).

max 2
2
max max
1: Initially, set LB=0
2:WHILE , and Do
1
3: (1 ) , ( ) 0,
4 : WHILE ( ) 0 and 0 and Do
5: Find a node close to with ( )>0
6
>
i
i
ij
ii i i
ij
i
E
DiN
d
E E d D D i LB LB D
Di E i N
ji Dj
β
β
∈
+
=− + = = +
=> ∈
22
(1)
2
(1)
2
2
(1)
: let min( , )
21
7 : (2 ) ,
(2 ) ,
( ) ( ) ,
8: WHILE Do
9 : if
1
ii
ij
in ij
ii
in ij
jj
ij ij
ij ij
i
in
EE
f
dd
EE d f
EE d f
Dj Dj f LB LB f
iN
E
D
d
ββ
β
β
β
+
+
+
=
++
=− +
=− +
=− = +
∈
<
+
2
(1)
() then
10: LB= LB +
1
11: else LB=LB+ ( )
12: LB is the Onehop_LB
i
in
i
E
d
Di
β
+
+

Procedure 3.4.1 (Finding Onehop_LB)

31
3.5 Distributed Algorithms

The centralized algorithm assumes we can compute the step-size globally and gets the
optimal flow
ij
f from (3.5) at times t=0, 1, 2, 3… In this section, we extend the
centralized algorithm to a distributed model. In the experimental section below we show
that we maintain in part the quick convergence of the sub-gradient algorithm in this
distributed implementation

The centralized algorithms discussed above are coordinated in two steps: in
determining
*
( ) f t , the optimal solution of (()) B pt , and in setting the step size. The
steps of computing the sub-gradient and the new price can be done separately at each
node.

To solve the
*
ij
f , we have to solve the following problem that has a linear objective
function, which can be separated grouping all outgoing arcs of each node.
2
{( , }
() max[ ( )]
ij
ii j
ij ij
ij A
fX
Bp f pd p p β
∈
∈
=−−−
∑

Hence the only coordination has to do with the flow constraints . Our distributed
algorithm approximately solves this problem by increasing or decreasing the flow at each
arc
f X ∈
f
ij
independently according to the sign of the objective cost coefficient
32
2ij
ij ij
i
vpd p β =− − −p , while maintaining a close to feasible flow with the information
available. We can use the following heuristic way to get an approximate optimal solution:

Condition 3.4.1
1)
2
()
ij
ij
i
if p d p p β−−− >0,
We can increase flow rate
ij
f at the next period, and let
max 2
{|(,)} {|(,)}
min{ max{ ,1}, }
1
i
ij
i
ii
jji A j i j A
j j ji ij
E
ff D f f
d β
∈∈
=+ + −
+
∑∑

(2)
2
(
i j
ij
i
if p d p p β−−− <)0,
We can decrease flow rate
ij
f at the next period, and let
{} {} |( , ) |( , )
max{ max{ ,1 }, 0}
ij ij ij ji
ji j A j ji A
ff f f
∈∈
=− −
∑∑

(3)
2
( )
ii j
ij
if p pd p0,
(3.18)
β − − −=
ij
We let f unchanged at the next period.

To determine the step size in a distributed algorithm, we select a predetermined value at
every iteration
0
*
t
t
aa λ = , (
t
m
mt
λ =
+
, 0
t
λ1 < < ), where and m are fixed parameters.
This rule for selecting step sizes has been shown to lead to convergent sub-gradient
algorithms, see Low and Lapsey (1999), Bertsekas (1999).
0
a

33

Algorithm 3.2: Distributed algorithms

1: At t=0, set t=0, () 0,
i
pt i =∀∈N , or other init values
2: While and |( tITLIM ≤ () ( ( 1))| Dpt D pt TOL − −> do
3: Compute objective coefficients
2i
ij ij
i
vpdp β =− − −
j
p
4: Update f
ij
according to (3.18)
5: Adjust so f X
ij
∈ by modifying f
ij
with v 0
ij
=
6: Set
{( , }
(()) max[ ]
ij
ij ij
ij A
fX
B pt f v
∈
∈
=
∑

7: Set D(p(t))= ( ( )) ( )
ii
iN
B pt p t E
∈
+
∑

8: Set
0 t
m
aa
mt
=
+
9: Set
2
{|( , ) } {|( ,) }
() (1 ) ( ) ()
i
ij ij ji
ji j A j ji A
i
ftd ft gt β
∈∈
E ++− =
∑∑

10: Compute a new price, (1) [ () * ()]
t
ii i
ptpt gt α
+
+=−
11: Set t=t+1

The distributed algorithm above suggests the following protocol which can be
implemented on each sensor node i. Algorithm 3.2 running on each node leads to a
synchronized protocol since it requires that all nodes update flows before broadcasting
new prices.

34
Algorithm 3.3: Synchronous Distributed Protocol at node i:

1: At t=0, set t=0, ( ) 0,
i
pt iN = ∀∈ , for \{ } jN i ∈
2: While 0 do
i
E >
3: Broadcasts price ( )
i
p t and receives ( )
j
p t from node j via arc (i, j).
4: Compute vp
2ij
ij ij
i
dpp β =− − −
5: Computes the new flow rate f
ij
for \{ } jN i ∈ from (3.18)
6: Transmit the ( ) f t
ij
through arc and receives (, ) ij ( ) f t
ji

7: Set
2
{|( , ) } {|( ,) }
() (1 ) ( ) ()
i
ij ij ji
ji j A j ji A
i
ftd ft gt β
∈∈
E ++− =
∑∑

8: Set
0 t
m
aa
mt
=
+

9: Set a new price, (1) [ () * ()]
t
ii i
pt p t g t α
+
+=−
10: Set t=t+1

3.6 Evaluation Criteria and Simulation Results
3.6.1 Evaluation Criteria
We evaluate each scenario by its performance in the speed of convergence, and we
consider the number of iterations to reach the some range close to optimal results such
that
Current value _ ( ( )) Optimal value
100%
Optimal value
Dp t −
×
.
As is shown in Figure 3.5, the simulation consists of 50 nodes randomly deployed in the
0.5km*0.5km area, the sink node located at (0.25km, 0.5km). Each sensor node has
limited power and the ability to transmit data directly to the sink node. We used the
defined radio model before. In order to send data, the sensor has to run its transmitter and
amplifier circuitry. We assume that a reception of a single byte consumes one unit of
35
energy. From the value of
elec
ε , it can be that 0.1J of energy allows 250,000 receptions.
Thus, in our simulations, we set the maximum energy to 250,000. The maximum data is
set to 10,000 (approximately 10k). We also count the number of transmissions and
receptions due to energy consumed in the various experiments.

In each experiment, we use the average value of about 30 random rounds to reduce the
error. We use the AMPL and LOQO 6.03 tools to do some experiments for Algorithm 3.1.

Figure 3.5: Random 50-node topology for a 0.5 km* 0.5 km
network. BS is located at (0.25, 0.5) km
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5
36
3.6.2 Simulation Results

In this section, we present Homogeneous and Heterogeneous Experiments for Centralized
Algorithms, Homogeneous and Heterogeneous Experiments for Distributed Algorithms
(DA), Centralized Algorithms (CA).

(A) Homogeneous Experiment for Centralized Algorithms

All nodes are homogeneous, which means all nodes that have same energy and same data
in one experiment. In my simulation, each node has the same energy = 25,000 and the
same available data = 10,000 (approximately 10k). Note that, we set different start
price to accelerate the converge speed.
i
E
max
i
D
(0)
i
p
M1: Algorithm 3.1 with D(P*)
M2: Algorithm 3.1 with D(P*) instead by Lower bound 1
M3: Algorithm 3.1 with D(P*) instead by Lower bound 2

37

0%
5%
10%
15%
20%
25%
1 5 10 20 30
Iterations
Avg.Rate
M1
M2
M3

Figure 3.6: Homogeneous Experiments for Centralized Algorithms

Figure 3.6 shows these algorithms achieve close to optimal performance quickly
(achieving 10% of optimal in less than 5 iterations in most experiments) for
homogeneous experiments. Lowerbound2 can get the values more close to the optimal
value D(P*), Method M3 performs better than Method M2.

38
(B) Heterogeneous Experiment for Centralized Algorithms
All nodes are heterogeneous with different ratios of
max
i
i
E
D
, which means all nodes that
don’t have same energy and data in one experiment. In our simulation three types of
nodes are considered: high energy nodes with a ratio of 2,500, with = 250,000 and
= 100; medium energy nodes with ratio 2.5 for = 25,000 and = 10,000;
and low energy nodes with ratio 0.5 for = 2,500 and = 5,000.
i
E
max
i
D
i
E
max
i
D
i
E
max
i
D
0%
5%
10%
15%
20%
25%
1 5 10 20 30
Iterations
Avg.Rate
M1
M2
M3

Figure 3.7 Heterogeneous Experiments for Centralized Algorithms

The type of node in a heterogeneous scenario is selected randomly maintaining even
proportions: 17 high energy nodes, 17 medium energy nodes, and 16 low energy nodes. It
39
is clear that high energy nodes have residual energy but no data, while low energy nodes
are exhausted even with substantial data left to transmit to sink node.
Figure 3.7 also shows these algorithms achieve close to optimal performance quickly
(achieving 10% of optimal in less than 10 iterations in most experiments) for
heterogeneous experiments. Lowerbound2 can get the values more close to the optimal
value D(P*), Method M3 performs better than Method M2.

