# Gradient Conjugate

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Research Report No. 2007:09

Optimization Algorithms with

Applications to Unicast QoS

Routing in Overlay Networks

Dragos Ilie

Department of Telecommunication Systems,

School of Engineering,

Blekinge Institute of Technology,

S371 79 Karlskrona, Sweden

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Copyright c September 2007, Dragos Ilie. All rights reserved.

Blekinge Institute of TechnologyResearch Report No. 2007:09ISSN 1103-1581Published 2007Printed by Kaserntryckeriet ABKarlskrona 2007Sweden

This publication was typeset using LATEX.

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Abstract

The research report is focused on optimization algorithms with application to quality ofservice (QoS) routing. A brief theoretical background is provided for mathematical tools inrelation to optimization theory.

The rest of the report provides a survey of different types of optimization algorithms:several numerical methods, a heuristics and a metaheuristic. In particular, we discuss ba-sic descent methods, gradient-based methods, particle swarm optimization (PSO) and aconstrained-path selection algorithm called Self-Adaptive Multiple Constraints Routing Algo-rithm (SAMCRA).

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Contents

1 Introduction 1

1.1 Quality of Service . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Overlay Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Overlay Routing Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 QoS Routing Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Theoretical Foundation 9

2.1 Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Optimization Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3 Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4 Routing Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 282.5 Algorithms and Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Descent Methods 33

3.1 Steepest Descent Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2 Newtons Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Conjugate Gradient Method 37

5 Simplex Method 41

6 Gradient Projection Method 47

7 Particle Swarm Optimization 49

8 SAMCRA 53

9 Final Remarks 59

A Acronyms 61

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List of Algorithms

1 General descent method (GDM) . . . . . . . . . . . . . . . . . . . . . . . . . 332 Steepest descent method (SDM) . . . . . . . . . . . . . . . . . . . . . . . . . 343 Newtons method (NM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Conjugate gradient method (CGM) . . . . . . . . . . . . . . . . . . . . . . . . 385 Conjugate gradient method with quadratic approximation (Q-CGM) . . . . . 386 Fletcher-Reeves conjugate gradient method (FR-CGM) . . . . . . . . . . . . 397 Polak-Ribiere conjugate gradient method (PR-CGM) . . . . . . . . . . . . . . 398 Two-phase simplex method (2-SM) . . . . . . . . . . . . . . . . . . . . . . . . 449 Simplex method (SM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510 Gradient projection method (GPM) . . . . . . . . . . . . . . . . . . . . . . . 4811 Global best particle swarm optimization (GB-PSO) . . . . . . . . . . . . . . . 5112 Local best particle swarm optimization (LB-PSO) . . . . . . . . . . . . . . . 5213 Self-Adaptive Multiple Constraints Routing Algorithm (SAMCRA) . . . . . . 5414 INITIALIZE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

15 FEASIBILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5616 UPDATEQUEUE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

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Chapter 1

Introduction

The path towards establishing the Internet as a platform for multimedia content distributionhas shown to be a source of great challenges for the Internet research community. Thetransition from Internet Protocol (IP)s best-effort service to a service that incorporates thenotion of QoS is probably one of the biggest challenge to solve before the Internet can becomea reliable platform for multimedia services.

Multimedia services have requirements on the underlaying network that go beyond whatshortest-path routing can offer. In particular, live or interactive multimedia communica-tions have stringent constraints on the path between sender and receiver. Examples of suchconstraints are bottleneck bandwidth, packet delay, packet delay variation and packet lossrate.

The goal of the research report is to survey several algorithms that can be used for selecting

a path or sets of paths subject to multiple constraints. We focus on algorithms that can beused online, in the sense that they have the ability to react almost in real-time to relevantevents in the network. Furthermore, we are also planning to implement these algorithms in anoverlay network. This places additional requirements on the performance of the algorithms.

The report is organized as follows. The remainder of this chapter discusses the notion ofQoS, existing QoS architectures for IP networks and the main characteristics of QoS routing.Additionally, we provide an overview of the Overlay Routing Protocol (ORP), which is aframework for unicast QoS routing under development at Blekinge Institute of Technology(BTH) in Karlskrona, Sweden.

Chapter 2 outlines the theoretical background required for the remainder of the report.The chapter begins with definitions and notations for matrix algebra, graph theory and op-

timization theory. We conclude the chapter with a discussion about algorithms and compu-tation complexity.

In Chapter 3 we describe basic descent methods such as steepest descent method andNetwons method, which are fundamental algorithms in optimization theory.

The remainder of the report allocates one chapter for each optimization method consid-ered: conjugate gradient, simplex method, gradient projection, PSO and SAMCRA.

1.1 Quality of Service

QoS is one of the most debated topics in the areas of computer network engineering and

research. It is generally understood that a network that provides QoS has the ability to

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CHAPTER 1. INTRODUCTION

allocate resources for the purpose of implementing services better than best-effort. The majorsource of debate is on how to provide QoS in IP-based networks [Wan00, PD00].

At one extreme of the debate it is argued that no new mechanisms are required to provideQoS in the Internet, and simply increasing the amount of available bandwidth will suffice.

The people at the other extreme of the debate point out that it is doubtful that bandwidthover-provisioning alone takes care of QoS issues such as packet loss and delay. History hasshown that whenever bandwidth has been added to the networks, new killer applicationswere developed to use most of it. Furthermore, over-provisioning may not be an economicallyviable solution for developing countries and in the long run it may prove to be very expensiveeven for developed countries.

The first proposed QoS architectures to be used on top of IP is called Integrated Services(IntServ) [BCS94]. IntServ senders allocate resources along a path using the Resource Reser-vation Protocol (RSVP) [ZDE+93, Wro97]. IntServ performs per-flow resource management.This has led to skepticism towards IntServs ability to scale, since core routers in the In-ternet must handle several hundred thousands flows simultaneously [TMW97]. A newer re-port [FML+03] corroborates this number. However, the authors of the report argue thatper-flow management is feasible in these conditions due to advances in network processors,which allow over a million concurrent flows to be handled simultaneously.

A new architecture called Differentiated Services (DiffServ) [BBC+98] was developed, dueto concerns about IntServs scalability. DiffServ attempts to solve the scalability problemby dividing the traffic into separate forwarding classes. Each forwarding class is allocatedresources proportional to the service level expected by the customer (e.g., bronze, silver,gold and platinum). Packets are classified and mapped to a specific forwarding class at

the edge of the network. Inside the core, the routers handle the packets according to theirforwarding class. Since routers do not have to store state information for every flow, butonly have to peek at certain locations in the packet header, it is expected that DiffServ scalesmuch

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