Nonlinear optimization: models and applications
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Boca Raton
Chapman & Hall/CRC
2020
Boca Raton CRC Press 2021 |
| Edition: | 1. ed. |
| Series: | Textbooks in mathematics
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| Physical Description: | xxi, 394 Seiten illustrations (black and white) 24 cm |
| ISBN: | 9780367444150 |
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| adam_text | Contents Preface: Nonlinear Optimization—Models and Applications........................xiii Acknowledgments...................................................................................................... xix Author........................................................................................................................... xxi 1 Introduction to Optimization Models........................................................... 1 1.1 Introduction....................................................................... 1 1.1.1 History...........................................................................................2 1.1.2 Applications of Optimization.............................. 3 1.1.3 Modeling.......................................................................................3 1.2 Classifying Optimization Problems..........................................................4 1.3 Review of Mathematical Programming with Excel Technology.........8 1.3.1 Excel Using the Solver................................................................10 1.3.2 Examples for Integer, Mixed-Integer, and Nonlinear Optimization...................................................20 1.4 Exercises.................................................................................................... 25 1.5 Review of the Simplex Method in Excel UsingRevised Simplex.......26 1.5.1 Steps of the Simplex Method.................................................... 29 References and Suggested Further Reading...................................................... 33 2 Review of Differential
Calculus.................................................................... 35 2.1 Limits.........................................................................................................35 2.2 Continuity.................................................................................................39 2.3 Differentiation..........................................................................................40 2.3.1 Increasing and Decreasing Functions..................................... 43 2.3.2 Higher Derivatives.....................................................................43 2.4 Convex and Concave Functions............................................................43 Exercises................................................................................................................ 48 References and Suggested Reading....................................................................49 3 Single-Variable Unconstrained Optimization............................................ 51 3.1 Introduction.............................................................................................. 51 3.2 Single-Variable Optimization and Basic Theory.................................. 52 3.3 Basic Applications of Max-Min Theory................................................54 Exercises................................................................................................................ 57 VII
viii ■ Contents 3.4 Applied Single-Variable Optimization Models.................................. 59 Exercises..........................................................................................................65 Projects...........................................................................................................68 References and Suggested Reading................................................................69 4 Numerical Search Techniques in Single-VariableOptimization........... 71 4.1 Single-Variable Techniques.................................................................71 4.1.1 Unrestricted Search................................................................73 4.1.2 Exhaust íve S earch..................................................................74 4.1.3 Dichotomous Search..............................................................74 4.1.4 Golden Section Search...........................................................76 4.1.5 Finding the Maximum of a Function over an Interval with Golden Section.......................................... 77 4.1.6 Golden Section Search with Technology...............................79 4.1.6.1 Excel Golden Search..............................................79 4.1.6.2 Maple Golden Search.............................................80 4.1.6.3 MATLAB Golden Search..................................... 82 4.1.7 Illustrious Examples with Technology.................................. 85 4.1.8 Fibonacci’s Search..................................................................88 4.1.8.1 Finding the Maximum of a
Function over an Interval with the Fibonacci Method................ 88 4.2 Interpolation with Derivatives: Newton’s Method............................ 91 4.2.1 Finding the Critical Points (Roots) of a Function............... 91 4.2.2 The Basic Application........................................................... 92 4.2.3 Newton’s Method to Find Critical Points with Technology....................................................................94 4.2.4 Excel: Newton’s Method....................................................... 94 4.2.5 Maple: Newton’s Method...................................................... 94 4.2.6 Newton’s Method for Critical Points with MATLAB.........96 4.2.7 The Bisection Method with Derivatives...............................98 Exercises........................................................................................................ 100 Projects......................................................................................................... 100 References and Suggested Further Readings................................................ 101 5 Review of Multivariable Differential Calculus...................................... 103 5.1 Introduction: Basic Theory and Partial Differentiation....................103 5.2 Directional Derivatives and the Gradient........................................109 Exercises........................................................................................................ 113 References and Suggested Reading.............................................................. 114 6 Models Using Unconstrained