0%
2%
4%
6%
8%
10%
12%
14%
16%
2 3 4 5 10 20 30
Iterations
Rate
CA
DA

Figure 3.8 Homogeneous Experiments for DA and CA Algorithms

(C) Homogeneous Experiment for Distributed Algorithms (DA), and Centralized
Algorithms (CA)
From the first set of experiments we know that the centralized algorithm converges to
within 10% of the optimal value in 10 iterations. Based on the step sizes used in these
40
centralized methods, we set the step size for the distributed method as
7
0
0.5 10 α
−
=× ,
with m = 1. We run both algorithms on each random problem created and average the
results.

As is shown in Figure3.8 for the homogeneous experiments, it was observed that CA and
DA algorithms can converge to within 10% of the optimal in 10 iterations. Since the DA
algorithms use the priori step size
7
0
0.5 10 α
−
=× in this case, DA can converge to
optimal value more quickly than CA in the first 20 iterations, while CA is better after 20
iterations.

(D) Heterogeneous Experiments for DA and CA algorithms

We set initial value for high energy nodes i, (0) 0
i
p = (0) 0.01
j
p = for other nodes j to
do heterogeneous experiments. Figure3.9 shows that for heterogeneous experiments the
behavior is similar. Algorithms CA and DA both converge achieving 10% of the optimal
on average in less than 10 iterations. In addition, DA algorithms use the priori step-size in
this case, DA can converge to optimal value more quickly than CA in the first 10
iterations, while CA is better after 20 iterations.

41

0%
5%
10%
15%
20%
25%
30%
35%
2 3 4 5 10 20 30
Iterations
Rate
CA
DA

Figure 3.9 Heterogeneous Experiments for DA and CA Algorithms
3.7 Conclusions

In this chapter, we formulate the maximal data extraction problem in energy limited
WSNs as a linear programming problem. By using Lagrangian relaxation methodology,
we formulate a related dual problem amenable to a solution via a sub-gradient projection
method and we present both a centralized and a distributed algorithms version. Then, we
show the results of the centralized and distributed algorithms based on Lagrangian
relaxation on the energy constraints through computational experiments that these
algorithms achieve close to optimal performance quickly, achieving 10% of optimal in
42
less than 10 iterations on average. Although convergence of sub-gradient methods can be
slow in theory, for our problem they quickly arrive at an efficient routing heuristic.

The distributed algorithm in particular performs better in the heterogeneous than in the
homogeneous experiments. The reason for this is that in a heterogeneous network there
are a few nodes that are critical in an efficient overall performance. The distributed
lagrangian relaxation method manages to identify these nodes and set correct prices
achieving quickly a good solution. In the homogeneous case it is more difficult to
identify these high value elements of the solution. More experiments should be done to
shed light into how the topology of the network influences the convergence speed of
distribution algorithms.
43

Chapter Four: Robust Models under Distance Uncertainty

Since sensors have limited energy, it becomes important to identify efficient operating
polices. This has lead to research on a number of different problems, such as identifying
the maximum data that can be extracted for a given amount of energy, minimizing the
energy consumed to extract certain amount of data and maximizing network lifetime. For
these problems, distance is an important factor that influences the consumption of energy,
and hence the efficiency with which the network operates. In many applications, the
distance measurements are subject to uncertainty as they might have been indirectly
estimated through signal strength or due to unfriendly conditions during the WSN’s
deployment or operation. The effect of ignoring distance uncertainty in the efficiency of
the operation of a WSN is unclear, however optimized operating practices can turn out to
be inefficient if the problem parameters change. A more successful strategy can be a
solution that is less optimal for a particular distance vector but obtains efficient solutions
for all likely distance measurements.
4.1 Problem Definition

We consider the problems with the same parameters as described in Section 3
4.1.1 Maximum Data Extraction Problem
We use the same model for maximum data extraction problem discussed in Section 3.
44
{, 1|( , ) }
2
{|( , ) } {|( , ) }
max
{|( , ) } {|( , ) }
{|( , ) } {|( , ) }
max
s.t.
(1 )
0
0(,)
ij
ij n i j A
i
ij ij ji
ji j A j ji A
i
ij ji
ji j A j ji A
ij ji
ji j A j ji A
ij
f
f dfE
ffD iN
ff iN
fijA
β
=+ ∈
∈∈
∈∈
∈∈
++ ≤ ∀∈
− ≤ ∀ ∈
− ≥ ∀ ∈
≥ ∀ ∈
∑
∑∑
∑∑
∑∑
iN

Where
amp
max
,
i
i
elec elec
E
E
ε
β
ε ε
==
In this problem, the first constraint is the energy constraint. The amount of data
transmitted and received by a sensor is limited by the energy of the sensor. The second
and third constraints represent conservation of flow constraints. The difference of the
amount of data transmitted and received at each node must be less than or equal to the
amount of the data collected at that node, and also must be greater than or equal to 0. The
last constraint states the non-negativity of flow.
4.1.2 Minimum Energy Consumption Problem
Minimizing energy consumption problem is closely related to the problem of maximizing
data extraction in energy limited WSNs. The Minimum Energy Consumption Problem
can be written as a linear programming problem. The objective of the problem is to
45
minimize the amount of energy consumed to extract
min
f information to the sink node.
The
min
f could represent a requirement to extract a certain fraction of the total
information available, where
max
i
i
D
∑
is the maximum amount of the information at the
node i.
Our minimum energy consumption model is given by the linear program below: i
2
{|( , ) } { |( , ) }
min max
{|( , ) }
max
{|( , ) } {|( , ) }
{|( , ) } {|( , ) }
min ( (1 ) )
s.t.
0
0
ij ij ji
ijijA jjiA
i
ij
ji j A i
i
ij ji
ji j A j ji A
ij ji
ji j A j ji A
ij
fd f
ff p D i N
ffD iN
f fi
f
β
∈∈
∈
∈∈
∈∈
++
≥= ∀∈
− ≤ ∀ ∈
− ≥ ∀ ∈
≥
∑∑ ∑
∑∑
∑∑
∑∑
(, ) ij A ∀ ∈
N

Where
amp
max
,
i
i
elec elec
E
E
ε
β
ε ε
==
and is the percentage of the total information that reaches the sink node. p
The information requirement can enforce, for example, that a given percentage p of the
available information must reach the sink. In this case we set
min max
i
i
f pD =
∑
.
46
In this problem, the first constraint references the minimum data extraction constraint: we
should guarantee that at least
min
f information is extracted to the sink node. The
polyhedral constraint represents the conservation of flow and non-negativity of flow
constraints, similar to the constraints in the maximum data extraction problem.

4.1.3 Maximum Lifetime Problem

Sankar and Liu (2004), Madan and Lall(2004), Hua and Yum (2005) have published
papers to discuss the problem of maximizing the life time of WSN. In these papers, the
lifetime of node i is defined as the expected time for the energy to be exhausted,
because each node i has a limited energy . We define the lifetime T of the system to
time of the first sensor i is drained of its energy, that is to say, the system lifetime T of a
sensor network as the minimum lifetime of all nodes of the network, T = min
{ }.
i
T
i
E
i
E
12
, ,...,
N
TT T
Then Maximum Lifetime Problem is described as the following optimization problem:

47
min max
{|( , ) }
max
{|( , ) } {|( , ) }
{|( , ) } {|( , ) }
{|( , ) }
max
s.t.
0
((1
i
ij
ji j A i
i
ij ji
ji j A j ji A
ij ji
ji j A j ji A
ij
ji j A
T
f fp D iN
f fD i N
f fi
Tf β
∈
∈∈
∈∈
∈
≥ = , ∀ ∈
− ≤ , ∀ ∈
− ≥ , ∀ ∈
+
∑∑
∑∑
∑∑
∑
2
{|( , ) }
)),
0(,)
i
ij ji
jji A
ij
dfE
fijA
∈
+≤∀
≥ ∀ ∈
∑
N
iN∈

Where
amp
max
,
i
i
elec elec
E
E
ε
β
ε ε
==
The constraints include the minimum data extraction constraint, energy constraints, and
the same polyhedral constraints representing conservation of flow and non-negativity.
Obviously, the above program is not linear because of the product terms . It is
natural only to consider for all
ij
T f ×
0
i
E > i N ∈ , otherwise simply remove the node from the
network. This implies that and we can obtain an equivalent linear program using a
new variable
0 T >
1
q
T
= . The objective function becomes to minimize q , and the non-linear
constraint can be equivalently written in linear constraint, yielding the linear program:
48

min max
{|( , ) }
max
{|( , ) } {|( , ) }
{|( , ) } {|( , ) }
2
{|(,)} {|(,)}
min
s.t.
0
(1 ) ,
i
ij
ji j A i
i
ij ji
ji j A j ji A
ij ji
ji j A j ji A
i
ij ij ji
ji j A j ji A
ij
q
ff p D i N
ffD iN
ff iN
f dfqE
f
β
∈
∈∈
∈∈
∈∈
≥ = ∀ ∈
− ≤ ∀ ∈
− ≥ ∀ ∈
+ + ≤ ∀ ∈

