Engineering optimization : theory and practice /: theory and practice. (2019)

• Record Type: Book
• Further Information: Note: Singiresu S. Rao.
• Authors: Rao, Singiresu S, 1944-
• Contents: Preface xvii Acknowledgment xxi About the Author xxiii About the Companion Website xxv 1 Introduction to Optimization 1 1.1 Introduction 1 1.2 Historical Development 3 1.2.1 Modern Methods of Optimization 4 1.3 Engineering Applications of Optimization 5 1.4 Statement of an Optimization Problem 6 1.4.1 Design Vector 6 1.4.2 Design Constraints 7 1.4.3 Constraint Surface 7 1.4.4 Objective Function 8 1.4.5 Objective Function Surfaces 9 1.5 Classification of Optimization Problems 14 1.5.1 Classification Based on the Existence of Constraints 14 1.5.2 Classification Based on the Nature of the Design Variables 14 1.5.3 Classification Based on the Physical Structure of the Problem 15 1.5.4 Classification Based on the Nature of the Equations Involved 18 1.5.5 Classification Based on the Permissible Values of the Design Variables 27 1.5.6 Classification Based on the Deterministic Nature of the Variables 28 1.5.7 Classification Based on the Separability of the Functions 29 1.5.8 Classification Based on the Number of Objective Functions 31 1.6 Optimization Techniques 33 1.7 Engineering Optimization Literature 34 1.8 Solutions Using MATLAB 34 References and Bibliography 34 Review Questions 40 Problems 41 2 Classical Optimization Techniques 57 2.1 Introduction 57 2.2 Single-Variable Optimization 57 2.3 Multivariable Optimization with no Constraints 62 2.3.1 Definition: r th Differential of f 62 2.3.2 Semidefinite Case 67 2.3.3 Saddle Point 67 2.4 Multivariable Optimization with Equality Constraints 69 2.4.1 Solution by Direct Substitution 69 2.4.2 Solution by the Method of Constrained Variation 71 2.4.3 Solution by the Method of Lagrange Multipliers 77 2.5 Multivariable Optimization with Inequality Constraints 85 2.5.1 Kuhn–Tucker Conditions 90 2.5.2 Constraint Qualification 90 2.6 Convex Programming Problem 96 References and Bibliography 96 Review Questions 97 Problems 98 3 Linear Programming I: Simplex Method 109 3.1 Introduction 109 3.2 Applications of Linear Programming 110 3.3 Standard form of a Linear Programming Problem 112 3.3.1 Scalar Form 112 3.3.2 Matrix Form 112 3.4 Geometry of Linear Programming Problems 114 3.5 Definitions and Theorems 117 3.5.1 Definitions 117 3.5.2 Theorems 120 3.6 Solution of a System of Linear Simultaneous Equations 122 3.7 Pivotal Reduction of a General System of Equations 123 3.8 Motivation of the Simplex Method 127 3.9 Simplex Algorithm 128 3.9.1 Identifying an Optimal Point 128 3.9.2 Improving a Nonoptimal Basic Feasible Solution 129 3.10 Two Phases of the Simplex Method 137 3.11 Solutions Using MATLAB 143 References and Bibliography 143 Review Questions 143 Problems 145 4 Linear Programming II: Additional Topics and Extensions 159 4.1 Introduction 159 4.2 Revised Simplex Method 159 4.3 Duality in Linear Programming 173 4.3.1 Symmetric Primal–Dual Relations 173 4.3.2 General Primal–Dual Relations 174 4.3.3 Primal–Dual Relations when the Primal Is in Standard Form 175 4.3.4 Duality Theorems 176 4.3.5 Dual Simplex Method 176 4.4 Decomposition Principle 180 4.5 Sensitivity or Postoptimality Analysis 187 4.5.1 Changes in the Right-Hand-Side Constants bi 188 4.5.2 Changes in the Cost Coefficients cj 192 4.5.3 Addition of New Variables 194 4.5.4 Changes in the Constraint Coefficients aij 