Optimization: Maximization and Minimization with Several Variables......................................................115 6.1 Introduction....................................................................................... 115 6.2 The Hessian Matrix........................................................................... 117
Contents ■ ix 6.3 Unconstrained Optimization............................................................128 Exercises........................................................................................................136 6.4 Eigenvalues........................................................................................ 139 Exercises........................................................................................................140 Reference and Further Suggested Reading...................................................141 7 Multivariate Optimization Search Techniques..................................... 143 7.1 Introduction.......................................................................................143 7.2 Gradient Search Methods................................................................. 143 7.3 Examples of Gradient Search............................................................ 151 7.4 Modified Newton’s Method............................................................. 158 7.4.1 Modified Newton with Technology....................................162 Exercises........................................................................................................166 7.5 Comparisons of Methods................................................................. 166 7-5.1 Maple Code for Steepest Ascent Method (See Fox and Richardson)................................................... 166 7.5.2 Newton’s Method for Optimization in Maple....................168
Exercises........................................................................................................ 170 Projects Chapter 7........................................................................................ 171 References and Suggested Reading.............................................................. 171 8 Optimization with Equality Constraints............................................... 173 8.1 Introduction....................................................................................... 173 8.2 Equality Constraints Method of Lagrange Multipliers.....................173 8.3 Introduction and Basic Theory......................................................... 174 8.4 Graphical Interpretation of Lagrange Multipliers............................176 8.5 Computational Method of Lagrange Multipliers ........................... 178 Lagrange Method with Technology...................................................180 8.6 Applications with Lagrange Multipliers........................................... 188 Exercises........................................................................................................ 191 Projects......................................................................................................... 193 References and Suggested Reading.............................................................. 194 9 Inequality Constraints: Necessary/ Sufficient Kuhn—Tucker Conditions (KTC)...........................................195 9.1 Introduction to
KTC.........................................................................195 9.2 Basic Theory of Constrained Optimization.....................................196 9.2.1 Necessary and Sufficient Conditions...................................197 9.3 Geometric Interpretation of KTC....................................................200 9.3.1 Spanning Cones (Optional)................................................200 9.4 Computational KTC with Maple....................................................204 9.5 Modeling and Application with KTC.............................................. 218 Exercises........................................................................................................225
x ■ Contents Project...........................................................................................................228 Manufacturing.................................................................................. 228 References and Suggested Reading ............................................................ 229 10 Specialized Nonlinear Optimization Methods............. 231 10.1 Introduction......................................................................................231 10.1.1 Numerical and Heuristic Methods......................................231 10.1.2 Technology.......................................................................... 234 10.2 Method of Feasible Directions......................................................... 234 Exercises....................................................................................................... 238 10.3 Quadratic Programming................................ 239 Exercises....................................................................................................... 246 10.4 Separable Programming...................................................................247 10.4.1 Adjacency Assumptions......................................................248 10.4.2 Linearization Property........................................................ 248 Exercises........................................................................................................257 References and Suggested Reading..............................................................257 11 Dynamic