∑∑
∑∑
∑∑
∑∑
0(,) ij A ≥ ∀ ∈
iN

4.2 Methodology and Uncertainty Sets

We address the uncertainty in problems through the robust optimization methodology.
4.2.1 Uncertainty Sets
Distance measurements among nodes in sensor networks can suffer from uncertainty due
to a variety of reasons: distance measurement methods, unfriendly conditions, or
existence of arbitrary noise and rugged land surface. Here, we consider the uncertainty in
the vector of distances between nodes by allowing it to belong to an
uncertainty set d . The set U defines distance vectors that are a given deviation from
a given estimate of the distance vector between nodes .
()
()
ij ij A
dd
∈
=
U ∈
0
d

We notice that we can replace the distance uncertainty in our problem with alternate
49
models of how uncertainty affects the energy consumption. For example, we can consider
identity uncertainty in communication costs due to the uncertainty in the radio channel.
The uncertainty models are linear in terms of the term that appears in the energy equation
(in our case the objective function). We pay attention that we now present uncertainty
sets which are polyhedral sets for the estimated distance vectors .
0
d
12 02
{| ( ) | , 0}
ij ij
Udd d uMuqu ≡= + ≤ ≥
≤
≥
(4.1)
Or
22 02
{| ( ) | 1 }
T
ij ij
Udd d uuQu ≡= + (4.2)
In the definition, M is matrix for uncertaintyU , Q is described as positive definite

An example for this polyhedral uncertainty set defined by equation (4.1) is an
uncertainty set U in which each distance measurement can be larger than estimated
distance and the overall squared deviation is bounded. The idea is that the errors in
distances may vary independently of each other, however we omit unrealistic and overly
conservative situations in which every distance value increases to its upper bound.
1
U
3202
{|( ) ( ) , , 0}
ij ij ij ij ij
ij
Udd d R M ξξτξ ≡= + ≤
∑∑
. (4.3)
Or
4202
{| ( ) ( ) , , 0}
ij ij ij ij i ij
j
Udd d R M ξξτ ξ ≡ = + ≤ ≥
∑
(4.4)
These sets consider that the square of every distance measurement can vary up-wards by
at most R . To exclude overly conservative distance measurements, the sets limit either
50
the total variation by τ M or the variation out of every node by τ Mi, where τ ∈ [0, 1] is
a parameter that controls the size of the uncertainty set.

4.2.2 Robust Counterpart Problems
In this section we formulate robust counterpart problems, following the robust
optimization methodology, for the different problems of interest in WSNs data-centric
networks: maximum data extraction for a given amount of energy, minimum energy
consumption for a given amount of data transfer, and maximum network lifetime for a
given data transfer. We present three propositions in these WSNs problems and explain
how to express the robust counterpart problem of each problem. We note that the robust
counterpart problem in each case is of the same form as the original problem, therefore
the complexity of solving for the robust solution is the same as the complexity of solving
a deterministic problem.

A. Maximum Data Extraction
Proposition 4.1: Consider the maximum data extraction problem for a given amount of
energy with uncertainty distance, where given by equation (4.4). The robust
counterpart of this problem is equivalent to
4
d U ∈

51
{, 1|( , ) }
02
{|( , ) } {|( , ) }
max
{|( , ) } {|( , ) }
{|( , ) } {|( , ) }
max
s.t.
[1 ( ) ]
0
(,
ij
ij n i j A
i
ij ij i i ji
ji j A j ji A
i
ij ji
ji j A j ji A
ij ji
ji j A j ji A
iij
f
f dM fE i
ffD iN
ff iN
fR ij
βθτ
θβ
=+ ∈
∈∈
∈∈
∈∈
+ + + ≤ ∀ ∈
− ≤ ∀ ∈
− ≥ ∀ ∈
≥ ∀
∑
∑∑
∑∑
∑∑
)
0
0(,)
i
ij
A
iN
fijA
θ
∈
≥ ∀ ∈
≥ ∀ ∈
N

Proof: The robust counterpart of the maximum data extraction is the problem of
maximizing
{, 1|( , ) }
ij
ij n i j A
f
=+ ∈
∑
under the robust energy constraints due to the distance
uncertainty
and flow
02
{|( , ) } {|( , ) }
[1(() )] ,,0,
i
ij ij ij ji ij i ij
ji j A j ji A j
fd R fE MiN βξ ξτ ξ
∈∈
+ + + ≤ , ∀ ≤ ∈ ≥
∑∑∑
ij
f belong to the conservation of flow and non-negativity of flow constraints.
Then, from strong LP duality we have that for every iN ∈
{|( , ) }
0
max
ij i
j
ij
ij ij
ji j A
M
f R
ξτ
ξ
βξ
∈
≤
≥
∑
∑
has the same optimal value as
..
0
min
iij
i
i
st
fR
i
M
θβ
θ
θ τ
≥
≥
. That is to say,
52

02
{|( , ) } {|( , ) }
02
..
{|( , ) } {|( , ) }
0
,0
arg max { [1 (( ) )] } ,
min [1 ( ) ]
iij
i
ij i ij
j
i
ij ij ij ji
ji j A j ji A
ij ij i i ji
st
ji j A j ji A
fR
M
f dR f Ei
fd M fE
θβ
θ
ξτ ξ
βξ
βθτ
∈∈
∈∈
≥
≥
≤ ≥
++ + ≤∀
⇓
++ + ≤
∑
∑∑
∑∑
02
{|( , ) } {|( , ) }
, and 0, we have
[1 ( ) ]

i
ii ij i
i
ij ij i i ji
ji j A j ji A
iN
fR
fd M fE iN
θθ β θ
βθτ
∈∈
∀ ∈
∃≥ ≥
+ + + ≤ ∀ ∈

∑∑
N∈

Considering the conservation of flow and non-negativity of flow constraints, we prove
the Proposition 4.1.
Proposition 4.1 shows that the (RC) problem of a maximum data extraction problem for a
given amount of energy with distance uncertainty given by set (4.4) is similar to the
deterministic problem without uncertainty. Different from the deterministic problem, the
(RC) has n additional non-negative variables
4
U
i
θ , |A| + n additional constraints
and 0
iij i
fR θ β θ ≥ ≥ , and the energy is reduced to ,for
i
ii
E M i θτ N − ∈ . But this (RC)
problem is still a linear programming problem, which also implies that solving the (RC)
is just as difficult as solving the deterministic problem.

B. Minimum Energy Consumption
Lemma 4.1 Consider the Minimum Energy Consumption Problem for a given amount of
data transfer with uncertainty distance, where
1
d U ∈ given by equation (4.1). The robust
counterpart of this problem is equivalent to

53
02
{|( , ) } {|( , ) }
(1) min max
{|( , 1) }
max
{|( , ) } {|( , ) }
{|( , ) }
min (1 ( ) )
s.t.
(, )
0
T
ij ij ji
ijij A jji A
i
in
ii n A i
T
ij
i
ij ji
ji j A j ji A
ij j
ji j A
fdf
ff pD iN
Mf ij A
ffD iN
ff
q β θ
θβ
θ
∈∈
+
+∈
∈∈
∈
++ +
≥= ∀∈
≥ ∀ ∈
≥
− ≤ ∀ ∈
−
∑∑ ∑
∑∑
∑∑
∑
{|( , ) }
0
0(,)
i
jji A
ij
iN
fijA
∈
≥ ∀ ∈
≥ ∀ ∈
∑

Proof：The (RC) problem of the minimum energy consumption model is:
02
{|( , ) } {|( , ) }
0
min ( (1 ( ) ) min(max )
ij ij ji ij ij
Mu q
iN j i j A j ji A i j
u
fdf f ββ
≤
∈∈ ∈
≥
++ )+
∑∑ ∑ ∑∑
u
with the deterministic problem constraints. Then, from LP duality, we have that the dual
of
0
max
ij ij
Mu q
ij
u
f u β
≤
≥
∑∑

is equivalent to
0
min
T
ij
T
Mf
q
θβ
θ
θ
≥
≥

Therefore the objective function becomes
02
{|( , ) } { |( , ) }
0
min ( (1 ( ) ) min
T
ij
T
ij ij ji
Mf
iN j i j A j ji A
fdf
θβ
θ
q β θ
≥
∈∈ ∈
≥
++ )+
∑∑ ∑

Considering the following deterministic problem constraints,
54

f ∈
(1) min max
{|( , 1) }
max
{|( , ) } {|( , ) }
{|( , ) } {|( , ) }
0
0
i
in
iin A i
i
ij ji
ji j A j ji A
ij ji
ji j A j ji A
ij
f fp D iN
f fD i N
f fi
f
+
+∈
∈∈
∈∈
⎧
≥= ∀∈
⎪
⎪
⎪
− ≤ ∀ ∈
⎪
⎪
⎨
⎪
⎪ − ≥ ∀ ∈
⎪
⎪
≥
⎪
⎩
∑∑
∑∑
∑∑
N

we prove lemma 4.1.

