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Continuous functions minimization by dynamic random search technique下载
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2019-08-02 11:00:18
dynamic random search technique
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Continu
ous
funct
ion
s
min
imizat
ion
by
dynamic
random
search
technique
dynamic
random
search
technique
Introduct
ion
_to_Optimum_Design.pdf
Preface ix Chapter 1 Introduct
ion
to Design 1 1.1 The Design Process 2 1.2 Engineering Design versus Engineering Analysis 4 1.3 Convent
ion
al versus Optimum Design Process 4 1.4 Optimum Design versus Optimal Control 6 1.5 Basic Ter
min
ology and Notat
ion
7 1.5.1 Sets and Points 7 1.5.2 Notat
ion
for Constraints 9 1.5.3 Superscripts/Subscripts and Summat
ion
Notat
ion
9 1.5.4 Norm/Length of a Vector 11 1.5.5
Funct
ion
s 11 1.5.6 U.S.-British versus SI Units 12 Chapter 2 Optimum Design Problem Formulat
ion
15 2.1 The Problem Formulat
ion
Process 16 2.1.1 Step 1: Project/Problem Statement 16 2.1.2 Step 2: Data and Informat
ion
Collect
ion
16 2.1.3 Step 3: Identificat
ion
/Definit
ion
of Design Variables 16 2.1.4 Step 4: Identificat
ion
of a Criter
ion
to Be Optimized 17 2.1.5 Step 5: Identificat
ion
of Constraints 17 2.2 Design of a Can 18 2.3 Insulated Spherical Tank Design 20 2.4 Saw Mill Operat
ion
22 2.5 Design of a Two-Bar Bracket 24 2.6 Design of a Cabinet 30 2.6.1 Formulat
ion
1 for Cabinet Design 30 2.6.2 Formulat
ion
2 for Cabinet Design 31 2.6.3 Formulat
ion
3 for Cabinet Design 31 xi 2.7
Min
imum Weight Tubular Column Design 32 2.7.1 Formulat
ion
1 for Column Design 33 2.7.2 Formulat
ion
2 for Column Design 34 2.8
Min
imum Cost Cylindrical Tank Design 35 2.9 Design of Coil Springs 36 2.10
Min
imum Weight Design of a Symmetric Three-Bar Truss 38 2.11 A General Mathematical Model for Optimum Design 41 2.11.1 Standard Design Opt
imizat
ion
Model 42 2.11.2 Max
imizat
ion
Problem Treatment 43 2.11.3 Treatment of “Greater Than Type” Constraints 43 2.11.4 Discrete and Integer Design Variables 44 2.11.5 Feasible Set 45 2.11.6 Active/Inactive/Violated Constraints 45 Exercises for Chapter 2 46 Chapter 3 Graphical Opt
imizat
ion
55 3.1 Graphical Solut
ion
Process 55 3.1.1 Profit Max
imizat
ion
Problem 55 3.1.2 Step-by-Step Graphical Solut
ion
Procedure 56 3.2 Use of Mathematica for Graphical Opt
imizat
ion
60 3.2.1 Plotting
Funct
ion
s 61 3.2.2 Identificat
ion
and Hatching of Infeasible Reg
ion
for an Inequality 62 3.2.3 Identificat
ion
of Feasible Reg
ion
62 3.2.4 Plotting of Objective
Funct
ion
Contours 63 3.2.5 Identificat
ion
of Optimum Solut
ion
63 3.3 Use of MATLAB for Graphical Opt
imizat
ion
64 3.3.1 Plotting of
Funct
ion
Contours 64 3.3.2 Editing of Graph 64 3.4 Design Problem with Multiple Solut
ion
s 66 3.5 Problem with Unbounded Solut
ion
66 3.6 Infeasible Problem 67 3.7 Graphical Solut
ion
for
Min
imum Weight Tubular Column 69 3.8 Graphical Solut
ion
for a Beam Design Problem 69 Exercises for Chapter 3 72 Chapter 4 Optimum Design Concepts 83 4.1 Definit
ion
s of Global and Local
Min
ima 84 4.1.1
Min
imum 84 4.1.2 Existence of