Programming........................................................................... 259 11.1 Introduction: Basic Concepts and Theory........................................259 11.1.1 Characteristics of Dynamic Programming.........................261 11.1.2 Working Backwards............................................................ 261 11.2 Continuous DP................................................................................ 262 11.3 Modeling and Applications of Continuous DP...............................264 Exercises....................................................................................................... 266 11.4 Models of Discrete Dynamic Programming................................... 267 11.5 Modeling and Applications of Discrete DP.................................... 270 Exercises........................................................................................................276 References and Suggested Readings.................. 278 12 Data Analysis with Regression Models, Advanced Regression Models, and Machine Learning through Optimization...................... 279 12.1 Introduction and Machine Learning...............................................279 12.1.1 Machine Learning............................................................... 280 12.1.1.1 Data Cleaning and Breakdown...........................281 12.1.1.2 Engineering.......................................................... 282 12.1.1.3 Model Fitting....................................................... 282 12.2 The Different Curve Fitting
Criterion.............................................282 12.2.1 Fitting Criterion 1: Least Squares....................................... 282 12.2.2 Fitting Criterion 2: Minimize the Sum of the Absolute Deviations............................................................ 284 12.2.3 Fitting Criterion 3: Chebyshev’s Criterion or Minimize the Largest Error............................ 285 Exercises........................................................................................................285
Contents ■ xi 12.3 Introduction to Simple Linear andPolynomial Regression..............287 12.3.1 Excel..........................................................................................288 12.3.2 Regression in Maple................................................................ 289 12.3.3 MATLAB.................................................................................290 Exercises.............................................................................................................. 291 12.4 Diagnostics in Regression..................................................................... 291 12.4.1 Example for the Common Sense Test..................................294 12.4.1.1 Exponential Decay Example.................................294 12.4.2 Multiple Linear Regression.................................................... 296 Exercises..............................................................................................................296 12.5 Nonlinear Regression through Optimization................................... 296 12.5.1 Exponential Regression...........................................................297 12.5.1.1 Newton-Raphson Algorithm...............................298 12.5.2 Sine Regression Using Optimization.................................... 307 12.5.3 Illustrative Examples................................................................312 12.5.3.1 Nonlinear Regression (Exponential Decay)........312 Exercises.............................................................................................................. 322 12.6 One-
Predictor Logistic and One-Predictor Poisson Regression Models................................................................................. 322 12.6.1 Logistic Regression and Poisson Regression with Technology...................................................................... 323 12.6.1.1 Logistic Regression with Technology.................. 323 12.6.1.2 Simple PoissonRegression with Technology.......330 12.6.2 Logistic Regression Illustrious Examples..............................335 12.6.3 Poisson Regression Discussion and Examples...................... 337 12.6.3.1 Normality Assumption Lost.................................. 338 12.6.3.2 Estimates of Regression Coefficients................... 342 12.6.4 Illustrative Poisson Regression Examples..............................343 12.6.4.1 Maple........................................................................ 343 Exercises............................................................................................................. 354 Projects............................................................................................................... 358 12.7 Conclusions and Summary...................................................................359 References and Suggested Reading..................................................................359 Answers to Selected Problems........................................................................... 361 Index.................................................................................................................... 389
Contents Preface: Nonlinear Optimization—Models and Applications........................xiii Acknowledgments...................................................................................................... xix Author........................................................................................................................... xxi 1 Introduction to Optimization Models........................................................... 1 1.1 Introduction....................................................................... 1 1.1.1 History...........................................................................................2 1.1.2 Applications of Optimization.............................. 3 1.1.3 Modeling.......................................................................................3 1.2 Classifying Optimization Problems..........................................................4 1.3 Review of Mathematical Programming with Excel Technology.........8 1.3.1 Excel Using the Solver................................................................10 1.3.2 Examples for Integer, Mixed-Integer, and Nonlinear Optimization...................................................20 1.4 Exercises.................................................................................................... 25 1.5 Review of the Simplex Method in Excel UsingRevised Simplex.......26 1.5.1 Steps of the Simplex Method.................................................... 29 References and Suggested Further Reading...................................................... 33 2 Review of Differential