Lemma 4.2 Consider the Minimum Energy Consumption Problem with uncertainty
distance, where given by the equation (4.2). The robust counterpart of this
problem is equivalent to
2
d U ∈
02
{|( , ) } {|( ,) }
(1) min max
{|( , 1) }
max
{|( , ) } {|( ,) }
{|( , ) } {|( ,) }
min (1 ( ) )
s.t.
0
T
ij ij ji
iN j i j A j ji A
i
in
iin A i
i
ij ji
ji j A j ji A
ij ji
ji j A j ji A
fdf f
ff pD iN
ffD iN
ff iN
f
ββ
∈∈ ∈
+
+∈
∈∈
∈∈
++ +
≥= ∀∈
− ≤ ∀ ∈
− ≥ ∀ ∈
∑∑ ∑
∑∑
∑∑
∑∑
0
ij
≥
Qf

Proof: The (RC) problem of the minimum energy consumption model is:
02
1
{|( , ) } { |( ,) }
min ( (1 ( ) ) min max
T
ij ij ji ij ij
uQu
iN j i j A j ji A i j
fdf fu β β
≤
∈∈ ∈
++ )+
∑∑ ∑ ∑∑
with the deterministic
problem constraints.
55
Then, from Karush-Kuhn-Tucker Optimimality Conditions, we have that the optimal
solution value of
1
max
T
ij ij
uQu
ij
f u β
≤
∑∑

equals to
min
T
f Qf β
Therefore the objective function becomes
02
{|( , ) } {|( ,) }
min (1 ( ) )
T
ij ij ji
iN j i j A j ji A
fdf f ββ
∈∈ ∈
++ +
∑∑ ∑
Qf
Considering the problem deterministic constraints, we prove lemma 4.2.

Proposition 4.2: Consider the Minimum Energy Consumption Problem for a given
amount of data transfer with uncertainty distance, where given by equation (4.3).
The robust counterpart of this problem is equivalent to
3
d U ∈

56
02
{|(,)} {|(,)} {|(,)}
(1) min max
{|( , 1) }
max
{|( , ) } {|( , ) }
{|( , ) } {|( , ) }
min ( )
s.t.
0
ij ij ij ji
ijij A i jij A jji A
i
in
ii n A i
i
ij ji
ji j A j ji A
ij ji
ji j A j ji A
fdM f
ff pD iN
ffD iN
ff
βθτ
∈∈
+
+∈
∈∈
∈∈
f
∈
+++
≥= ∀∈
− ≤ ∀ ∈
− ≥
∑∑ ∑∑ ∑
∑∑
∑∑
∑∑
(, )
0
0(,)
ij
ij
iN
fR i j A
fijA
θβ
θ
∀ ∈
≥ ∀ ∈
≥
≥ ∀ ∈

Proof: It is the easy to know that the robust counterpart of the minimum energy
consumption model is the problem
02
..
{|( , ) } {|( , ) }
0
min max
{|( , ) }
max
{|( , ) } {|( , ) }
{|( , ) } {|(
min (1 ( ) ) min min max
s.t.
ij
ij
ij
ij
ij ij ji ij ij
f st
ijij A jji A
M
i
ij
ji j A i
i
ij ji
ji j A j ji A
ij ji
ji j A j
fdf f
ff p D i N
ffD iN
ff
ξτ
ξ
ββ
∈∈
≤
≥
∈
∈∈
∈
+ + +
∑∑
≥= ∀∈
− ≤ ∀ ∈
−
∑∑ ∑
∑∑
∑∑
∑
,) }
0
0(,)
ji A
ij
iN
fijA
∈
≥ ∀ ∈
≥ ∀ ∈
∑
Rξ

57
Considering the part
s.t.
0
max
ij
ij
ij
ij ij
M
f R
ξτ
ξ
βξ
≤
≥

∑∑

And from LP the dual theory, we get it’s dual. That is

It's dual
max min
. s.t.
0
0
ij ij
ij ij
ij
ij
fRM
st
fR M
βξ τθ
θβ ξτ
θ
ξ
⇒
≥ ≤ ∑∑
≥
≥

So the objective function becomes

02
{|( , ) } {|( ,) }
min (1 ( ) ) min min min
..
0
ij
ij ij ji
ijij A jji A
ij
f
fd f M
st
fR
β τθ
θβ
θ
∈∈
+ + +
≥
≥
∑∑ ∑

Considering the conservation of flow and non-negativity of flow constraints, we prove
the Proposition 4.2
Proposition 4.2 also shows that the (RC) problem of a minimum energy consumption
problem for a given amount of data transfer with uncertainty set given by (4.3) is
similar to the deterministic problem without uncertainty. Different from deterministic
problem without uncertainty, the (RC) has an additional nonnegative variable additional
constraints
3
U
i
θ , |A| additional constraints
ij
f R θ β ≥ , and the objective function has an
extra term M τ θ . But this (RC) problem is still a linear problem and this implies that
solving the (RC) is just as difficult as solving the deterministic problem.

58
C. Maximum Lifetime Problem
Proposition 4.3: Consider the Maximum Lifetime Problem for a given amount of data
transfer with uncertainty distance, where given by equation (4.4). The robust
counterpart of this problem is equivalent to
4
d U ∈

min max
{|( , ) }
max
{|( , ) } { |( , ) }
{|( , ) } { |( , ) }
02
{|( , ) } { |( , ) , 1}
min
s.t.
0
[1 ( ) ]
i
ij
ji j A i
i
ij ji
ji j A j ji A
ij ji
ji j A j ji A
i
ij ij i i ji
ji j A j ji Ai n
q
ff p D i N
ffD iN
ff iN
fd M fqE βθτ
∈
∈∈
∈∈
∈∈≠+

≥= ∀∈
− ≤ ∀ ∈
− ≥ ∀ ∈
++ + ≤
∑∑
∑∑
∑∑
∑∑
,
(, )
0
0(,)
iij
i
ij
iN
fR i j A
iN
fijA
θβ
θ
∀∈
≥ ∀ ∈
≥ ∀ ∈
≥ ∀ ∈

Proof: Analogous to the proof of Proposition 4.1
Proposition 4.3 shows that the (RC) problem of a maximum lifetime problem for a given
amount of data transfer with U given by set (4.4) is similar to the deterministic problem
without uncertainty. Different from deterministic problem, the (RC) has n additional non-
negative variables
i
θ , |A| + n additional constraints and 0
iij i
fR θ β θ ≥ ≥ , and the energy
is reduced to . But this (RC) problem is still a linear problem and
this implies that solving the (RC) is just as difficult as solving the deterministic problem.
for
i
ii
qE M i N θτ− ∈
59
The three propositions show robust counterpart problems in maximum data extraction for
a given amount of energy, minimum energy consumption for a given amount of data
transfer, and maximum network lifetime for a given data transfer are of the same form as
the original problem and imply the complexity of solving for the robust solution is the
same as the complexity of solving a deterministic problem without uncertainty.

4.4. Computational Experiments
We conducted computational experiments that investigate the relative merit of the robust
solution when compared to the deterministic solution of the problem without uncertainty
for all three type of problems considered. We present three types of experiments:
We investigate the maximum protection that the robust solution can provide in the worst
case and at what cost, (2) we conduct a simulation to observe the practical performance
of a robust and deterministic solutions, and (3) we study the effect of varying problem
parameters on the robust and deterministic solutions.

4.4.1 Experimental Set Up
These experiments were coded in AMPL and are solved with CPLEX 8.1. In our
simulation, there are 50 nodes randomly deployed in 0.5km * 0.5km area, the sink node
located at (0.25km, 0.5km). Each node has or units of energy
and units of data. We use the R=0.1 or 0.005 , or 1 (
250,000
i
E = 100,000
i
E =
max
10000
i
D =
2
km 2
i
M = i N ∀ ∈ ),
M=50, p=90% or 50% in the uncertainty set U .
60
We compare the performance of the robust solution to the deterministic solution obtained
for , some nominal (average estimate) of the uncertain distances. In case of a
minimization problem, we consider a pair of ratios that compare the robust and
deterministic solutions on their respective worst case and on the nominal data:
0
d
() (
()
WC
wc
WC
Dsol d Rsol d
R
Dsol d
−
=
)

and

00
0
() ()
()
ac
Rsol d Dsol d
R
Dsol d
−
=
where

Dsol( ): optimal value of the deterministic solution.
0
d
Rsol( ): optimal value of the robust solution. d
Dsol( ): objective value of the deterministic solution in its worst case scenario.
WC
d
Rsol( ): objective value of the robust solution in the deterministic scenario.
0
d
The first ratio
wc
R measures the relative increase of the deterministic solution in the
worst case, while the second ratio
ac
R quantifies the relative loss of optimality of the
robust solution on the nominal data. Therefore the ratio
wc
R measures the maximum
protection that a robust solution can provide, while
ac
R is the percent increase in cost for
this protection. Similar ratios are defined for a maximization problem simply by flipping
the signs.
61
For each problem, we solve the deterministic problems for the nominal distance scenario
and robust problems with appropriate distance uncertainty sets to get the objective
value Dsol( ) and Rsol(d). Obtaining the other values requires to solve related
optimization problems in which:
0
d
0
d
We fix the deterministic solution to determine its worst distance case for Dsol( ) and
compute the value of the robust solution in the deterministic scenario in Rsol( ). We
compute the ratios above for 30 randomly generated networks and report the mean ratio
values.
wc
d
0
d
We also present numerical results that illustrate the performance in practice of the robust
and deterministic solutions on a fixed network. Given a robust and deterministic routing
solutions,
R
f and
D
f respectively, we conduct the following simulation to compute
their practical performance. We randomly generate the uncertainty parameters
ij
ξ in the
uncertainty set (4.4) to obtain an actual sampled distance vector (where we fix ).
The robust and deterministic solutions are scaled,
ij
d
0
ij
d
R
f α and
D
f α respectively, to
guarantee feasibility and efficiency for the problem defined with the sampled distance
values . We conduct 100 random experiments and report mean and standard deviation
of these simulations for the robust and deterministic solutions.
d
In each of the next three subsections we present the results for each of the three problems
considered in this work: the maximum data extraction, the minimum energy consumption,
and the maximum lifetime problems. For each problem we present the three types of
experiments: a study of the trade-offs of the robust solution (ratios
ac
R and
wc
R ), the
62
simulated performance, and the sensitivity of the robust solution for changes in the
problem parameters.