Min
imum 89 4.2 Review of Some Basic Calculus Concepts 89 4.2.1 Gradient Vector 90 4.2.2 Hessian Matrix 92 4.2.3 Taylor’s Expans
ion
93 4.2.4 Quadratic Forms and Definite Matrices 96 4.2.5 Concept of Necessary and Sufficient Condit
ion
s 102 xii Contents 4.3 Unconstrained Optimum Design Problems 103 4.3.1 Concepts Related to Optimality Condit
ion
s 103 4.3.2 Optimality Condit
ion
s for
Funct
ion
s of Single Variable 104 4.3.3 Optimality Condit
ion
s for
Funct
ion
s of Several Variables 109 4.3.4 Roots of Nonlinear Equat
ion
s Using Excel 116 4.4 Constrained Optimum Design Problems 119 4.4.1 Role of Constraints 119 4.4.2 Necessary Condit
ion
s: Equality Constraints 121 4.4.3 Necessary Condit
ion
s: Inequality Constraints— Karush-Kuhn-Tucker (KKT) Condit
ion
s 128 4.4.4 Solut
ion
of KKT Condit
ion
s Using Excel 140 4.4.5 Solut
ion
of KKT Condit
ion
s Using MATLAB 141 4.5 Postoptimality Analysis: Physical Meaning of Lagrange Multipliers 143 4.5.1 Effect of Changing Constraint Limits 143 4.5.2 Effect of Cost
Funct
ion
Scaling on Lagrange Multipliers 146 4.5.3 Effect of Scaling a Constraint on Its Lagrange Multiplier 147 4.5.4 Generalizat
ion
of Constraint Variat
ion
Sensitivity Result 148 4.6 Global Optimality 149 4.6.1 Convex Sets 149 4.6.2 Convex
Funct
ion
s 151 4.6.3 Convex Program
min
g Problem 153 4.6.4 Transformat
ion
of a Constraint 156 4.6.5 Sufficient Condit
ion
s for Convex Program
min
g Problems 157 4.7 Engineering Design Examples 158 4.7.1 Design of a Wall Bracket 158 4.7.2 Design of a Rectangular Beam 162 Exercises for Chapter 4 166 Chapter 5 More on Optimum Design Concepts 175 5.1 Alternate Form of KKT Necessary Condit
ion
s 175 5.2 Irregular Points 178 5.3 Second-Order Condit
ion
s for Constrained Opt
imizat
ion
179 5.4 Sufficiency Check for Rectangular Beam Design Problem 184 Exercises for Chapter 5 185 Chapter 6 Linear Program
min
g Methods for Optimum Design 191 6.1 Definit
ion
of a Standard Linear Program
min
g Problem 192 6.1.1 Linear Constraints 192 6.1.2 Unrestricted Variables 193 6.1.3 Standard LP Definit
ion
193 Contents xiii 6.2 Basic Concepts Related to Linear Program
min
g Problems 195 6.2.1 Basic Concepts 195 6.2.2 LP Ter
min
ology 198 6.2.3 Optimum Solut
ion
for LP Problems 201 6.3 Basic Ideas and Steps of the Simplex Method 201 6.3.1 The Simplex 202 6.3.2 Canonical Form/General Solut
ion
of Ax = b 202 6.3.3 Tableau 203 6.3.4 The Pivot Step 205 6.3.5 Basic Steps of the Simplex Method 206 6.3.6 Simplex Algorithm 211 6.4 Two-Phase Simplex Method—Artificial Variables 218 6.4.1 Artificial Variables 219 6.4.2 Artificial Cost
Funct
ion
219 6.4.3 Definit
ion
of Phase I Problem 220 6.4.4 Phase I Algorithm 220 6.4.5 Phase II Algorithm 221 6.4.6 Degenerate Basic Feasible Solut
ion
226 6.5 Postoptimality Analysis 228 6.5.1 Changes in Resource Limits 229 6.5.2 Ranging Right Side Parameters 235 6.5.3 Ranging Cost Coefficients 239 6.5.4 Changes in the Coefficient Matrix 241 6.6 Solut
ion
of LP Problems Using Excel Solver 243 Exercises for Chapter 6 246 Chapter 7 More on Linear Program
min
g Methods for Optimum Design 259 7.1 Derivat
