Book number: 9787302266082

Authors: Yongheng Zhang et al.

Pricing: 34 yuan

Release date: 20 1 1-9-5

Publishing House: Tsinghua University Publishing House

brief Introduction of the content

Based on engineering examples, this book comprehensively introduces the theory and application of optimal design with MATLAB as a tool. The main contents of this book include: the basic model of optimal design; Basic knowledge of optimization design mathematics; One-dimensional search method; Classical algorithms for unconstrained optimization problems and constrained optimization problems; Heuristic optimization algorithms, including ant colony algorithm, particle swarm optimization algorithm, genetic algorithm, simulated annealing algorithm, tabu algorithm and artificial neural network algorithm; MATLAB optimization toolbox function and its application: engineering application example of optimization algorithm and basic knowledge of MATLAB. The book is equipped with a complete MATLAB program. This book can be used as a teaching material and reference book for related courses of professional optimization design in higher engineering colleges, and can also be used as a reference for teachers, students and engineers of related majors.

order

Optimal design is an ancient and emerging theory with strong application background and solid mathematical foundation. Its mathematical foundation can be traced back to the calculus theory founded by Newton (Newton, 1642- 1727) and Leibniz (W. Leibniz, 1646- 17 16). Optimal design is closely related to operational research, the former is the extension and development of the latter in the direction of nonlinear planning. Optimal design mainly studies the optimal value of single-objective function or multi-objective function of continuous function under constrained and unconstrained conditions, and operational research mainly studies the planning and management problems that can be expressed quantitatively in economic activities and military activities. With the development of science and technology and production, operational research has penetrated into many fields and is developing continuously, including many branches of mathematics, such as mathematical programming (including linear programming, nonlinear programming, integer programming, combinatorial programming, etc.). ), graph theory, network flow, decision analysis, queuing theory, reliability mathematics theory, inventory theory, game theory, search theory, simulation and so on. In operational research, famous Chinese scientists Qian Xuesen, Xu Guozhi and mathematician Hua all made important contributions. 1956, Qian Xuesen and Xu Guozhi jointly founded the first operational research organization in China. Since the 1960s, China has promoted the optimization method and the overall planning method for nearly 20 years, which has produced great economic benefits. Among them, the golden section search method used in optimization method is also a common method of one-dimensional search in optimization design. Heuristic algorithms or intelligent algorithms, such as genetic algorithm, ant colony algorithm, particle swarm optimization algorithm and neural network algorithm, can not only solve the optimization problem of continuous functions, but also solve the optimization problem of discrete functions. They combine optimization design with operational research.

Broadly speaking, the method used in optimization design is search method. The traditional optimization design method mainly adopts line search method, while the heuristic optimization method adopts multi-directional random search method. For nonlinear function, it can be approximated by quadratic function near the extreme point. If there is a minimum value, the function values near the extreme point are greater than those at the extreme point. To solve the extreme value problem of continuous function, some people may think of derivative, while others may directly determine the search interval and minimum value by comparison method instead of derivative. Compared with derivative method, direct search method is a more basic method in optimization design. From the mathematical model of optimization design, optimization design problems can be divided into constrained optimization problems and unconstrained optimization design problems; From the solution method, the optimization design method can be divided into derivative-based method and direct search method. Random direction method, composite method, Powell method and feasible direction method all belong to direct search method. It is worth noting that heuristic algorithms such as genetic algorithm, ant colony algorithm and particle swarm optimization all contain the basic connotation of random direction method.

Optimal design is widely used in aviation, automobile, chemical industry, electric power, construction, machinery manufacturing and many other fields. Due to the diversity of optimization problems, there are many optimization design methods, each of which has its own characteristics and scope of application. In practical application, especially for large-scale optimization design problems, a calculation result or a method result should not be regarded as the final optimal result.

