# Nonlinear Optimization in Electrical Engineering with Applications in MATLAB?

## Hardcover

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## Overview

Nonlinear Optimization in Electrical Engineering with Applications in MATLAB® provides an introductory course on nonlinear optimization in electrical engineering, with a focus on applications such as the design of electric, microwave, and photonic circuits, wireless communications, and digital filter design.

Basic concepts are introduced using a step-by-step approach and illustrated with MATLAB® codes that the reader can use and adapt. Topics covered include:

• classical optimization methods
• one dimensional optimization
• unconstrained and constrained optimization
• global optimization
• space mapping optimization

Nonlinear Optimization in Electrical Engineering with Applications in MATLAB® is essential reading for advanced students in electrical engineering.

## Product Details

ISBN-13: 9781849195430 Institution of Engineering and Technology (IET) 11/28/2013 Computing and Networks Series 328 7.20(w) x 10.20(h) x 0.90(d)

Mohamed Bakr is an Associate Professor at the Department of Electrical and Computer Engineering, McMaster University, Canada, where his research interests include optimization methods, computeraided design and modelling of microwave circuits, neural networks applications, smart analysis of microwave circuits and efficient optimization using time domain simulation methods.

Preface xi

Acknowledgments xv

1 Mathematical background 1

1.1 Introduction 1

1.2 Vectors 1

1.3 Matrices 3

1.4 The solution of linear systems of equations 6

1.5 Derivatives 11

1.5.1 Derivative approximation 11

1.5.3 The Jacobian 14

1.5.4 Second-order derivatives 15

1.5.5 Derivatives of vectors and matrices 16

1.6 Subspaces 18

1.7 Convergence rates 20

1.8 Functions and sets 20

1.9 Solutions of systems of nonlinear equations 22

1.10 Optimization problem definition 25

References 25

Problems 25

2 An introduction to linear programming 29

2.1 Introduction 29

2.2 Examples of linear programs 29

2.2.1 A farming example 29

2.2.2 A production example 30

2.2.3 Power generation example 31

2.2.4 Wireless communication example 32

2.2.5 A battery charging example 32

2.3 Standard form of an LP 33

2.4 Optimality conditions 37

2.5 The matrix form 39

2.6 Canonical augmented form 40

2.7 Moving from one basic feasible solution to another 42

2.8 Cost reduction 45

2.9 The classical Simplex method 46

2.10 Starting the Simplex method 49

2.10.1 Endless pivoting 51

2.10.2 The big M approach 51

2.10.3 The two-phase Simplex 52

A2.1 Minimax optimization 55

A2.1.1 Minimax problem definition 55

A2.1.2 Minimax solution using linear programming 57

A2.1.3 A microwave filter example 59

A2.1.4 The design of coupled microcavities optical filter 61

References 65

Problems 65

3 Classical optimization 69

3.1 Introduction 69

3.2 Single-variable Taylor expansion 69

3.3 Multidimensional Taylor expansion 71

3.4 Meaning of the gradient 73

3.5 Optimality conditions 76

3.6 Unconstrained optimization 76

3.7 Optimization with equality constraints 78

3.7.1 Method of direct substitution 79

3.7.2 Method of constrained variation 80

3.8 Lagrange multipliers 84

3.9 Optimization with inequality constraints 86

3.10 Optimization with mixed constraints 92

References 99

Problems 99

4 One-dimensional optimization-Line search 101

4.1 Introduction 101

4.2 Bracketing approaches 102

4.2.1 Fixed line search 103

4.2.2 Accelerated line search 104

4.3 Derivative-free line search 105

4.3.1 Dichotomous line search 105

4.3.2 The interval-halving method 106

4.3.3 The Fibonacci search 108

4.3.4 The Golden Section method 111

4.4 Interpolation approaches 112

4.4.2 Cubic interpolation 116

4.5 Derivative-based approaches 119

4.5.1 The classical Newton method 119

4.5.2 A quasi-Newton method 121

4.5.3 The Secant method 122

4.6 Inexact line search 123

A4.1 Tuning of electric circuits 124

A4.1.1 Tuning of a current source 125

A4.1.2 Coupling of nanowires 127

A4.1.3 Matching of microwave antennas 128

References 129

Problems 130

5 Derivative-free unconstrained techniques 131

5.1 Why unconstrained optimization? 131

5.2 Classification of mi constrained optimization techniques 131

5.3 The random jump technique 132

5.4 The random walk method 133

5.5 Grid search method 134

5.6 The univariate method 135

5.7 The pattern search method 137

5.8 The Simplex method 140

5.9 Response surface approximation 143

A5.1 Electrical application: impedance transformers 146

A5.2 Electrical application: the design of photonic devices 149

References 151

Problems 152

6 First-order unconstrained optimization techniques 153

6.1 Introduction 153

6.2 The steepest descent method 153

6.3 The conjugate directions method 156

6.3.1 Definition of conjugacy 157

6.3.2 Powell's method of conjugate directions 158

A6.1 Solution of large systems of linear equations 164

A6.2 The design of digital FIR filters 169

References 173

Problems 173

7 Second-order unconstrained optimization techniques 175

7.1 Introduction 175

7.2 Newton's method 175

7.3 The Levenberg-Marquardt method 178

7.4 Quasi-Newton methods 179

7.4.1 Broyden's rank-1 update 180

7.4.2 The Davidon-Fletcher-Powell (DFP) formula 182

7.4.3 The Broyden-Fletcher-Goldfarb-Shanno method 184

7.4.4 The Gauss-Newton method 185

A7.1 Wireless channel characterization 188

A7.2 The parameter extraction problem 189

A7.3 Artificial neural networks training 193

References 201

Problems 201

8 Constrained optimization techniques 203

8.1 Introduction 203

8.2 Problem definition 203

8.3 Possible optimization scenarios 204

8.4 A random search method 206

8.5 Finding a feasible starting point 208

8.6 The Complex method 210

8.7 Sequential linear programming 212

8.8 Method of feasible directions 215

8.9 Rosen's projection method 218

8.10 Barrier and penalty methods 221

A8.1 Electrical engineering application; analog filter design 224

A8.2 Spectroscopy 227

References 230

Problems 231

9 Introduction to global optimization techniques 233

9.1 Introduction 233

9.2 Statistical optimization 233

9.3 Nature-inspired global techniques 236

9.3.1 Simulated annealing 237

9.3.2 Genetic algorithms 240

9.3.3 Particle swarm optimization 246

A9.1 Least ph optimization of filters 247

A9.2 Pattern recognition 252

References 257

Problems 258

10.1 Introduction 261

10.2 Tellegen's theorem 262

10.4 Adjoint sensitivity analysts of a linear system of equations 278

10.5 Time-domain adjoint sensitivity analysis 282

A10.1 Sensitivity analysis of high-frequency structures 295

References 298

Problems 299

Index 303