ion
of the Simplex Method 259 7.1.1 Select
ion
of a Basic Variable That Should Become Nonbasic 259 7.1.2 Select
ion
of a Nonbasic Variable That Should Become Basic 260 7.2 Alternate Simplex Method 262 7.3 Duality in Linear Program
min
g 263 7.3.1 Standard Primal LP 263 7.3.2 Dual LP Problem 264 7.3.3 Treatment of Equality Constraints 265 7.3.4 Alternate Treatment of Equality Constraints 266 7.3.5 Deter
min
at
ion
of Primal Solut
ion
from Dual Solut
ion
267 7.3.6 Use of Dual Tableau to Recover Primal Solut
ion
271 7.3.7 Dual Variables as Lagrange Multipliers 273 Exercises for Chapter 7 275 Chapter 8 Numerical Methods for Unconstrained Optimum Design 277 8.1 General Concepts Related to Numerical Algorithms 278 8.1.1 A General Algorithm 279 8.1.2 Descent Direct
ion
and Descent Step 280 xiv Contents 8.1.3 Convergence of Algorithms 282 8.1.4 Rate of Convergence 282 8.2 Basic Ideas and Algorithms for Step Size Deter
min
at
ion
282 8.2.1 Definit
ion
of One-Dimens
ion
al
Min
imizat
ion
Subproblem 282 8.2.2 Analytical Method to Compute Step Size 283 8.2.3 Concepts Related to Numerical Methods to Compute Step Size 285 8.2.4 Equal Interval
Search
286 8.2.5 Alternate Equal Interval
Search
288 8.2.6 Golden Sect
ion
Search
289 8.3
Search
Direct
ion
Deter
min
at
ion
: Steepest Descent Method 293 8.4
Search
Direct
ion
Deter
min
at
ion
: Conjugate Gradient Method 296 Exercises for Chapter 8 300 Chapter 9 More on Numerical Methods for Unconstrained Optimum Design 305 9.1 More on Step Size Deter
min
at
ion
305 9.1.1 Polynomial Interpolat
ion
306 9.1.2 Inaccurate Line
Search
309 9.2 More on Steepest Descent Method 310 9.2.1 Properties of the Gradient Vector 310 9.2.2 Orthogonality of Steepest Descent Direct
ion
s 314 9.3 Scaling of Design Variables 315 9.4
Search
Direct
ion
Deter
min
at
ion
: Newton’s Method 318 9.4.1 Classical Newton’s Method 318 9.4.2 Modified Newton’s Method 319 9.4.3 Marquardt Modificat
ion
323 9.5
Search
Direct
ion
Deter
min
at
ion
: Quasi-Newton Methods 324 9.5.1 Inverse Hessian Updating: DFP Method 324 9.5.2 Direct Hessian Updating: BFGS Method 327 9.6 Engineering Applicat
ion
s of Unconstrained Methods 329 9.6.1
Min
imizat
ion
of Total Potential Energy 329 9.6.2 Solut
ion
of Nonlinear Equat
ion
s 331 9.7 Solut
ion
of Constrained Problems Using Unconstrained Opt
imizat
ion
Methods 332 9.7.1 Sequential Unconstrained
Min
imizat
ion
Technique
s 333 9.7.2 Multiplier (Augmented Lagrangian) Methods 334 Exercises for Chapter 9 335 Chapter 10 Numerical Methods for Constrained Optimum Design 339 10.1 Basic Concepts and Ideas 340 10.1.1 Basic Concepts Related to Algorithms for Constrained Problems 340 10.1.2 Constraint Status at a Design Point 342 10.1.3 Constraint Normalizat
ion
343 Contents xv 10.1.4 Descent
Funct
ion
345 10.1.5 Convergence of an Algorithm 345 10.2 Linearizat
ion
of Constrained Problem 346 10.3 Sequential Linear Program
min
g Algorithm 352 10.3.1 The Basic Idea—Move Limits 352 10.3.2 An SLP Algorithm 353 10.3.3 SLP Algorithm: Some Observat
ion
s 357 10.4 Quadratic Program
min
g Subproblem 358 10.4.1 Definit
ion