Calculus.................................................................... 35 2.1 Limits.........................................................................................................35 2.2 Continuity.................................................................................................39 2.3 Differentiation..........................................................................................40 2.3.1 Increasing and Decreasing Functions..................................... 43 2.3.2 Higher Derivatives.....................................................................43 2.4 Convex and Concave Functions............................................................43 Exercises................................................................................................................ 48 References and Suggested Reading....................................................................49 3 Single-Variable Unconstrained Optimization............................................ 51 3.1 Introduction.............................................................................................. 51 3.2 Single-Variable Optimization and Basic Theory.................................. 52 3.3 Basic Applications of Max-Min Theory................................................54 Exercises................................................................................................................ 57 VII
viii ■ Contents 3.4 Applied Single-Variable Optimization Models.................................. 59 Exercises..........................................................................................................65 Projects...........................................................................................................68 References and Suggested Reading................................................................69 4 Numerical Search Techniques in Single-VariableOptimization........... 71 4.1 Single-Variable Techniques.................................................................71 4.1.1 Unrestricted Search................................................................73 4.1.2 Exhaust íve S earch..................................................................74 4.1.3 Dichotomous Search..............................................................74 4.1.4 Golden Section Search...........................................................76 4.1.5 Finding the Maximum of a Function over an Interval with Golden Section.......................................... 77 4.1.6 Golden Section Search with Technology...............................79 4.1.6.1 Excel Golden Search..............................................79 4.1.6.2 Maple Golden Search.............................................80 4.1.6.3 MATLAB Golden Search..................................... 82 4.1.7 Illustrious Examples with Technology.................................. 85 4.1.8 Fibonacci’s Search..................................................................88 4.1.8.1 Finding the Maximum of a
Function over an Interval with the Fibonacci Method................ 88 4.2 Interpolation with Derivatives: Newton’s Method............................ 91 4.2.1 Finding the Critical Points (Roots) of a Function............... 91 4.2.2 The Basic Application........................................................... 92 4.2.3 Newton’s Method to Find Critical Points with Technology....................................................................94 4.2.4 Excel: Newton’s Method....................................................... 94 4.2.5 Maple: Newton’s Method...................................................... 94 4.2.6 Newton’s Method for Critical Points with MATLAB.........96 4.2.7 The Bisection Method with Derivatives...............................98 Exercises........................................................................................................ 100 Projects......................................................................................................... 100 References and Suggested Further Readings................................................ 101 5 Review of Multivariable Differential Calculus...................................... 103 5.1 Introduction: Basic Theory and Partial Differentiation....................103 5.2 Directional Derivatives and the Gradient........................................109 Exercises........................................................................................................ 113 References and Suggested Reading.............................................................. 114 6 Models Using Unconstrained
Optimization: Maximization and Minimization with Several Variables......................................................115 6.1 Introduction....................................................................................... 115 6.2 The Hessian Matrix........................................................................... 117
Contents ■ ix 6.3 Unconstrained Optimization............................................................128 Exercises........................................................................................................136 6.4 Eigenvalues........................................................................................ 139 Exercises........................................................................................................140 Reference and Further Suggested Reading...................................................141 7 Multivariate Optimization Search Techniques..................................... 143 7.1 Introduction.......................................................................................143 7.2 Gradient Search Methods................................................................. 143 7.3 Examples of Gradient Search............................................................ 151 7.4 Modified Newton’s Method............................................................. 158 7.4.1 Modified Newton with Technology....................................162 Exercises........................................................................................................166 7.5 Comparisons of Methods................................................................. 166 7-5.1 Maple Code for Steepest Ascent Method (See Fox and Richardson)................................................... 166 7.5.2 Newton’s Method for Optimization in Maple....................168