Optimal design is the product of the combination of optimization principle and calculation technology based on engineering design problems. From the perspective of learning and application, practice is very important. Practice is both the end and the starting point of learning. This book places special emphasis on the combination of theory and practice. Practice includes many aspects. The most basic thing is to calculate and verify the algorithm by hand through simple examples, then realize and verify the optimization algorithm by computer programming, and finally establish an optimization design model for engineering design problems and choose the appropriate optimization algorithm to solve the design problems. MATLAB is not only a high-level computer language for numerical calculation, but also a simulation software for solving many engineering and mathematical problems. This book takes MATLAB as the programming language and practical environment, and writes learning programs for each algorithm to facilitate readers' learning. These programs are mainly to verify the optimization algorithm, and readers can write their own programs on this basis. MATLAB itself contains an optimization toolbox of command format and GUI format, and new optimization algorithms are constantly added with the upgrade of the version. Chapter 1 1 of this book briefly introduces various optimization functions of the command format of MATLAB optimization toolbox, which provides great convenience for optimization design. However, from the perspective of learning, in order to deeply understand and master the optimization algorithm, we should try our best to program ourselves.

The revised edition of this book retains the contents of the original book, modifies some contents and improves the exercises in each chapter. This book comes with an electronic lesson plan. If necessary, you can contact Tsinghua University Publishing House.

This book is edited by Yongheng Zhang, reviewed by Cai Huilin and Chu Yandong, and compiled by He Wei, Ma Bin, Zhu Lingyun (Lanzhou Jiaotong University) and Yan Jun (Northwest Normal University). Chapter 1 and section 12. 1~ 12.3 were written by Yongheng Zhang; Chapter 9 and section 12. 10 were written by He Wei; Chapters 5, 6 and 7 were written by Ma Bin. The second, fourth and eighth chapters were written by Zhu Lingyun. Chapter III 10, 1 1, and sections 12.4~ 12.9 were written by Yan Jun; These exercises were written by Yongheng Zhang, Ma Bin and Zhu Lingyun. In the process of compiling, Zhang Peng, Liu Jinping, Cheng Ming, Zhou Zhiyong, Ning Zhen, Liu Junqiang and Tang Qiang have completed the debugging of some programs, and I would like to express my gratitude. In the process of writing, I refer to the information about the author on the internet, and I would like to express my gratitude.

Due to the limited level of the author, there must be many mistakes and shortcomings in the book. Please give your valuable advice.

catalogue

catalogue

Chapter 1 Introduction 1

1. 1 optimization problem proposition 1

1.2 Classification of optimization problems 4

1.3 Diagram of Optimization Model 5

1.4 Examples of finite element method 10

1.5 introduction of multidisciplinary design optimization integration software iSIGHT 12

Exercise 16 Chapter II Mathematical Basis of Optimal Design 18

2. 1 norm of vector sum matrix 18

Norm 2. 1. 1 vector 18

2. 1.2 norm of matrix 18

2.2 directional derivative and gradient 19

2.2. 1 directional derivative 19

Gradient 20

2.3 Taylor series expansion of function 2 1

2.4 Extreme conditions for unconstrained optimization problems 22

2.5 Convex Sets and Convex Functions 25

2.5. 1 convex set 25

Convex function 25

2.6 Extreme conditions for constrained optimization problems 27

2.6. 1 Extreme conditions for equality constrained optimization problems 27

2.6.2 Extreme conditions for inequality constrained optimization problems 29

Exercise 36 Chapter 3 Linear Programming 37

3. 1 standard form of linear programming 37

3.2 Simplex Method 38

3.2. 1 Basic solution and basic feasible solution 38

3.2.2 Transformation of Basic Feasible Solution 42

3.2.3 Simplex Method Calculation Step 44

3.2.4 Simplex Method List Calculation 47

3.3 MATLAB program of simplex method and examples 49

3.4 modified simplex method 5 1

3.4. 1 modified simplex method's basic ideas 52

3.4.2 Calculation Steps of modified simplex method 52

3.5 modified simplex method's MATLAB program and example 55

Exercise 57 Chapter 4 One-dimensional Search Method 60

4. 1 Method for determining initial unimodal interval-advance and retreat method 60

4. 1. 1 advance and retreat principle 60

4. 1.2 program block diagram of advance and retreat method and MATLAB program 6 1

4.2 Golden Section 63

4.2. 1 the basic principle of the golden section 63

4.2.2 Calculation method of golden section method

4.2.3 golden section calculation block diagram and MATLAB program 64

4.3 Lagrange Interpolation Polynomial 66

4.3. 1 linear interpolation 66

4.3.2 Quadratic Function Interpolation 66

4.3.3 ? n？ Sub-Lagrangian interpolation polynomial 70

4.4 Other interpolation and fitting methods 7 1

4.4. 1 difference quotient and Newton interpolation 7 1

4.4.2 Levi's interpolation method 72

4.4.3 Least Square Method of Curve Fitting 75

4.4.4 Orthogonal Polynomial and Its Application in Curve Fitting 76

4.5 Find the roots of univariate and multivariate nonlinear equations 8 1

4.5. 1 Find the root of one-dimensional nonlinear equation 8 1

4.5.2 Finding the Roots of Multivariate Nonlinear Equations 84

Exercise 85 Chapter 5 Derivative Solution of Unconstrained Optimization Problem 87