of QP Subproblem 358 10.4.2 Solut
ion
of QP Subproblem 361 10.5 Constrained Steepest Descent Method 363 10.5.1 Descent
Funct
ion
364 10.5.2 Step Size Deter
min
at
ion
366 10.5.3 CSD Algorithm 368 10.5.4 CSD Algorithm: Some Observat
ion
s 368 10.6 Engineering Design Opt
imizat
ion
Using Excel Solver 369 Exercises for Chapter 10 373 Chapter 11 More on Numerical Methods for Constrained Optimum Design 379 11.1 Potential Constraint Strategy 379 11.2 Quadratic Program
min
g Problem 383 11.2.1 Definit
ion
of QP Problem 383 11.2.2 KKT Necessary Condit
ion
s for the QP Problem 384 11.2.3 Transformat
ion
of KKT Condit
ion
s 384 11.2.4 Simplex Method for Solving QP Problem 385 11.3 Approximate Step Size Deter
min
at
ion
388 11.3.1 The Basic Idea 388 11.3.2 Descent Condit
ion
389 11.3.3 CSD Algorithm with Approximate Step Size 393 11.4 Constrained Quasi-Newton Methods 400 11.4.1 Derivat
ion
of Quadratic Program
min
g Subproblem 400 11.4.2 Quasi-Newton Hessian Approximat
ion
403 11.4.3 Modified Constrained Steepest Descent Algorithm 404 11.4.4 Observat
ion
s on the Constrained Quasi-Newton Methods 406 11.4.5 Descent
Funct
ion
s 406 11.5 Other Numerical Opt
imizat
ion
Methods 407 11.5.1 Method of Feasible Direct
ion
s 407 11.5.2 Gradient Project
ion
Method 409 11.5.3 Generalized Reduced Gradient Method 410 Exercises for Chapter 11 411 Chapter 12 Introduct
ion
to Optimum Design with MATLAB 413 12.1 Introduct
ion
to Opt
imizat
ion
Toolbox 413 12.1.1 Variables and Express
ion
s 413 xvi Contents 12.1.2 Scalar, Array, and Matrix Operat
ion
s 414 12.1.3 Opt
imizat
ion
Toolbox 414 12.2 Unconstrained Optimum Design Problems 415 12.3 Constrained Optimum Design Problems 418 12.4 Optimum Design Examples with MATLAB 420 12.4.1 Locat
ion
of Maximum Shear Stress for Two Spherical Bodies in Contact 420 12.4.2 Column Design for
Min
imum Mass 421 12.4.3 Flywheel Design for
Min
imum Mass 425 Exercises for Chapter 12 429 Chapter 13 Interactive Design Opt
imizat
ion
433 13.1 Role of Interact
ion
in Design Opt
imizat
ion
434 13.1.1 What Is Interactive Design Opt
imizat
ion
? 434 13.1.2 Role of Computers in Interactive Design Opt
imizat
ion
434 13.1.3 Why Interactive Design Opt
imizat
ion
? 435 13.2 Interactive Design Opt
imizat
ion
Algorithms 436 13.2.1 Cost Reduct
ion
Algorithm 436 13.2.2 Constraint Correct
ion
Algorithm 440 13.2.3 Algorithm for Constraint Correct
ion
at Constant Cost 442 13.2.4 Algorithm for Constraint Correct
ion
at Specified Increase in Cost 445 13.2.5 Constraint Correct
ion
with
Min
imum Increase in Cost 446 13.2.6 Observat
ion
s on Interactive Algorithms 447 13.3 Desired Interactive Capabilities 448 13.3.1 Interactive Data Preparat
ion
448 13.3.2 Interactive Capabilities 448 13.3.3 Interactive Decis
ion
Making 449 13.3.4 Interactive Graphics 450 13.4 Interactive Design Opt
imizat
ion
Software 450 13.4.1 User Interface for IDESIGN 451 13.4.2 Capabilities of IDESIGN 453 13.5 Examples of Interactive Design Opt
imizat
ion
454 13.5.1 Formulat
ion
of Spring Design Problem 454 13.5.2 Optimum Solut
ion
for the Spring Design Problem 455 13.5.3 Interactive Solut
ion
