Optimization techniques and applications with examples. (2018)

• Record Type: Book
• Further Information: Note: Xin-She Yang.
• Authors: Yang, Xin-She
• Contents: List of Figures xiii List of Tables xvii Preface xix Acknowledgements xxi Acronyms xxiii Introduction xxv Part I Fundamentals 1 1 Mathematical Foundations 3 1.1 Functions and Continuity 3 1.1.1 Functions 3 1.1.2 Continuity 4 1.1.3 Upper and Lower Bounds 4 1.2 Review of Calculus 6 1.2.1 Differentiation 6 1.2.2 Taylor Expansions 9 1.2.3 Partial Derivatives 12 1.2.4 Lipschitz Continuity 13 1.2.5 Integration 14 1.3 Vectors 16 1.3.1 Vector Algebra 17 1.3.2 Norms 17 1.3.3 2D Norms 19 1.4 Matrix Algebra 19 1.4.1 Matrices 19 1.4.2 Determinant 23 1.4.3 Rank of a Matrix 24 1.4.4 Frobenius Norm 25 1.5 Eigenvalues and Eigenvectors 25 1.5.1 Definiteness 28 1.5.2 Quadratic Form 29 1.6 Optimization and Optimality 31 1.6.1 Minimum and Maximum 31 1.6.2 Feasible Solution 32 1.6.3 Gradient and Hessian Matrix 32 1.6.4 Optimality Conditions 34 1.7 General Formulation of Optimization Problems 35 Exercises 36 Further Reading 36 2 Algorithms, Complexity, and Convexity 37 2.1 What Is an Algorithm? 37 2.2 Order Notations 39 2.3 Convergence Rate 40 2.4 Computational Complexity 42 2.4.1 Time and Space Complexity 42 2.4.2 Class P 43 2.4.3 Class NP 44 2.4.4 NP-Completeness 44 2.4.5 Complexity of Algorithms 45 2.5 Convexity 46 2.5.1 Linear and Affine Functions 46 2.5.2 Convex Functions 48 2.5.3 Subgradients 50 2.6 Stochastic Nature in Algorithms 51 2.6.1 Algorithms with Randomization 51 2.6.2 Random Variables 51 2.6.3 Poisson Distribution and Gaussian Distribution 54 2.6.4 Monte Carlo 56 2.6.5 Common Probability Distributions 58 Exercises 61 Bibliography 62 Part II Optimization Techniques and Algorithms 63 3 Optimization 65 3.1 Unconstrained Optimization 65 3.1.1 Univariate Functions 65 3.1.2 Multivariate Functions 68 3.2 Gradient-Based Methods 70 3.2.1 Newton’s Method 71 3.2.2 Convergence Analysis 72 3.2.3 Steepest Descent Method 73 3.2.4 Line Search 77 3.2.5 Conjugate Gradient Method 78 3.2.6 Stochastic Gradient Descent 79 3.2.7 Subgradient Method 81 3.3 Gradient-Free Nelder–Mead Method 81 3.3.1 A Simplex 81 3.3.2 Nelder–Mead Downhill Simplex Method 82 Exercises 84 Bibliography 84 4 Constrained Optimization 87 4.1 Mathematical Formulation 87 4.2 Lagrange Multipliers 87 4.3 Slack Variables 91 4.4 Generalized Reduced Gradient Method 94 4.5 KKT Conditions 97 4.6 PenaltyMethod 99 Exercises 101 Bibliography 101 5 Optimization Techniques: Approximation Methods 103 5.1 BFGS Method 103 5.2 Trust-Region Method 105 5.3 Sequential Quadratic Programming 107 5.3.1 Quadratic Programming 107 5.3.2 SQP Procedure 107 5.4 Convex Optimization 109 5.5 Equality Constrained Optimization 113 5.6 Barrier Functions 115 5.7 Interior-PointMethods 119 5.8 Stochastic and Robust Optimization 121 Exercises 123 Bibliography 123 Part III Applied Optimization 125 6 Linear Programming 127 6.1 Introduction 127 6.2 Simplex Method 129 6.2.1 Slack Variables 129 6.2.2 Standard Formulation 130 6.2.3 Duality 131 6.2.4 Augmented Form 132 6.3 Worked Example by Simplex Method 133 6.4 