Exercises........................................................................................................ 170 Projects Chapter 7........................................................................................ 171 References and Suggested Reading.............................................................. 171 8 Optimization with Equality Constraints............................................... 173 8.1 Introduction....................................................................................... 173 8.2 Equality Constraints Method of Lagrange Multipliers.....................173 8.3 Introduction and Basic Theory......................................................... 174 8.4 Graphical Interpretation of Lagrange Multipliers............................176 8.5 Computational Method of Lagrange Multipliers ........................... 178 Lagrange Method with Technology...................................................180 8.6 Applications with Lagrange Multipliers........................................... 188 Exercises........................................................................................................ 191 Projects......................................................................................................... 193 References and Suggested Reading.............................................................. 194 9 Inequality Constraints: Necessary/ Sufficient Kuhn—Tucker Conditions (KTC)...........................................195 9.1 Introduction to
KTC.........................................................................195 9.2 Basic Theory of Constrained Optimization.....................................196 9.2.1 Necessary and Sufficient Conditions...................................197 9.3 Geometric Interpretation of KTC....................................................200 9.3.1 Spanning Cones (Optional)................................................200 9.4 Computational KTC with Maple....................................................204 9.5 Modeling and Application with KTC.............................................. 218 Exercises........................................................................................................225
x ■ Contents Project...........................................................................................................228 Manufacturing.................................................................................. 228 References and Suggested Reading ............................................................ 229 10 Specialized Nonlinear Optimization Methods............. 231 10.1 Introduction......................................................................................231 10.1.1 Numerical and Heuristic Methods......................................231 10.1.2 Technology.......................................................................... 234 10.2 Method of Feasible Directions......................................................... 234 Exercises....................................................................................................... 238 10.3 Quadratic Programming................................ 239 Exercises....................................................................................................... 246 10.4 Separable Programming...................................................................247 10.4.1 Adjacency Assumptions......................................................248 10.4.2 Linearization Property........................................................ 248 Exercises........................................................................................................257 References and Suggested Reading..............................................................257 11 Dynamic
Programming........................................................................... 259 11.1 Introduction: Basic Concepts and Theory........................................259 11.1.1 Characteristics of Dynamic Programming.........................261 11.1.2 Working Backwards............................................................ 261 11.2 Continuous DP................................................................................ 262 11.3 Modeling and Applications of Continuous DP...............................264 Exercises....................................................................................................... 266 11.4 Models of Discrete Dynamic Programming................................... 267 11.5 Modeling and Applications of Discrete DP.................................... 270 Exercises........................................................................................................276 References and Suggested Readings.................. 278 12 Data Analysis with Regression Models, Advanced Regression Models, and Machine Learning through Optimization...................... 279 12.1 Introduction and Machine Learning...............................................279 12.1.1 Machine Learning............................................................... 280 12.1.1.1 Data Cleaning and Breakdown...........................281 12.1.1.2 Engineering.......................................................... 282 12.1.1.3 Model Fitting....................................................... 282 12.2 The Different Curve Fitting
Criterion.............................................282 12.2.1 Fitting Criterion 1: Least Squares....................................... 282 12.2.2 Fitting Criterion 2: Minimize the Sum of the Absolute Deviations............................................................ 284 12.2.3 Fitting Criterion 3: Chebyshev’s Criterion or Minimize the Largest Error............................ 285 Exercises........................................................................................................285