for Spring Design Problem 455 13.5.4 Use of Interactive Graphics 457 Exercises for Chapter 13 462 Chapter 14 Design Opt
imizat
ion
Applicat
ion
s with Implicit
Funct
ion
s 465 14.1 Formulat
ion
of Practical Design Opt
imizat
ion
Problems 466 14.1.1 General Guidelines 466 14.1.2 Example of a Practical Design Opt
imizat
ion
Problem 467 Contents xvii 14.2 Gradient Evaluat
ion
for Implicit
Funct
ion
s 473 14.3 Issues in Practical Design Opt
imizat
ion
478 14.3.1 Select
ion
of an Algorithm 478 14.3.2 Attributes of a Good Opt
imizat
ion
Algorithm 478 14.4 Use of General-Purpose Software 479 14.4.1 Software Select
ion
480 14.4.2 Integrat
ion
of an Applicat
ion
into General- Purpose Software 480 14.5 Optimum Design of Two-Member Frame with Out-of-Plane Loads 481 14.6 Optimum Design of a Three-Bar Structure for Multiple Performance Requirements 483 14.6.1 Symmetric Three-Bar Structure 483 14.6.2 Asymmetric Three-Bar Structure 484 14.6.3 Comparison of Solut
ion
s 490 14.7 Discrete Variable Optimum Design 491 14.7.1
Continu
ous
Variable Opt
imizat
ion
492 14.7.2 Discrete Variable Opt
imizat
ion
492 14.8 Optimal Control of Systems by Nonlinear Program
min
g 493 14.8.1 A Prototype Optimal Control Problem 493 14.8.2
Min
imizat
ion
of Error in State Variable 497 14.8.3
Min
imum Control Effort Problem 503 14.8.4
Min
imum Time Control Problem 505 14.8.5 Comparison of Three Formulat
ion
s for Optimal Control of System Mot
ion
508 Exercises for Chapter 14 508 Chapter 15 Discrete Variable Optimum Design Concepts and Methods 513 15.1 Basic Concepts and Definit
ion
s 514 15.1.1 Definit
ion
of Mixed Variable Optimum Design Problem: MV-OPT 514 15.1.2 Classificat
ion
of Mixed Variable Optimum Design Problems 514 15.1.3 Overview of Solut
ion
Concepts 515 15.2 Branch and Bound Methods (BBM) 516 15.2.1 Basic BBM 517 15.2.2 BBM with Local
Min
imizat
ion
519 15.2.3 BBM for General MV-OPT 520 15.3 Integer Program
min
g 521 15.4 Sequential Linearizat
ion
Methods 522 15.5 Simulated Annealing 522 15.6
Dynamic
Rounding-off Method 524 15.7 Neighborhood
Search
Method 525 15.8 Methods for Linked Discrete Variables 525 15.9 Select
ion
of a Method 526 Exercises for Chapter 15 527 Chapter 16 Genetic Algorithms for Optimum Design 531 16.1 Basic Concepts and Definit
ion
s 532 16.2 Fundamentals of Genetic Algorithms 534 xviii Contents 16.3 Genetic Algorithm for Sequencing-Type Problems 538 16.4 Applicat
ion
s 539 Exercises for Chapter 16 540 Chapter 17 Multiobjective Optimum Design Concepts and Methods 543 17.1 Problem Definit
ion
543 17.2 Ter
min
ology and Basic Concepts 546 17.2.1 Criter
ion
Space and Design Space 546 17.2.2 Solut
ion
Concepts 548 17.2.3 Preferences and Utility
Funct
ion
s 551 17.2.4 Vector Methods and Scalarizat
ion
Methods 551 17.2.5 Generat
ion
of Pareto Optimal Set 551 17.2.6 Normalizat
ion
of Objective
Funct
ion
s 552 17.2.7 Opt
imizat
ion
Engine 552 17.3 Multiobjective Genetic Algorithms 552 17.4 Weighted Sum Method 555 17.5 Weighted
Min
-Max Method 556 17.6 Weighted Global Criter
ion
Method 556 17.7 Lexicographic Method 558 17.8 Bounded Objective
Funct
ion