4.4.2 Results for Maximum Data Extraction Problem
(1) The trade-off between robust solutions and deterministic solutions through the
comparison of the ratios
ac
R and
wc
R .
In Figure 4.1 we present
ac
R and
wc
R for different energy =250,000 and
=100,000. We observe that the robust solutions are able to improve the worst case
scenario with relatively little loss in optimality. With increasing
1
i
E
2
i
E
τ , we observe a faster
increase in
wc
R than
ac
R , which shows that the robust solutions become more attractive
as the distance uncertainty increases and can be able to compensate for uncertainty while
suffering small performance losses.
63

Figure 4.1: The comparison of
ac
R and
wc
R in maximum data extraction
problems ( =250000, =100000, =10000, R=0.005) 1
i
E 2
i
E
max
i
D

(2) A comparison of the practical performance for robust solutions and deterministic
solutions for a given network under different uncertainty levels
Figure 4.2 presents the mean and standard deviations for the robust solutions and
deterministic solutions over the 100 samples of all uncertain distance values. Given a
sampled vector of distances we scale the robust and deterministic routing solutions so
that they are feasible and maximize the objective value. This plot shows that the mean
value of the adjusted robust solution is always better than the mean value of the adjusted
deterministic solution. In addition we note that the standard deviation, represented by the
64
vertical segments is much smaller for the robust solution. This shows that on a simulation
the robust solution is observed to perform better on average and with a smaller standard
deviation than a deterministic solution for any uncertainty level.

Figure 4.2: The mean and standard deviations of objective value for the
deterministic and robust solutions in maximum data extraction
problem for different uncertainty level ( =250000, =10000,
R=0.005)
i
E
max
i
D

(3) The effect of varying on the robust and deterministic solutions.
i
E
For the maximum data extraction problem we conduct a sensitivity analysis on the
available energy and observe its effect on the trade-off ratios. Figure 4.3 shows the
i
E
65
rate values
wc
R and
ac
R for 10.1 τ = and 20.9 τ = and different available energy E on
each node. Notice that,
wc
R is higher than
ac
R for any energy level, with most ratios
remaining below 5%, except for a middle range of energy. In this middle range the
robust solution appears more attractive as there is a bigger gap between
wc
R and
ac
R .
Note also that the ratios
wc
R and
ac
R are almost 0 when the available energy E is small
or large. In these two cases the levels of energy in the nodes encourage every node to
route directly to sink for both the deterministic and robust solutions. This occurs because,
either the node does not have enough energy to transmit someone else's information or
has enough energy to transmit everything directly to the sink.

Figure 4.3: The comparison of
ac
R and
wc
R in maximum data extraction problems
as a function of the available energy E of each node. ( =10000,
max
i
D
1 τ =0.1, and 2 τ =0.9}

66
4.4.3 Result for Minimum Energy Consumption Problem
(1) The trade-off between robust solutions and deterministic solutions through the
comparison of the ratios
ac
R and
wc
R
In Figure 4.4 we present the ratios
ac
R and
wc
R for different information extraction
requirements
min max
i
iN
f pD
∈
=
∑
. We present the ratios
ac
R and
wc
R for p1=90% and
p2=50%. The graph presents the ratios
ac
R and
wc
R as a function of τ . We observe that
ratio
wc
R is larger than
ac
R with this difference being accentuated as τ increases.

Figure 4.4: The comparison of
ac
R and
wc
R for objective value in minimum energy
consumption problems (p1=90%, p2=50%, =250000$, =10000,
R=0.1)
i
E
max
i
D
67
This shows that the robust solutions become more attractive as the distance uncertainty
increases and can be able to compensate for uncertainty while suffering small
performance losses.

(2) The comparison of the practical performance for robust solutions and deterministic
solutions under different uncertainty levels
We present the mean and standard deviation for these simulation results in Figure 4.5.
We observe that, similarly to the maximal data extraction problem results showed in
Figure 4.2, the mean value of the adjusted robust solution is always better than the mean
value of the adjusted deterministic solution, also the standard deviation of the simulation
is smaller for the robust solution. This shows that on a simulation the robust solution is
observed to perform better on average and with a smaller standard deviation than a
deterministic solution for any uncertainty level.
68

Figure 4.5: The mean and standard deviations of objective value for the
deterministic and robust solutions in minimum energy consumption
problem for different uncertainty level ( =250000, =10000,
p=90%,
i
E
max
i
D
i
M =2, R=0.1)

(3) The effect of varying percentage p on the robust and deterministic solutions
We conduct a sensitivity analysis on the trade-off ratios,
ac
R and
wc
R , as we vary the
minimal amount of information that must be sent. In Figure 4.6, we present the ratios
ac
R
and
wc
R for different percentages p of the total information that must be sent to the sink,
min max
i
iN
f pD
∈
=
∑
. We observe that
wc
R is always higher than
ac
R for every level of
minimal amount of information, which means that the robust solutions are attractive as
69
they provide a higher protection in the worst case than additional cost on the nominal
data. Note that the benefit of the robust solution increases for a larger uncertaintyτ .

Figure 4.6: The comparison of
ac
R and
wc
R of the objective value in minimum
energy consumption problem on different percentage p of total data
that must be sent to the sink. ( =250000, =10000,
i
E
max
i
D 1 τ =0.1, and
2 τ =0.9}

The comparison of the practical performance for robust solutions and deterministic
solutions under different percentage of total data that must be sent to the sink node p
We present the mean and standard deviation for these simulation results in Figure 4.7.
We observe that, the mean value of the adjusted robust solution is always better than the
mean value of the adjusted deterministic solution, also the standard deviation of the
simulation is smaller for the robust solution. This shows that on a simulation the robust
70
solution is observed to perform better on average and with a smaller standard deviation
than a deterministic solution for any uncertainty level.

Figure 4.7: The mean and standard deviations of objective value for the
deterministic and robust solutions in minimum energy consumption
problem for different percentage P of total data that must be sent to
the sink.
max
250000, 10000, 1 0.1
ii
ED τ == =

We conduct the experiments for different location topologies of the sink node about the
trade-off between robust solutions and deterministic solutions through the comparison of
the ratios
ac
R and
wc
R . For example, in Figure 4.8, we present the ratios
ac
R and
wc
R for
p1 = 90% and p2 = 50%, the sink node is located at the center (0.25, 0.25), which has the
most number of neighbor nodes, the side of the border (0.25, 0.5), which has the second
highest number of neighbor nodes and the corner (0, 0.25), which has the less number of
71
neighbor nodes around the sink nodes. We observed that the number of neighboring
nodes around the sink node does not affect the performance significantly. We observe
that ratio
wc
R is larger than
ac
R with this difference being accentuated as τ increases for
all the three location of the sink node. This shows that the robust solutions become more
attractive as the distance uncertainty increases and can be able to compensate for
uncertainty while suffering small performance losses for all the three location scenario of
the sink node.

Figure 4.8: The comparison of
ac
R and
wc
R for objective value in minimum energy
consumption problems in different location of the sink node (p1 = 90%,
p2 = 50%, = 250000, = 10000, R =0.1,
i
E
max
i
D τ = 0.5)
4.4.4 Results for Maximum Lifetime problem
(1) The trade-off between robust solutions and deterministic solutions through the
comparison of the ratios
ac
R and
wc
R
72
In Figure 4.9, we present the ratios
ac
R and
wc
R for different information extraction
requirements. We plot the ratios
ac
R and
wc
R for p1=90% and p2=50%. The graph
presents the ratios
ac
R and
wc
R as a function of τ . The results here mimic the results
found in the other two examples, that is the robust solution can significantly reduce the
worst case cost, as the uncertainty increases, while increasing at a slower rate the loss of
optimality of the robust solution in the nominal case, which shows the robust solutions
can be able to compensate for uncertainty while suffering small performance losses.