Contents ■ xi 12.3 Introduction to Simple Linear andPolynomial Regression..............287 12.3.1 Excel..........................................................................................288 12.3.2 Regression in Maple................................................................ 289 12.3.3 MATLAB.................................................................................290 Exercises.............................................................................................................. 291 12.4 Diagnostics in Regression..................................................................... 291 12.4.1 Example for the Common Sense Test..................................294 12.4.1.1 Exponential Decay Example.................................294 12.4.2 Multiple Linear Regression.................................................... 296 Exercises..............................................................................................................296 12.5 Nonlinear Regression through Optimization................................... 296 12.5.1 Exponential Regression...........................................................297 12.5.1.1 Newton-Raphson Algorithm...............................298 12.5.2 Sine Regression Using Optimization.................................... 307 12.5.3 Illustrative Examples................................................................312 12.5.3.1 Nonlinear Regression (Exponential Decay)........312 Exercises.............................................................................................................. 322 12.6 One-
Predictor Logistic and One-Predictor Poisson Regression Models................................................................................. 322 12.6.1 Logistic Regression and Poisson Regression with Technology...................................................................... 323 12.6.1.1 Logistic Regression with Technology.................. 323 12.6.1.2 Simple PoissonRegression with Technology.......330 12.6.2 Logistic Regression Illustrious Examples..............................335 12.6.3 Poisson Regression Discussion and Examples...................... 337 12.6.3.1 Normality Assumption Lost.................................. 338 12.6.3.2 Estimates of Regression Coefficients................... 342 12.6.4 Illustrative Poisson Regression Examples..............................343 12.6.4.1 Maple........................................................................ 343 Exercises............................................................................................................. 354 Projects............................................................................................................... 358 12.7 Conclusions and Summary...................................................................359 References and Suggested Reading..................................................................359 Answers to Selected Problems........................................................................... 361 Index.................................................................................................................... 389
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Contents Preface: Nonlinear Optimization—Models and Applications.xiii Acknowledgments. xix Author. xxi 1 Introduction to Optimization Models. 1 1.1 Introduction. 1 1.1.1 History.2 1.1.2 Applications of Optimization. 3 1.1.3 Modeling.3 1.2 Classifying Optimization Problems.4 1.3 Review of Mathematical Programming with Excel Technology.8 1.3.1 Excel Using the Solver.10 1.3.2 Examples for Integer, Mixed-Integer, and Nonlinear Optimization.20 1.4 Exercises. 25 1.5 Review of the Simplex Method in Excel UsingRevised Simplex.26 1.5.1 Steps of the Simplex Method. 29 References and Suggested Further Reading. 33 2 Review of Differential
Calculus. 35 2.1 Limits.35 2.2 Continuity.39 2.3 Differentiation.40 2.3.1 Increasing and Decreasing Functions. 43 2.3.2 Higher Derivatives.43 2.4 Convex and Concave Functions.43 Exercises. 48 References and Suggested Reading.49 3 Single-Variable Unconstrained Optimization. 51 3.1 Introduction. 51 3.2 Single-Variable Optimization and Basic Theory. 52 3.3 Basic Applications of Max-Min Theory.54 Exercises. 57 VII
viii ■ Contents 3.4 Applied Single-Variable Optimization Models. 59 Exercises.65 Projects.68 References and Suggested Reading.69 4 Numerical Search Techniques in Single-VariableOptimization. 71 4.1 Single-Variable Techniques.71 4.1.1 Unrestricted Search.73 4.1.2 Exhaust íve S earch.74 4.1.3 Dichotomous Search.74 4.1.4 Golden Section Search.76 4.1.5 Finding the Maximum of a Function over an Interval with Golden Section. 77 4.1.6 Golden Section Search with Technology.79 4.1.6.1 Excel Golden Search.79 4.1.6.2 Maple Golden Search.80 4.1.6.3 MATLAB Golden Search. 82 4.1.7 Illustrious Examples with Technology. 85 4.1.8 Fibonacci’s Search.88 4.1.8.1 Finding the Maximum of a
Function over an Interval with the Fibonacci Method. 88 4.2 Interpolation with Derivatives: Newton’s Method. 91 4.2.1 Finding the Critical Points (Roots) of a Function. 91 4.2.2 The Basic Application. 92 4.2.3 Newton’s Method to Find Critical Points with Technology.94 4.2.4 Excel: Newton’s Method. 94 4.2.5 Maple: Newton’s Method. 94 4.2.6 Newton’s Method for Critical Points with MATLAB.96 4.2.7 The Bisection Method with Derivatives.98 Exercises. 100 Projects. 100 References and Suggested Further Readings. 101 5 Review of Multivariable Differential Calculus. 103 5.1 Introduction: Basic Theory and Partial Differentiation.103 5.2 Directional Derivatives and the Gradient.109 Exercises. 113 References and Suggested Reading. 114 6 Models Using Unconstrained
Optimization: Maximization and Minimization with Several Variables.115 6.1 Introduction. 115 6.2 The Hessian Matrix. 117
Contents ■ ix 6.3 Unconstrained Optimization.128 Exercises.136 6.4 Eigenvalues. 139 Exercises.140 Reference and Further Suggested Reading.141 7 Multivariate Optimization Search Techniques. 143 7.1 Introduction.143 7.2 Gradient Search Methods. 143 7.3 Examples of Gradient Search. 151 7.4 Modified Newton’s Method. 158 7.4.1 Modified Newton with Technology.162 Exercises.166 7.5 Comparisons of Methods. 166 7-5.1 Maple Code for Steepest Ascent Method (See Fox and Richardson). 166 7.5.2 Newton’s Method for Optimization in Maple.168