195 4.5.5 Addition of Constraints 197 4.6 Transportation Problem 199 4.7 Karmarkar’s Interior Method 202 4.7.1 Statement of the Problem 203 4.7.2 Conversion of an LP Problem into the Required Form 203 4.7.3 Algorithm 205 4.8 Quadratic Programming 208 4.9 Solutions Using Matlab 214 References and Bibliography 214 Review Questions 215 Problems 216 5 Nonlinear Programming I: One-Dimensional Minimization Methods 225 5.1 Introduction 225 5.2 Unimodal Function 230 Elimination Methods 231 5.3 Unrestricted Search 231 5.3.1 Search with Fixed Step Size 231 5.3.2 Search with Accelerated Step Size 232 5.4 Exhaustive Search 232 5.5 Dichotomous Search 234 5.6 Interval Halving Method 236 5.7 Fibonacci Method 238 5.8 Golden Section Method 243 5.9 Comparison of Elimination Methods 246 Interpolation Methods 247 5.10 Quadratic Interpolation Method 248 5.11 Cubic Interpolation Method 253 5.12 Direct Root Methods 259 5.12.1 Newton Method 259 5.12.2 Quasi-Newton Method 261 5.12.3 Secant Method 263 5.13 Practical Considerations 265 5.13.1 How to Make the Methods Efficient and More Reliable 265 5.13.2 Implementation in Multivariable Optimization Problems 266 5.13.3 Comparison of Methods 266 5.14 Solutions Using MATLAB 267 References and Bibliography 267 Review Questions 267 Problems 268 6 Nonlinear Programming II: Unconstrained Optimization Techniques 273 6.1 Introduction 273 6.1.1 Classification of Unconstrained Minimization Methods 276 6.1.2 General Approach 276 6.1.3 Rate of Convergence 276 6.1.4 Scaling of Design Variables 277 Direct Search Methods 280 6.2 Random Search Methods 280 6.2.1 Random Jumping Method 280 6.2.2 Random Walk Method 282 6.2.3 Random Walk Method with Direction Exploitation 283 6.2.4 Advantages of Random Search Methods 284 6.3 Grid Search Method 285 6.4 Univariate Method 285 6.5 Pattern Directions 288 6.6 Powell’s Method 289 6.6.1 Conjugate Directions 289 6.6.2 Algorithm 293 6.7 Simplex Method 298 6.7.1 Reflection 298 6.7.2 Expansion 301 6.7.3 Contraction 301 Indirect Search (Descent) Methods 304 6.8 Gradient of a Function 304 6.8.1 Evaluation of the Gradient 306 6.8.2 Rate of Change of a Function Along a Direction 307 6.9 Steepest Descent (Cauchy) Method 308 6.10 Conjugate Gradient (Fletcher–Reeves) Method 310 6.10.1 Development of the Fletcher–Reeves Method 310 6.10.2 Fletcher–Reeves Method 311 6.11 Newton’s Method 313 6.12 Marquardt Method 316 6.13 Quasi-Newton Methods 317 6.13.1 Computation of [Bi ] 318 6.13.2 Rank 1 Updates 319 6.13.3 Rank 2 Updates 320 6.14 Davidon–Fletcher–Powell Method 321 6.15 Broyden–Fletcher–Goldfarb–Shanno Method 327 6.16 Test Functions 330 6.17 Solutions Using Matlab 332 References and Bibliography 333 Review Questions 334 Problems 336 7 Nonlinear Programming III: Constrained Optimization Techniques 347 7.1 Introduction 347 7.2 Characteristics of a Constrained Problem 347 Direct Methods 350 7.3 Random Search Methods 350 7.4 Complex Method 351 7.5 Sequential Linear Programming 353 7.6 Basic Approach in the Methods of Feasible Directions 360 7.7 Zoutendijk’s Method of
• Edition: Fifth edition
• Publisher Details: Hoboken, New Jersey : John Wiley & Sons, Inc
• Publication Date: 2019
• Extent: 1 online resource
• Subjects: 620.0042; Engineering design; Mathematical optimization
• Languages: English
• ISBNs: 9781119454793
• Related ISBNs: 9781119454762
• Notes: Note: Description based on CIP data; resource not viewed.
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.