Method 558 17.9 Goal Program
min
g 559 17.10 Select
ion
of Methods 559 Exercises for Chapter 17 560 Chapter 18 Global Opt
imizat
ion
Concepts and Methods for Optimum Design 565 18.1 Basic Concepts of Solut
ion
Methods 565 18.1.1 Basic Concepts 565 18.1.2 Overview of Methods 567 18.2 Overview of Deter
min
istic Methods 567 18.2.1 Covering Methods 568 18.2.2 Zoo
min
g Method 568 18.2.3 Methods of Generalized Descent 569 18.2.4 Tunneling Method 571 18.3 Overview of Stochastic Methods 572 18.3.1 Pure
Random
Search
573 18.3.2 Multistart Method 573 18.3.3 Clustering Methods 573 18.3.4 Controlled
Random
Search
575 18.3.5 Acceptance-Reject
ion
Methods 578 18.3.6 Stochastic Integrat
ion
579 18.4 Two Local-Global Stochastic Methods 579 18.4.1 A Conceptual Local-Global Algorithm 579 18.4.2 Domain Eli
min
at
ion
Method 580 18.4.3 Stochastic Zoo
min
g Method 582 18.4.4 Operat
ion
s Analysis of the Methods 583 18.5 Numerical Performance of Methods 585 18.5.1 Summary of Features of Methods 585 18.5.2 Performance of Some Methods Using Unconstrained Problems 586 Contents xix 18.5.3 Performance of Stochastic Zoo
min
g and Domain Eli
min
at
ion
Methods 586 18.5.4 Global Opt
imizat
ion
of Structural Design Problems 587 Exercises for Chapter 18 588 Appendix A Economic Analysis 593 A.1 Time Value of Money 593 A.1.1 Cash Flow Diagrams 594 A.1.2 Basic Economic Formulas 594 A.2 Economic Bases for Comparison 598 A.2.1 Annual Base Comparisons 599 A.2.2 Present Worth Comparisons 601 Exercises for Appendix A 604 Appendix B Vector and Matrix Algebra 611 B.1 Definit
ion
of Matrices 611 B.2 Type of Matrices and Their Operat
ion
s 613 B.2.1 Null Matrix 613 B.2.2 Vector 613 B.2.3 Addit
ion
of Matrices 613 B.2.4 Multiplicat
ion
of Matrices 613 B.2.5 Transpose of a Matrix 615 B.2.6 Elementary Row–Column Operat
ion
s 616 B.2.7 Equivalence of Matrices 616 B.2.8 Scalar Product–Dot Product of Vectors 616 B.2.9 Square Matrices 616 B.2.10 Partit
ion
ing of Matrices 617 B.3 Solut
ion
of n Linear Equat
ion
s in n Unknowns 618 B.3.1 Linear Systems 618 B.3.2 Deter
min
ants 619 B.3.3 Gaussian Eli
min
at
ion
Procedure 621 B.3.4 Inverse of a Matrix: Gauss-Jordan Eli
min
at
ion
625 B.4 Solut
ion
of m Linear Equat
ion
s in n Unknowns 628 B.4.1 Rank of a Matrix 628 B.4.2 General Solut
ion
of m ¥ n Linear Equat
ion
s 629 B.5 Concepts Related to a Set of Vectors 635 B.5.1 Linear Independence of a Set of Vectors 635 B.5.2 Vector Spaces 639 B.6 Eigenvalues and Eigenvectors 642 B.7 Norm and Condit
ion
Number of a Matrix 643 B.7.1 Norm of Vectors and Matrices 643 B.7.2 Condit
ion
Number of a Matrix 644 Exercises for Appendix B 645 Appendix C A Numerical Method for Solut
ion
of Nonlinear Equat
ion
s 647 C.1 Single Nonlinear Equat
ion
647 C.2 Multiple Nonlinear Equat
ion
s 650 Exercises for Appendix C 655 xx Contents Appendix D Sample Computer Programs 657 D.1 Equal Interval
Search
657 D.2 Golden Sect
ion
Search
660 D.3 Steepest Descent Method 660 D.4 Modified Newton’s Method 669 References 675 Bibliography 683 Answers to Selected Problems 687 Index 695 Contents
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