Exercises. 170 Projects Chapter 7. 171 References and Suggested Reading. 171 8 Optimization with Equality Constraints. 173 8.1 Introduction. 173 8.2 Equality Constraints Method of Lagrange Multipliers.173 8.3 Introduction and Basic Theory. 174 8.4 Graphical Interpretation of Lagrange Multipliers.176 8.5 Computational Method of Lagrange Multipliers . 178 Lagrange Method with Technology.180 8.6 Applications with Lagrange Multipliers. 188 Exercises. 191 Projects. 193 References and Suggested Reading. 194 9 Inequality Constraints: Necessary/ Sufficient Kuhn—Tucker Conditions (KTC).195 9.1 Introduction to
KTC.195 9.2 Basic Theory of Constrained Optimization.196 9.2.1 Necessary and Sufficient Conditions.197 9.3 Geometric Interpretation of KTC.200 9.3.1 Spanning Cones (Optional).200 9.4 Computational KTC with Maple.204 9.5 Modeling and Application with KTC. 218 Exercises.225
x ■ Contents Project.228 Manufacturing. 228 References and Suggested Reading . 229 10 Specialized Nonlinear Optimization Methods. 231 10.1 Introduction.231 10.1.1 Numerical and Heuristic Methods.231 10.1.2 Technology. 234 10.2 Method of Feasible Directions. 234 Exercises. 238 10.3 Quadratic Programming. 239 Exercises. 246 10.4 Separable Programming.247 10.4.1 Adjacency Assumptions.248 10.4.2 Linearization Property. 248 Exercises.257 References and Suggested Reading.257 11 Dynamic
Programming. 259 11.1 Introduction: Basic Concepts and Theory.259 11.1.1 Characteristics of Dynamic Programming.261 11.1.2 Working Backwards. 261 11.2 Continuous DP. 262 11.3 Modeling and Applications of Continuous DP.264 Exercises. 266 11.4 Models of Discrete Dynamic Programming. 267 11.5 Modeling and Applications of Discrete DP. 270 Exercises.276 References and Suggested Readings. 278 12 Data Analysis with Regression Models, Advanced Regression Models, and Machine Learning through Optimization. 279 12.1 Introduction and Machine Learning.279 12.1.1 Machine Learning. 280 12.1.1.1 Data Cleaning and Breakdown.281 12.1.1.2 Engineering. 282 12.1.1.3 Model Fitting. 282 12.2 The Different Curve Fitting
Criterion.282 12.2.1 Fitting Criterion 1: Least Squares. 282 12.2.2 Fitting Criterion 2: Minimize the Sum of the Absolute Deviations. 284 12.2.3 Fitting Criterion 3: Chebyshev’s Criterion or Minimize the Largest Error. 285 Exercises.285
Contents ■ xi 12.3 Introduction to Simple Linear andPolynomial Regression.287 12.3.1 Excel.288 12.3.2 Regression in Maple. 289 12.3.3 MATLAB.290 Exercises. 291 12.4 Diagnostics in Regression. 291 12.4.1 Example for the Common Sense Test.294 12.4.1.1 Exponential Decay Example.294 12.4.2 Multiple Linear Regression. 296 Exercises.296 12.5 Nonlinear Regression through Optimization. 296 12.5.1 Exponential Regression.297 12.5.1.1 Newton-Raphson Algorithm.298 12.5.2 Sine Regression Using Optimization. 307 12.5.3 Illustrative Examples.312 12.5.3.1 Nonlinear Regression (Exponential Decay).312 Exercises. 322 12.6 One-
Predictor Logistic and One-Predictor Poisson Regression Models. 322 12.6.1 Logistic Regression and Poisson Regression with Technology. 323 12.6.1.1 Logistic Regression with Technology. 323 12.6.1.2 Simple PoissonRegression with Technology.330 12.6.2 Logistic Regression Illustrious Examples.335 12.6.3 Poisson Regression Discussion and Examples. 337 12.6.3.1 Normality Assumption Lost. 338 12.6.3.2 Estimates of Regression Coefficients. 342 12.6.4 Illustrative Poisson Regression Examples.343 12.6.4.1 Maple. 343 Exercises. 354 Projects. 358 12.7 Conclusions and Summary.359 References and Suggested Reading.359 Answers to Selected Problems. 361 Index. 389
Contents Preface: Nonlinear Optimization—Models and Applications.xiii Acknowledgments. xix Author. xxi 1 Introduction to Optimization Models. 1 1.1 Introduction. 1 1.1.1 History.2 1.1.2 Applications of Optimization. 3 1.1.3 Modeling.3 1.2 Classifying Optimization Problems.4 1.3 Review of Mathematical Programming with Excel Technology.8 1.3.1 Excel Using the Solver.10 1.3.2 Examples for Integer, Mixed-Integer, and Nonlinear Optimization.20 1.4 Exercises. 25 1.5 Review of the Simplex Method in Excel UsingRevised Simplex.26 1.5.1 Steps of the Simplex Method. 29 References and Suggested Further Reading. 33 2 Review of Differential
Calculus. 35 2.1 Limits.35 2.2 Continuity.39 2.3 Differentiation.40 2.3.1 Increasing and Decreasing Functions. 43 2.3.2 Higher Derivatives.43 2.4 Convex and Concave Functions.43 Exercises. 48 References and Suggested Reading.49 3 Single-Variable Unconstrained Optimization. 51 3.1 Introduction. 51 3.2 Single-Variable Optimization and Basic Theory. 52 3.3 Basic Applications of Max-Min Theory.54 Exercises. 57 VII
viii ■ Contents 3.4 Applied Single-Variable Optimization Models. 59 Exercises.65 Projects.68 References and Suggested Reading.69 4 Numerical Search Techniques in Single-VariableOptimization. 71 4.1 Single-Variable Techniques.71 4.1.1 Unrestricted Search.73 4.1.2 Exhaust íve S earch.74 4.1.3 Dichotomous Search.74 4.1.4 Golden Section Search.76 4.1.5 Finding the Maximum of a Function over an Interval with Golden Section. 77 4.1.6 Golden Section Search with Technology.79 4.1.6.1 Excel Golden Search.79 4.1.6.2 Maple Golden Search.80 4.1.6.3 MATLAB Golden Search. 82 4.1.7 Illustrious Examples with Technology. 85 4.1.8 Fibonacci’s Search.88 4.1.8.1 Finding the Maximum of a
Function over an Interval with the Fibonacci Method. 88 4.2 Interpolation with Derivatives: Newton’s Method. 91 4.2.1 Finding the Critical Points (Roots) of a Function. 91 4.2.2 The Basic Application. 92 4.2.3 Newton’s Method to Find Critical Points with Technology.94 4.2.4 Excel: Newton’s Method. 94 4.2.5 Maple: Newton’s Method. 94 4.2.6 Newton’s Method for Critical Points with MATLAB.96 4.2.7 The Bisection Method with Derivatives.98 Exercises. 100 Projects. 100 References and Suggested Further Readings. 101 5 Review of Multivariable Differential Calculus. 103 5.1 Introduction: Basic Theory and Partial Differentiation.103 5.2 Directional Derivatives and the Gradient.109 Exercises. 113 References and Suggested Reading. 114 6 Models Using Unconstrained