Figure 4.9: The comparison of
ac
R and
wc
R for objective value q (q=1/T) in
maximum lifetime problems (p1=90%, p2=50%, =250000,
=10000, R=0.1)
i
E
max
i
D

(2) The comparison of the practical performance for robust solutions and deterministic
solutions under different uncertainty levels
73

Figure 4.10 presents the mean and standard deviation for these simulation results. We
observe that, as it was with the previous two types of problems, the robust solution
outperforms the deterministic solution in mean value for any uncertainty level, also the
standard deviation of the objective function is smaller for the robust solution. This shows
that under a simulation the robust solution of the maximum lifetime problem is observed
to perform better on average and with smaller standard deviation than a deterministic
solution for any uncertainty level.

Figure 4.10: The mean and standard deviations for the deterministic and robust
solutions of objective value T in maximum lifetime problem for
different uncertainty level. T=1/q, =250000, =10000, p=90%,
i
E
max
i
D
i
M =2, R=0.1

74
(3) The effect of varying percentage p on the robust and deterministic solutions.
For the maximum lifetime problem, we study the sensitivity of the ratios
ac
R and
wc
R
to the percentage p of total data that must be sent to the sink,
min max
i
iN
f pD
∈
=
∑
. Figure
4.11 shows that
wc
R is much higher than
ac
R for problems with little uncertainty10.1 τ = .
However, the two ratios are comparable for large uncertainty sets 20.9 τ = , with
ac
R
beating
wc
R for small percentages of information p ≤0.3. Therefore, for this problem the
benefits of a robust solution depend on the amount of uncertainty and amount of
information that is being sent to the sink.

Figure 4.11 The comparison of
ac
R and
wc
R of the objective value q (q=1/T) for
maximum lifetime problem for different percentage P of total data to
sink. =250000, =10000,
i
E
max
i
D 1 τ =0.1, 2 τ =0.9}

75
There are conditions where the robust solution becomes overly conservative, costing in
the nominal case more than the protection it can provide in the worst case.

4.5 Conclusions
Many planning and operational problems on energy limited wireless sensor networks
must operate in conditions with significant uncertainty in distances between nodes.
Optimal solutions that do not take into consideration this uncertainty may be inefficient
solutions in practice. In this chapter, we present robust optimization models to address
distance uncertainty for three optimization problems related to the operation of energy
limited wireless sensor networks: the maximum data extraction, minimum energy
consumption, and maximum lifetime problems. For these three problems we proved that
computing the robust solution, i.e. the solution with best worst case objective over the
uncertainty set, is no harder than solving the deterministic version of the problem. The
specific form of the uncertainty sets considered is fundamental to be able to compute the
robust solution efficiently.
Our computational experiments investigate whether a robust solution can be an attractive
solution in practice. We find that the robust solution can provide significant worst case
protection while often incurring in a small additional expense over the optimal solution
for a nominal data instance.
In addition we showed through simulations that a robust solution can exhibit better mean
objective value and smaller standard deviations, and that these results hold for a wide
setting of problem parameters.
76
This work showed that for a specific uncertainty set, the robust solution can be computed
efficiently and it can be an attractive solution in practice. Future work will study what are
representative models of the uncertainty faced by sensor networks in different
applications and develop the problem formulations and algorithms to compute the robust
solution efficiently.
Our computational experiments also showed that for some problem instances the robust
solution could be overly conservative, incurring in a large cost over a nominal optimal
solution. An important future research direction is to develop a method to identify when a
robust solution will be competitive from the problem instance and uncertainty set
considered.
77

Chapter Five: Robust Localization of Sensor Nodes
Sensor’s location estimation plays an important role in many applications of sensor
networks. However, most existing localization methods are vulnerable to malicious
attacks in an unfriendly environment. Recently, numerous position and distance estimate
techniques for WSNs have been proposed by Liu et al. (2005), Li et al. (2005), Priyantha
et al. (2000), and Bahl et al. (2000). However, these technologies assume the correct
position of nodes is known and the system is in non-adversarial environments. The study
of the effect of malicious attacks has been mostly ignored.
We use robust optimization to solve the problem of node localization in adversarial
domains to mitigate the vulnerabilities of the estimation to the malicious attack. This
method, as me mentioned in Section 2, is based on obtaining the solution that achieves
the best objective value under its worst case distance data. We consider the uncertain
distance information where the bad distance is not specified exactly and it is only known
to belong to a given uncertainty set U. In this method, the attack-tolerant mechanisms are
tentatively given to protect the localization structures from threats or lies from the
attacked objects. The method is to live with all information including tampered
information. It is challenging to apply robust techniques to sensor localization.

78
5.1 Problem Description

The network is a graph with n sensor nodes randomly placed in a square region with the
known positions ( ,
i i
x y ), and an attacker with the unknown positions ( 1, 2,..., i = n
0 0
, x y )
who enters the region. Different nodes will use different methods to detect the distance
from the attacker, while the attacker tries to manipulate sensor nodes and the attacker will
use unknown ways to fake the n sensing nodes. So some nodes can receive correct
distance references, while other nodes do not. Then, although we know there are some
errors in the distance data, we do not know which information has been tampered with.
In Figure 5.1, sensors intend to detect the 2-D location of the attacker.
Let (
12 3 n
S , S , S , ...,S
11 2 2 3 3
(x ,y ), (x ,y ), (x ,y ), ..., ,
n n
x y ) be the known locations of ,
and the be the position of the attacker to be determined. Each sensor
can get , the distance between the node
12 3 n
S , S , S , ...,S
0 0
(x , y )
( 1,..., ) ii n =
i
d ( 1, 2..., ) ii n = and the attacker,
by using localization techniques. For example, S1 can send at time a signal to attacker
which immediately echoes a response received by S1 at time , S1 can then estimate its
distance to attacker as , where c is the speed of light.
a
t
b
t
1
( )/
ab
dt tc ≈− 2

79

Attack
S1(x1, y1)
S2(x2, y2)
S3(x3, y3)
d1
d2
d3
No attack

Attack
S1(x1, y1)
S2(x2, y2)
S3(x3, y3)
d1
d2
d3’
d3’ has been reduced by attacker

S1(x1, y1)
S2(x2, y2)
S3(x3, y3)
d1
d2
d3’’

d3’ has been enlarged by attacker

Figure 5.1: The topology description of malicious distance by attackers.

Assume the sink node collects the distance from the
sensorsS , then the position (,
12
, ,...,
n
dd d
12 3 n
, S , S , ...,S )
0 0
x y of the attacker can be obtained via
Minimum Mean-Square Error (MMSE) method which selects the position ( , x y ) that
minimizes
2
1
[( )( )
n
ii i
i
dxx yy
=
−− + −
∑
22
] . We refer to this method of determining
80
the position of the attacker as Method-1, and the position is the minimizer, given by:
2
00
,
1
(, ) argmin [ ( ) ( )]
n
ii i
xy
i
xy d x x y y
=
=−−+−
∑
22
. (5.1)
This is a non-convex nonlinear squares problem. Local optimal solutions for this problem
can be obtained by nonlinear techniques such as gradient descent or Newton method,
which converge to a Karush-Khun-Tucker (KKT) point. There is typically guarantee on
how close to the global optimum the solution found is, but multi starts with different
initial solutions can be used as a heuristic to improve the solution.
While, attackers send the fake information to reduce or enlarge distance estimates as
shown in the above figure, so as to maliciously increase the location inaccurately. Both
the distance reduction and enlargement attacks may make the location estimate of
attacker far from its true location, shown in Figure 5.1. In this dissertation, we will talk
about different methods to mitigate the impact of such attacks.
5.2 Methodologies

In this section, we first describe the uncertainty of distance and the uncertainty sets used
to represent the distance uncertainty, then we present two methods to estimate the
location that address uncertainty in the distance measurements. These methods will be
compared with Method-1, which ignores that the distance data may be inaccurate and
solves MMSE Problem (equation 5.1) above using all available data.
The second method we propose uses robust optimization to account for the
inaccurate distance data, while our third method is based on solving the minimum error
square problems with the removal of some location measurements.
81

5.2.1 Uncertainty Sets

There exist two sources of uncertainty, one is the defect distance information and the
malicious perturbation from attacker, the other is from error in measurement.
The uncertainty set of the malicious perturbation from attacker is defined as the distance
exist in the set d
, (5.2)
00
[(1 ), (1 )], [1.. ]
ii
dd d i N αβ ∈− + ∈
where is the true distance between detecting sensor i and the attacker.
0
i
d [0,1] α ∈
means the scale of reduction error that the attacker fake sensors i , 0 β ≥ means the
maximum enlargement error that the attacker want to fake sensors i . If 1, 1 α β = = then
the uncertainty set ; or If
0
[0,2 ]
i
d d ∈
i
1, 2 α β = = then the uncertainty set .
0
[0,3 ]
ii
dd ∈
We use binary variable , which is defined as 1, indicating that the link is the bad
information between the send node and the attacker, or 0, indicating the link is the
good information between the send node and the attacker. If there are K perturbations
or bad information in the measure, noted that
i
z
i
s
i
s
1
n
i
i
zK
=
=
∑
.
The uncertainty set of the error in the measurement is defined as the distance d existing
in the set , (5.3)
00
[(1 ), (1 )], [1.. ]
ii
dd d i N εε ∈− + ∈
where ε is error of the measure such as 5%, 10%.
82
5.2.2 Method-2: Robust Optimization (RO) method