Interior-PointMethod for LP 136 Exercises 139 Bibliography 139 7 Integer Programming 141 7.1 Integer Linear Programming 141 7.1.1 Review of LP 141 7.1.2 Integer LP 142 7.2 LP Relaxation 143 7.3 Branch and Bound 146 7.3.1 How to Branch 153 7.4 Mixed Integer Programming 155 7.5 Applications of LP, IP, and MIP 156 7.5.1 Transport Problem 156 7.5.2 Product Portfolio 158 7.5.3 Scheduling 160 7.5.4 Knapsack Problem 161 7.5.5 Traveling Salesman Problem 161 Exercises 163 Bibliography 163 8 Regression and Regularization 165 8.1 Sample Mean and Variance 165 8.2 Regression Analysis 168 8.2.1 Maximum Likelihood 168 8.2.2 Regression 168 8.2.3 Linearization 173 8.2.4 Generalized Linear Regression 175 8.2.5 Goodness of Fit 178 8.3 Nonlinear Least Squares 179 8.3.1 Gauss–Newton Algorithm 180 8.3.2 Levenberg–Marquardt Algorithm 182 8.3.3 Weighted Least Squares 183 8.4 Over-fitting and Information Criteria 184 8.5 Regularization and Lasso Method 186 8.6 Logistic Regression 187 8.7 Principal Component Analysis 191 Exercises 195 Bibliography 196 9 Machine Learning Algorithms 199 9.1 Data Mining 199 9.1.1 Hierarchy Clustering 200 9.1.2 k-Means Clustering 201 9.1.3 Distance Metric 202 9.2 Data Mining for Big Data 202 9.2.1 Characteristics of Big Data 203 9.2.2 Statistical Nature of Big Data 203 9.2.3 Mining Big Data 204 9.3 Artificial Neural Networks 206 9.3.1 Neuron Model 207 9.3.2 Neural Networks 208 9.3.3 Back Propagation Algorithm 210 9.3.4 Loss Functions in ANN 212 9.3.5 Stochastic Gradient Descent 213 9.3.6 Restricted Boltzmann Machine 214 9.4 Support Vector Machines 216 9.4.1 Statistical Learning Theory 216 9.4.2 Linear Support Vector Machine 217 9.4.3 Kernel Functions and Nonlinear SVM 220 9.5 Deep Learning 221 9.5.1 Learning 221 9.5.2 Deep Neural Nets 222 9.5.3 Tuning of Hyper-Parameters 223 Exercises 223 Bibliography 224 10 Queueing Theory and Simulation 227 10.1 Introduction 227 10.1.1 Components of Queueing 227 10.1.2 Notations 228 10.2 Arrival Model 230 10.2.1 Poisson Distribution 230 10.2.2 Inter-arrival Time 233 10.3 Service Model 233 10.3.1 Exponential Distribution 233 10.3.2 Service Time Model 235 10.3.3 Erlang Distribution 235 10.4 Basic QueueingModel 236 10.4.1 M/M/1 Queue 236 10.4.2 M/M/s Queue 240 10.5 Little’s Law 242 10.6 Queue Management and Optimization 243 Exercises 245 Bibliography 246 Part IV Advanced Topics 249 11 Multiobjective Optimization 251 11.1 Introduction 251 11.2 Pareto Front and Pareto Optimality 253 11.3 Choice and Challenges 255 11.4 Transformation to Single Objective Optimization 256 11.4.1 Weighted Sum Method 256 11.4.2 Utility Function 259 11.5 The '�-Constraint Method 261 11.6 Evolutionary Approaches 264 11.6.1 Metaheuristics 264 11.6.2 Non-Dominated Sorting Genetic Algorithm 265 Exercises 266 Bibliography 266 12 Constraint-Handling Techniques 269 &
• Edition: 1st
• Publisher Details: Hoboken, New Jersey : John Wiley & Sons, Inc
• Publication Date: 2018
• Extent: 1 online resource
• Subjects: 519.6; Mathematical optimization
• Languages: English
• ISBNs: 9781119490623
• Related ISBNs: 9781119490609
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