5. 1 steepest descent method 87

5. 1. 1 Basic principle of steepest descent method 87

5. 1.2 MATLAB program of steepest descent method 89

5.2 Newton method 90

5.2. Basic principle of1Newton method 90

Damped Newton method 92

5.2.3 MATLAB program of damping Newton method 93

5.3 *** yoke gradient method 94

5.3. Concept of1* * * yoke direction 94

5.3.2 *** Relationship between yoke orientation and function extreme value 94

5.3.3 *** Several forms of yoke gradient method 95

MATLAB program of 5.3.4 *** yoke gradient method 99

5.4 variable proportion method 100

5.4. 1 variable scale 100

5.4.2 Establishment of Variable Scale Matrix 103

5.4.3 MATLAB program of variable scale method 106

Exercise 108 chapter 6 direct solution of unconstrained optimization problem 109

6. 1 coordinate rotation method 109

6. 1. 1 Basic principle of coordinate rotation method5438+009

6. 1.2 Determination of search direction and step size 109

6. 1.3 MATLAB program of coordinate rotation method 1 10

6.2 simplex replacement method 1 12

6.2. 1 simplex substitution method (I) 1 13

6.2.2 Simplex substitution method (2) 1 14

6.2.3 MATLAB program of simplex replacement method 1 15.

6.3 Powell method 1 19

Principle of Powell method 120

6.3.2 steps of Powell's basic algorithm 120

6.3.3 Improved Powell method 12 1

6.4 MATLAB program and example of Powell method 125

Exercise 127 chapter 7 direct solution of constrained optimization problem 129

7. 1 random direction method 129

7. 1. 1 Basic principle of random direction method5438+029

7. 1.2 step random direction method 129

7. 1.3 Random Direction Method 130 MATLAB Program

7.2 composite method 133

7.2. 1 steps of complex methods 133

7.2.2 MATLAB Program of Complex Method 135

7.3 feasible direction method 140

7.3. 1 feasible direction search strategy 140

7.3.2 feasible direction method 14 1

7.3.3 Rosen feasible direction method 144

7.3.4 MATLAB program of Rosen feasible direction method 146

Exercise 150 chapter 8 indirect solution of constrained optimization problem 152

8. 1 penalty function method 152

8. 1. 1 interior point penalty function method 152

8. 1.2 external penalty function method 156

8. 1.3 mixed penalty function method 158

8.2 Extended Multiplier Method 160

8.2. 1 Lagrange multiplier method

8.2.2 augmented multiplier method with equality constraints 162

8.2.3 augmented multiplier method with inequality constraints 165

Exercise 169 chapter 9 optimization design of multi-objective function 17 1

9. 1 multi-objective optimization problem 172

9. 1. 1 Mathematical model of multi-objective optimization problem 5438+072

9. 1.2 Types of multi-objective optimization design schemes 172

9.2 Method for Solving Multi-objective Optimization Problems 173

9.2. 1 linear combination method 173

9.2.2 Ideal Point Method 174

9.2.3 Method of multiplication and division 175

Exercise 175 Chapter 10 Heuristic Algorithm for Optimization Problems 177

10. 1 ant colony algorithm 177

10. 1 the basic principle of ant colony algorithm for solving TSP 177.

10. 1.2 Solving function optimization problems with ant colony algorithm 18 1

10.2 particle swarm optimization algorithm 185

10.2. 1 basic principle of particle swarm optimization algorithm 5438+085

10.2.2 Solving function optimization problems with particle swarm optimization algorithm 185

10.3 genetic algorithm 189

10.3. 1 Basic principle of genetic algorithm

10.3.2 hybrid genetic algorithm 196

10.3.3 decimal coding genetic algorithm 199

10.3.4 solving TSP problem with genetic algorithm 203

10.4 simulated annealing algorithm 204

10.5 artificial neural network algorithm 208

10.5. 1 the characteristics and classification of artificial neural network566661

10.5.2 BP network209

10.5.3 Hopfield neural network model 2 12

Exercise 222 Chapter 1 1 Introduction to MATLAB Optimization Toolbox 223

11.1MATLAB common internal mathematical function 223

11.2 main functions of MATLAB optimization toolbox 224

11.2.1Main functions of MATLAB in solving optimization problems 224

1 1.2.2 Optimize function control parameters 225

1 1.3 linear programming problem 226

1 1.4 optimization of univariate and multivariate functions 228

On the optimization of the univariate function 1 1.4. 1 20000.100000000606

Unconstrained optimization problem of 1 1.4.2 multivariate function2228.0000000000005

Constraint optimization problem of 1 1.4.3 multivariate function 230

1 1.4.4 quadratic programming problem 23 1

1 1.5 semi-infinite constrained multivariate function optimization problem 233

1 1.6 Multi-objective optimization problem 234

1 1.6. 1 ideal point method.50000.00000000016

1 1.6.2 linear weighted sum method 237

1 1.6.3 maximum and minimum methods 239

1 1.6.4 Method of achieving the goal 240

On the application of 1 1.7 least square method in optimization and data fitting120666.1000666666666

1 1.7. 1 constrained linear least squares 243

1 1.7.2 a least squares data (curve) fitting 244

1 1.7.3 least square data (curve) fitting Ⅱ 245

1 1.7.4 least square data (curve) fitting Ⅲ 246

Solutions of the nonlinear equation 1 1.8 247

1 1.8. 1 the solution of one-dimensional nonlinear equation 247

1 1.8.2 Solutions of the Nonlinear Equation 247

Exercise 25 1 Chapter 12 Engineering Optimization Design Example 254

12. 1 optimal design of planar linkage mechanism120066.100000606666

Optimal design mathematical model of 12. 1. 1 crank rocker mechanism50000.00000000056

MATLAB program for optimal design of crank-rocker mechanism and its running results 256

Optimal design of 12.2 cam s257

12.2. 1 objective function of cam profile optimization design 258

12.2.2 optimization function constraint 259

12.2.3 MATLAB program for optimal design of cam mechanism and calculation example 259

Optimal design of 12.3 bolt connection

12. 3. 1 bearing capacity analysis of bolted connection 26 1

12.3.2 Design variables, objective functions and constraints of bolted connections 262

12.3.3 optimization mathematical model of bolt connection 263

MATLAB program for optimal design of bolt connection and its running results

Optimum design of 12.4 cylindrical gear drive50000.00000000105

General process of fuzzy comprehensive evaluation of 12.4 5 1264363636

12.4.2 objective function and design variables for optimal design of cylindrical gear transmission

Constraints of Optimal Design of Cylindrical Gear Transmission

12.4.4 optimal cut set? λ? Determination of 269

12.4.5 MATLAB program and calculation results of optimal design of cylindrical gear transmission 270

Optimal design of 12.5 cylindrical helical spring 2772

12.5. 1 mathematical model for the optimal design of cylindrical helical spring506661

12.5.2 Optimal design example of cylindrical helical spring 274

Optimum design of 12.6 axle50000.00000000016

Optimum design of 12. 6. 1 torsion shaft40000.00000000016

Optimal design of 12.6.2 circular shaft with equal section 276

12.6.3 optimal design of lathe spindle

Optimal design of 12.7 truss 28 1

Optimum design of statically indeterminate truss56438+02.7.3338+05438+05437.36661

Optimum design of three-bar truss+02.7.25000.000000000005

Optimum design of 12.8 heat exchange50000.00000000056

12.8. 1 Optimal design of heat exchanger (1) 286

12.8.2 optimal design of heat exchanger (2) 289

12.9 numerical solution of boundary value problem of ordinary differential equation based on optimization method 29 1

Solution method based on MATLAB function 29 1

12.9.2 shooting method for solving two-point boundary value problem 292

12.9.3 boundary layer differential equations and similar solutions 293

12.9.4 solution of flow function equation and temperature equation 295

12. 10 Parameter Optimization Design of Mechanical System with 306 Gaps

12. 10. 1 mechanical model and differential equation of motion 307

Branches of 12. 10.2 system and the road to chaos. 46636.6666666666 1

12. 10.3 MATLAB program for system optimization design309

Exercise 3 12 Reference 3 16