In this section, we use the robust optimization methodology introduced by Ben-Tal and
Nemirovski (1998) into localization problems and refer to the literature review section for
a description. We now formulate robust counterpart problems of the localization problem
with distance uncertainty set given by (5.2) according to robust optimization approach.
We note that the robust counterpart problem in each case is of the same form as the
original problem, therefore the complexity of solving for the robust solution is the same
as the complexity of solving a deterministic problem.
Proposition 5.1: Consider uncertainty distance set given by equation (5.2), the robust
counterpart of this problem is equivalent to
00
02 2 2
00
1
(, )
00
min max | ( ) (( ) ( ) ) |
..
[(1 ) ,(1 ) ]
{0,1 }
N
ii i
i
xy z
iii ii
i
i
i
dxx yy
st
dzd zd
zK
z
αβ
=
−− + −
∈− +
=
∈
∑
∑

where means the number of the perturbations or bad information in the measure, k

And the objective is to minimize
02 2 2
0
1
|( ) (( ) ( ) )|
N
ii i
i
dxx yy
=
−− + −
0 ∑
, considered in our
robust optimization method.
Proof:
According to definition of uncertainty set, the malicious perturbation influenced by the
distance of measure exist in the uncertainty set d
83
,
00
[(1 ), (1 )], [1.. ]
ii
dd d i N αβ ∈− + ∈
we use binary variable , which is defined as 1, indicating that the link is the bad
information between the send node and the attacker, or 0, indicating the link is the
good information between the send node and the attacker, If there are perturbations
or bad information in the measure, The uncertainty set are defined as
,
i
z
i
s
i
s k
00
[(1 ), (1 )], [1.. ]
{0,1 }
ii i i i
i
i
i
dd z d z i N
zK
z
αβ ∈− + ∈
=
∈
∑
Based on the robust solution under uncertainty is defined as the solution that has the best
objective value in its worst case uncertainty
scenario. So the robust solution is obtained by solving the following Robust Counterpart
problem:
02 2 2
0
1
|( ) (( ) ( ) )|
N
ii i
i
dxx yy
=
−− + −
∑ 0
00
()(
ii i
00
02 2 2
00
1
(, )
00
min max | ( ) (( ) ( ) ) |
..
[(1 ) ,(1 ) ]
{0,1 }
N
ii i
i
xy z
iii ii
i
i
i
dxx yy
st
dzd zd
zK
z
αβ
=
−− + −
∈− +
=
∈
∑
∑

which has given the proof of Proposition 5.1.
To simplification the notation, we define
22 2
) xxyy ξ=− + − [1.. ] i N ∈ , . The robust
counterpart of this problem can be written as following
84

00
00
02 2 0 2 2
1,
(, ) .
[(1 ) ,(1 ) ]
1
{0,1}
min{| ((1 ) ) ) | max | ( ) |}
ii i i i
i
i
i
N
jj ii
ii j
xy st
dzd z d
zK
z
dd
αβ
αξ ξ
=≠
∈− +
=−
∈
−− + −
∑
∑

and
00
00
02 2 0 2 2
1,
(, ) .
[(1 ) ,(1 ) ]
1
{0,1}
min{| ((1 ) ) ) | max | ( ) |}
ii i i i
i
i
i
N
jj ii
ii j
xy st
dzd z d
zK
z
dd
αβ
βξ ξ
=≠
∈− +
=−
∈
+− + −
∑
∑

If there are only one bad information (K=1) existing in the problem, then we have the
following formulation, see lemma 5.1, which is a nonlinear programming problem.
Lemma 5.1: The robust counterpart problem can be written as follows when K=1
00
(, ),
02 2 0 2 2
11
2
02 2 0 2 2
11
2
02 2 0 2 2
22
1, 2
02 2 0 2 2
22
1, 2
02 2 0
min
..
|((1 )) | |() |
|((1 ) ) | |( ) |
|((1 )) | |() |
|((1 ) ) | |( ) |
...
|((1 ) ) | |( )
xy
N
ii
i
N
ii
i
N
ii
ii
N
ii
ii
NN i
st
dd
dd
dd
dd
dd
γ
γ
αξξγ
βξξγ
αξξγ
βξξ
βξ
=
=
=≠
=≠
−− + −≤
+−+ −≤
−− + −
+−+ −
+−+
∑
∑
∑
∑
1
22
1
|
N
i
i
γ
≤
≤
ξ γ
−
=
−≤
∑

Proof: when K=1, the robust counterpart problem is below
85
00
02 2 2
00
1
(, )
00
min max | ( ) (( ) ( ) ) |
s. .
[(1 ) ,(1 ) ]
1
{0,1 }
N
ii i
i
xy z
iii ii
i
i
i
dxx yy
t
dzd zd
z
z
αβ
=
−− + −

∈ − +
=
∈
∑
∑

02 2 2
00
1
00
max | ( ) (( ) ( ) ) |
..
[(1 ) ,(1 ) ]
1
{0,1 }
let
s
N
ii i
i
z
iii ii
i
i
i
dxx yy
t
dzd zd
z
z
γ
αβ
=
=−−+

∈ − +
=
∈
∑
∑
−

which can become

00
00
00
02 2 0 2 2
1,
(, ) .
[(1 ) ,(1 ) ]
0
{0,1}
02 2 0 2 2
1,
(, )
min
min{| ((1 ) ) ) | max | ( ) |}
min max{| ((1 ) ) ) | | ( ) |}
ii i i i
i
i
i
N
jj i i
ii j
xy st
dzd z d
z
z
N
jj ii
ii j
xy
dd
dd
αβ
γ
αξξ
αξ ξ
=≠
∈− +
=
∈
=≠
=− −+ −
∑
=− −+ −
∑
∑

or
00
00
00
02 2 0 2 2
1,
(, ) .
[(1 ) ,(1 ) ]
0
{0,1}
02 2 0 2 2
1,
(, )
min
min{|((1 )) )| max |() |}
min max{| ((1 ) ) ) | | ( ) |}
ii i i i
i
i
i
N
jj ii
ii j
xy st
dzd z d
z
z
N
jj i i
ii j
xy
dd
dd
αβ
γ
βξξ
βξ ξ
=≠
∈− +
=
∈
=≠
=+ −+ −
∑
=+ −+ −
∑
∑

which competes a proof of lemma 5.1.

86
5.2.3 Method-3 Minimizing Minimum Mean Square Error (MMMSE) method

In this method we use the MMSE Method-1 but do not consider all the data. We select a
subset of the available distance data, hoping to exclude the incorrect information. By
solving MMSE for multiple subsets of nodes and selecting the one with minimal error we
expect to obtain the true position of the attacker. In particular if there is no measurement
error and we consider a subset with no tampered data then this minimal MMSE obtains
the exact position. The equation is denoted as follows:
00
2
22
00
(, )
,| |
min min [ ( ) ( ) ]
ii i
xy
iS S m
dxx yy
∈=
⎧⎫
⎪⎪
−− + −
⎨⎬
⎪⎪
⎩⎭
∑
For each
00
2
2
0
(, )
,| |
min [ ( ) ( ) ]
ii i
xy
iS S m
dxx yy
∈=
−− + −
∑
2
0
, we need to solve a minimum square
error problem to get the estimated position (
0 0
, x y ). We need solve all minimum
square error problems, then compare which one causes the minimum error between the
true position and the estimate solutions, and we get the estimated position. When is big,
the workload for solving problem is heavy. If n =10, =3, we just need to solve
n
m
⎛ ⎞
⎜ ⎟
⎝⎠
n
m
10
10!
120
3 3!7!
⎛⎞
==
⎜⎟
⎝⎠
minimum square error problems; while n =100, we need to
solve
100
100!
161700
3 3!97!
⎛⎞
==
⎜⎟
⎝⎠
. This shows that if n is big then the problem size can be
very big. In my experiments, we use mn K = − to do our simulations.

87
5.3 Simulations and Results

We have conducted experiments of three models in WSNs to what performance robust
optimization methods are.

5.3.1 Experimented Set-Up

These experiments to study how the three models perform at localizing a given node
under adversarial conditions were coded by AMPL and solved with KNITRO 5.1. Since
these problems are nonlinear and non-convex problems, we use multi start points to
search best possible optimal solutions. In our simulation, there are 10 sensor nodes
randomly deployed in 0.5km* 0.5km area, which are used for the anchors to detect the
attacker’s positions. The true position (
0 0
, x y ) of the attacker is always (0.121, 0.197).
We let be the true distance between the sensor node
0
j
d
j
s and the attacker. The
uncertainty set is given by equation (5.1) and we randomly generate the ‘sensed distance
data’ in this interval. This happens as follows,
we first select k of the random nodes to have altered data, and generate a distance
j
d for
each of these nodes from uniform distribution on
0 0
[(1 ), (1 )
j j
dd ] α β −+ .
For all other nodes we sample
j
d uniformly from
0 0
[(1 ), (1 )]
j j
dd ε ε −+ .