Optimization: Maximization and Minimization with Several Variables.115 6.1 Introduction. 115 6.2 The Hessian Matrix. 117
Contents ■ ix 6.3 Unconstrained Optimization.128 Exercises.136 6.4 Eigenvalues. 139 Exercises.140 Reference and Further Suggested Reading.141 7 Multivariate Optimization Search Techniques. 143 7.1 Introduction.143 7.2 Gradient Search Methods. 143 7.3 Examples of Gradient Search. 151 7.4 Modified Newton’s Method. 158 7.4.1 Modified Newton with Technology.162 Exercises.166 7.5 Comparisons of Methods. 166 7-5.1 Maple Code for Steepest Ascent Method (See Fox and Richardson). 166 7.5.2 Newton’s Method for Optimization in Maple.168
Exercises. 170 Projects Chapter 7. 171 References and Suggested Reading. 171 8 Optimization with Equality Constraints. 173 8.1 Introduction. 173 8.2 Equality Constraints Method of Lagrange Multipliers.173 8.3 Introduction and Basic Theory. 174 8.4 Graphical Interpretation of Lagrange Multipliers.176 8.5 Computational Method of Lagrange Multipliers . 178 Lagrange Method with Technology.180 8.6 Applications with Lagrange Multipliers. 188 Exercises. 191 Projects. 193 References and Suggested Reading. 194 9 Inequality Constraints: Necessary/ Sufficient Kuhn—Tucker Conditions (KTC).195 9.1 Introduction to
KTC.195 9.2 Basic Theory of Constrained Optimization.196 9.2.1 Necessary and Sufficient Conditions.197 9.3 Geometric Interpretation of KTC.200 9.3.1 Spanning Cones (Optional).200 9.4 Computational KTC with Maple.204 9.5 Modeling and Application with KTC. 218 Exercises.225
x ■ Contents Project.228 Manufacturing. 228 References and Suggested Reading . 229 10 Specialized Nonlinear Optimization Methods. 231 10.1 Introduction.231 10.1.1 Numerical and Heuristic Methods.231 10.1.2 Technology. 234 10.2 Method of Feasible Directions. 234 Exercises. 238 10.3 Quadratic Programming. 239 Exercises. 246 10.4 Separable Programming.247 10.4.1 Adjacency Assumptions.248 10.4.2 Linearization Property. 248 Exercises.257 References and Suggested Reading.257 11 Dynamic
Programming. 259 11.1 Introduction: Basic Concepts and Theory.259 11.1.1 Characteristics of Dynamic Programming.261 11.1.2 Working Backwards. 261 11.2 Continuous DP. 262 11.3 Modeling and Applications of Continuous DP.264 Exercises. 266 11.4 Models of Discrete Dynamic Programming. 267 11.5 Modeling and Applications of Discrete DP. 270 Exercises.276 References and Suggested Readings. 278 12 Data Analysis with Regression Models, Advanced Regression Models, and Machine Learning through Optimization. 279 12.1 Introduction and Machine Learning.279 12.1.1 Machine Learning. 280 12.1.1.1 Data Cleaning and Breakdown.281 12.1.1.2 Engineering. 282 12.1.1.3 Model Fitting. 282 12.2 The Different Curve Fitting
Criterion.282 12.2.1 Fitting Criterion 1: Least Squares. 282 12.2.2 Fitting Criterion 2: Minimize the Sum of the Absolute Deviations. 284 12.2.3 Fitting Criterion 3: Chebyshev’s Criterion or Minimize the Largest Error. 285 Exercises.285
Contents ■ xi 12.3 Introduction to Simple Linear andPolynomial Regression.287 12.3.1 Excel.288 12.3.2 Regression in Maple. 289 12.3.3 MATLAB.290 Exercises. 291 12.4 Diagnostics in Regression. 291 12.4.1 Example for the Common Sense Test.294 12.4.1.1 Exponential Decay Example.294 12.4.2 Multiple Linear Regression. 296 Exercises.296 12.5 Nonlinear Regression through Optimization. 296 12.5.1 Exponential Regression.297 12.5.1.1 Newton-Raphson Algorithm.298 12.5.2 Sine Regression Using Optimization. 307 12.5.3 Illustrative Examples.312 12.5.3.1 Nonlinear Regression (Exponential Decay).312 Exercises. 322 12.6 One-
Predictor Logistic and One-Predictor Poisson Regression Models. 322 12.6.1 Logistic Regression and Poisson Regression with Technology. 323 12.6.1.1 Logistic Regression with Technology. 323 12.6.1.2 Simple PoissonRegression with Technology.330 12.6.2 Logistic Regression Illustrious Examples.335 12.6.3 Poisson Regression Discussion and Examples. 337 12.6.3.1 Normality Assumption Lost. 338 12.6.3.2 Estimates of Regression Coefficients. 342 12.6.4 Illustrative Poisson Regression Examples.343 12.6.4.1 Maple. 343 Exercises. 354 Projects. 358 12.7 Conclusions and Summary.359 References and Suggested Reading.359 Answers to Selected Problems. 361 Index. 389 |
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| discipline | Mathematik |
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| index_date | 2024-07-03T15:51:33Z |
| indexdate | 2024-07-10T08:59:23Z |
| institution | BVB |
| isbn | 9780367444150 |
| language | English |
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| spelling | Fox, William P. 1949- Verfasser (DE-588)1043166513 aut Nonlinear optimization models and applications William P. Fox 1. ed. 202010 Boca Raton Chapman & Hall/CRC 2020 Boca Raton CRC Press 2021 xxi, 394 Seiten illustrations (black and white) 24 cm txt rdacontent n rdamedia nc rdacarrier Textbooks in mathematics Nichtlineare Optimierung (DE-588)4128192-5 gnd rswk-swf Mathematical optimization Nonlinear theories Nichtlineare Optimierung (DE-588)4128192-5 s DE-604 9781000196962 ePub ebook 9781000196924 PDF ebook 9781000196948 Mobipocket ebook 9781003009573 ebook 9781000196962 ebook version 9781000196962 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032394443&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032394443&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fox, William P. 1949- Nonlinear optimization models and applications Nichtlineare Optimierung (DE-588)4128192-5 gnd |
| subject_GND | (DE-588)4128192-5 |
| title | Nonlinear optimization models and applications |
| title_auth | Nonlinear optimization models and applications |
| title_exact_search | Nonlinear optimization models and applications |
| title_exact_search_txtP | Nonlinear optimization models and applications |
| title_full | Nonlinear optimization models and applications William P. Fox |
| title_fullStr | Nonlinear optimization models and applications William P. Fox |
| title_full_unstemmed | Nonlinear optimization models and applications William P. Fox |
| title_short | Nonlinear optimization |
| title_sort | nonlinear optimization models and applications |
| title_sub | models and applications |
| topic | Nichtlineare Optimierung (DE-588)4128192-5 gnd |
| topic_facet | Nichtlineare Optimierung |
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