MATLAB Optimization Techniques (eBook)

César Perez Lopez is a Professor at the Department of Statistics and Operations Research at the University of Madrid. César Perez Lopez is also a Mathematician and Economist at the National Statistics Institute (INE) in Madrid, a body which belongs to the Superior Systems and Information Technology Department of the Spanish Government. César also currently works at the Institute for Fiscal Studies in Madrid.

Contents at a Glance 3
Contents 279
About the Author 283
Chapter 1: Introducing MATLAB and the MATLAB Working Environment 4
1.1 Introduction 4
1.1.1 Developing Algorithms and Applications 5
1.1.2 Data Access and Analysis 8
1.1.3 Data Visualization 9
1.1.4 Numerical Calculation 12
1.1.5 Publication of Results and Distribution of Applications 13
1.2 The MATLAB Working Environment 14
1.3 Help in MATLAB 19
Chapter 2: MATLAB Programming 25
2.1 MATLAB Programming 25
2.1.1 The Text Editor 25
2.1.2 Scripts 28
2.1.3 Functions and M-files. Eval and Feval 31
2.1.4 Local and Global Variables 34
2.1.5 Data Types 36
2.1.6 Flow Control: FOR, WHILE and IF ELSEIF Loops 37
FOR Loops 37
WHILE Loops 38
IF ELSEIF ELSE END Loops 39
SWITCH and CASE 41
CONTINUE 42
BREAK 42
TRY... CATCH 44
RETURN 44
2.1.7 Subfunctions 45
2.1.8 Commands in M-files 46
2.1.9 Functions Relating to Arrays of Cells 47
2.1.10 Multidimensional Array Functions 50
Chapter 3: Basic MATLAB Functions for Linear and Non-Linear Optimization 54
3.1 Solutions of Equations and Systems of Equations 54
3.2 Working with Polynomials 60
Chapter 4: Optimization by Numerical Methods: Solving Equations 68
4.1 Non-Linear Equations 68
4.1.1 The Fixed Point Method for Solving x = g(x) 68
4.1.2 Newton’s Method for Solving the Equation f(x) = 0 71
4.1.3 Schröder’s Method for Solving the Equation f(x) = 0 73
4.2 Systems of Non-Linear Equations 73
4.2.1 The Seidel Method 74
4.2.2 The Newton-Raphson Method 74
Chapter 5: Optimization Using Symbolic Computation 82
5.1 Symbolic Equations and Systems of Equations 82
Chapter 6: Optimization Techniques Via The Optimization Toolbox 86
6.1 The Optimization Toolbox 86
6.1.1 Standard Algorithms 86
6.1.2 Large Scale Algorithms 86
6.2 Minimization Algorithms 87
6.2.1 Multiobjective Problems 87
6.2.2 Non-Linear Scalar Minimization With Boundary Conditions 90
6.2.3 Non-Linear Minimization with Restrictions 90
6.2.4 Minimax Optimization: fminimax and fminuc 92
6.2.5 Minimax Optimization 93
6.2.6 Minimum Optimization: fminsearch and fminuc 94
6.2.7 Semi-Infinitely Constrained Minimization 94
6.2.8 Linear Programming 95
6.2.9 Quadratic programming 97
6.3 Equation Solving Algorithms 99
6.3.1 Solving Equations and Systems of Equations 99
6.4 Fitting Curves by Least Squares 101
6.4.1 Conditional Least Squares Problems 101
6.4.2 Non- Linear Least Squares Problems 101
6.4.3 Linear Non- Negative Least Squares Problems 102
Chapter 7: Differentiation in one and Several Variables. Applications to Optimization 110
7.1 Derivatives 110
7.2 Par tial Derivatives 112
7.3 Applications of Derivatives. Tangents, Asymptotes, Extreme Points and Turning Points 114
7.4 Differentiation of Functions of Several Variables 118
7.5 Maxima and Minima of Functions of Several Variables 123
7.6 Conditional Minima and Maxima. The Method of “Lagrange Multipliers” 131
7.7 Vector Differential Calculus 134
7.8 The Composite Function Theorem 135
7.9 The Implicit Function Theorem 136
7.10 The Inverse Function Theorem 137
7.11 The Change of Variables Theorem 139
7.12 Series Expansions in Several Variables 139
7.13 Vector Fields. Curl, Divergence and the Laplacian 140
Spherical, Cylindrical and Rectangular Coordinates 142
Chapter 8: Optimization of Functions of Complex Variables 165
8.1 Complex Numbers 165
8.2 General Functions of a Complex Variable 166
8.2.1 Trigonometric Functions of a Complex Variable 166
8.2.2 Hyperbolic Functions of a Complex Variable 167
8.2.3 Exponential and Logarithmic Functions of a Complex Variable 168
8.3 Specific Functions of a Complex Variable 169
8.4 Basic Functions with Complex Vector Arguments 170
8.5 Basic Functions with Complex Matrix Arguments 175
8.6 General Functions with Complex Matrix Arguments 181
8.6.1 Trigonometric Functions of a Complex Matrix Variable 181
8.6.2 Hyperbolic Functions of a Complex Matrix Variable 186
8.6.3 Exponential and Logarithmic Functions of a Complex Matrix Variable 190
8.6.4 Specific Functions of a Complex Matrix Variable 192
8.7 Matrix Operations with Real and Complex Variables 195
Chapter 9: Algebraic Expressions, Polynomials, Equations and Systems. Tools for Optimization 216
9.1 Expanding, Simplifying and Factoring Algebraic Expressions 216
9.2 Polynomials 219
9.3 Polynomial Interpolation 223
9.4 Solving Equations and Systems of Equations 231
9.4.1 General Methods 231
9.4.2 The Biconjugate Gradient Method 233
9.4.3 The Conjugate Gradients Method 236
9.4.4 The Residual Method 238
9.4.5 The Symmetric and Non-Negative Least Squares Method 241
9.5 Solving Linear Systems of Equations 243