88
5.3.2 Performance Criteria

In order to measure the effectiveness of our method on real sensor network problems, we
use the following criteria to measure different performance. The criteria by which we
evaluate the performance of the methods are how the computed estimated position differs
from the known true position. This error is expressed as
22
00 0 0
11
11
()( Avg_E r r or =
nn
i
ii
exx y
nn
==
=−+−
∑∑
)y
0
x , and
0
y is the measured value of the attacker position, and (
0
x , ) is the true value of
the attacker position. We do 10 computation simulations and get the average error from
the simulation results.
0
y

5.3.3 Experimented Results

We conducted computational experiments that investigate the comparison of three
methods considered. We present four types of experiments: (1) we investigate
experiments for different number of sensors n=6, 7, 8, 9, 10, (2) we conduct a simulation
to observe the practical performance when experiments with the known number of bad
information K=1, (3) we conduct a simulation to observe the practical performance when
experiments with the unknown number of bad information (4) we conduct a simulation to
observe the comparison of solving time when K=1, 2, 3, 4, 5

89
(1) Experiments with different number of sensors n=6, 7, 8, 9, 10
In this experiment scenario, we use 1, 1 α β = = and the error 0.1 ε = , which gives the
following intervals for the generation of the random distance data and
. We set the number of bad information K =1. We use the following ways
to generate 10 random experiments:
0
[0,2 ]
j
d
0
[0.9 ,1.1 ]
i i
d d
0
(0,0.5), (0,0.5) Uniform Uniform
i i
xy = = , the
number of sensors used to be measured may be 6, 7, 8, 9 or 10.

Figure 5.2: Experiments for different sensors n=6, 7, 8, 9, and 10 when K=1,
error=10%

Figure 5.2 shows the results of Method-2 (M2) and Method-3 (M3) are better than that of
Method-1 (M1) when the number of sensors n>7. As the increasing number of the
90
sensors, the solution of Method-2 is close to the solution of Method-3, which can
approximately get the true position, both are much better than that of Method-1.

(2) Experiments with the known number of bad information K=1
In these experiment scenarios, we assume the number of bad information is known, for
example K=1, then we use the M1, M2, and M3 models with K=1 to solve it.
Figure 5.3 shows the location solution in M2 and M3 are similar to close to true positions
and much better than that of M1 as the increasing of the uncertainty from 1 to 50 β
when error=0%. Average error of M1 is approximately linearly increasing as the
increasing of the uncertainty value β when 10 β ≤ , while it is increasing
approximately another linearly after 10 β ≥ since the objective function of M1 is
different
Figure 5.4 and Figure 5.5 also show the location solution in M2 and M3 are similar to
close to true positions and much better than that of M1 as the increasing of the
uncertainty from 5 to 50 β when error=10% and error =20%
But as the existence of the measurement error, there are some deviations between M2 and
M3. The solutions of M2 are a little worse than those of M3. In Figure 5.4, average error
of M1 is increasing quickly as the increasing of the uncertainty value β when 10 β ≤ ,
while the slope of the line becomes more flat after 10 β ≥ , since the objective function
of M1 is different.

91

Figure 5.3: M1, M2 and M3 for different uncertainty β , when Error=0%, K=1 and
n=10

Figure 5.4: M1, M2 and M3 for different uncertainty β when Error =10%, K=1
and n=10

92

Figure 5.5: M1, M2 and M3 for different uncertainty β when Error =20%, K=1
and n=10

(3) Experiments with the unknown number of bad information (K is unknown)
In these experiment scenarios, we assume the number of bad information is unknown, for
example there are two bad information K=2 in the real application, but we don’t know
how many bad information exist in hostile environment and we just know there exist one
bad information. Then we use the M1, M2, and M3 models with K=1 to solve it.
Figure 5.6 shows the location solutions in M2 and M3 are similarly better than that of M1
as the increasing of the uncertainty from 1 to 50 β when error=0%, but the solutions
of M2 is better than those of M3.
93
Figure 5.7 and Figure 5.8 also show the location solutions in M2 and M3 are similarly
better than those of M1 as the increasing of the uncertainty from 5 to 50 β when error
=10% and error=20%, and the solutions of M2 are better than those of M3, which show
the robust solution can get the solution close to the true location solution when we don’t
know specific bad information existing.

Figure 5.6: M1, M2 and M3 for different uncertainty β when Error =0%, and n=10

94

Figure 5.7: M1, M2 and M3 for different uncertainty β when Error=10% and n=10

95

Figure 5.8: M1, M2 and M3 for different uncertainty β when Error=20%, K=1 and
n=10

(4)Comparison for solution time, quality of solutions when K=1, 2, 3, 4, 5
Figure 5-9 shows that different solving time for the same problem in different methods
M1, M2 and M3. When the number of bad information K is small, the solving time of all
three methods are similar close, but as the increasing of the bad information K, the
solving time for M3 is much longer than M2 and M1.
96

Figure 5.9: Solution time of M1, M2 and M3 for different bad information K when
Error=0%and n=10

Figure 5.10 shows the location solution in M2 and M3 are similar to close to true
positions and much better than that of M1 as the increasing of the uncertainty
from 1 to 50 β when error=0%, when the number of bad information K=5, which has
similar results for the number of bad information K=1, as shown in Figure 5.3.

97

Figure 5.10: M1, M2 and M3 for different uncertainty β when Error=0%, n=10,
K=5

5.4 Conclusions

In this section, we have focused on three methods localizing the attacker in a sensor
network with fake information in the hostile distance measurement. We have presented
three methods to make the estimated location attack-tolerant, the first is called Minimum
Mean Square Error method (MMSE), the second is Robust Optimization method (RO) by
introducing the robust optimization method into localization problems in sensor network
applications, and the third is Minimizing Minimum Mean Square Error
method(MMMSE). The constraints in RO are a little complicated and we need solve
98
only one problem to get the estimated solution, while in MMMSE methods we need to
solve minimum square error problems to get the solutions.
n
m
⎛⎞
⎜ ⎟
⎝⎠
Our simulations show, if there exist known fake information and three correct models
are used to solve it, the performance of RO and MMMSE methods both can get much
better attack-tolerant result than MMSE methods but the solutions of RO is little worse
than that of MMMSE, However in the case of unknown fake information, for example, if
we use the three methods models with K=1 to solve the problems with K=2, RO methods
outperform MMMSE methods to get the attack-tolerant location estimation with less time
for some certain number of sensors in hostile environment.

99
Chapter Six: Contributions and Extensions

In this dissertation, we have done three main contributions in the core subjects of
Wireless Sensor Networks

(1) First, we are the first authors to introduce the Lagrangian relaxation method and the
sub-gradient projection method to achieve the energy efficiency routing protocol in
maximum data extraction models in energy limited wireless sensor network, which can
operate very quickly. Computational experiments show both centralized and distributed
versions of the algorithm arrive at routing solutions that are on average better than 10%
from optimal after only a few iterations.

(2) Second, we are the first authors to introduce robust optimization methodology and
build the robust models for three critical problems in Wireless Sensor Networks such as
maximizing data extraction, minimizing energy consumption, and maximizing the
lifetime of network problems. Our computational experiments show that as the
uncertainty increases a robust solution for these problems provides a significant
improvement in worst case performance at the expense of a small loss in optimality when
compared to the optimal solution of a fixed scenario.

(3) Thirdly, we consider the robust analysis in secure localization problems in energy
limited wireless sensor networks under hostile environment. We present three methods to
100
make the estimated location attack-tolerant: Minimum Mean Square Error (MMSE),
Robust Optimization (RO) and Minimizing Minimum Mean Square Error (MMMSE).
This is the first work to present robust optimization methods to make the location
estimate attack-tolerant.
In addition, possible extensions of our work included: research on representative models
of the uncertainty faced by sensor networks in different applications and problem
formulations and algorithms to compute the robust solution efficiently, since wireless
sensor networks are always used in hostile environments or unfriendly environments and
the distance measurements are subject to different uncertainty during the WSN’s different
deployments or operations. We can also develop securing localization methods for the
mobile base location service for cell phone applications.

101

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Abstract (if available)

Abstract Wireless Sensor Networks (WSNs) is an area of active research in industry and academia. WSNs can be used in a wide array of applications such as, battlefield surveillance, aerospace exploration, environmental monitoring, products tracking and supply chain management, homeland security applications, and so on. In this dissertation, we study algorithms that address two challenges faced by WSNs in applications: the need to operate distributedly and to take into account uncertain conditions. We first work on constructing efficient distributed routing algorithms for maximal data extraction problem, the second part of this thesis focuses on the effect of considering uncertain conditions for routing in WSN. The last part of the thesis introduces methodologies that address uncertainty to achieve secure localization in hostile environments.

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Asset Metadata

Creator Ye, Wei (author)

Core Title Models and algorithms for energy efficient wireless sensor networks

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Industrial and Systems Engineering

Publication Date 04/23/2009

Defense Date 11/21/2006

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag nonlinear optimization,OAI-PMH Harvest,operations research,robust optimization,wireless sensor networks

Language English

Advisor Ordonez, Fernando (committee chair), Dessouky, Maged M. (committee member), Krishnamachari, Bhaskar (committee member), Moore, James Elliott, II (committee member)

Creator Email yewei@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m447

Unique identifier UC1490809

Identifier etd-Ye-20070423 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-483963 (legacy record id),usctheses-m447 (legacy record id)

Legacy Identifier etd-Ye-20070423.pdf

Dmrecord 483963

Document Type Dissertation

Rights Ye, Wei

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu