Introduction to Optimum Design


  • Jasbir Arora, Professor, Department of Civil and Environmental Engineering & Department of Mechanical Engineering, University of Iowa, iowa City, IA, USA
  • Jasbir Arora, Professor, Department of Civil and Environmental Engineering & Department of Mechanical Engineering, University of Iowa, iowa City, IA, USA


Book information

  • Published: August 2011
  • ISBN: 978-0-12-381375-6


"I feel that Dr. Arora presented significant amounts of material in a clear and straightforward manner. The book is definitely a reference that practitioners would like to have and depend upon, especially with the plethora of examples and applications. As an educator, Dr. Arora’s book also has a tremendous number of problems at the end of the chapters and examples that I would try to use in class...the book is a solid introduction to optimization algorithms." - Georges Fadel, Associate Editor, Journal of Mechanical Design "Arora’s introduction of a much-anticipated second edition of Introduction to Optimum Design will not only satisfy established users of his well-received first edition, but moreover, significant updates, supplementary material, and fine-tuning of the pedagogical aspects of the presentation will certainly broaden its appeal…among some of the distinguishing characteristics of Arora’s book are its adaptability to audiences with diverse backgrounds, as well as the extent to which it makes the topic clear and approachable...The book would also be excellent as a self-study reference for the practicing engineer…In summary, when considering the pedagogical refinements of the book, the expanded and updated software examples, as well as the extended survey of emerging computational methods, Arora’s Introduction to Optimum Design, 2nd Ed., furthers its goal of describing engineering design optimization in a rigorous yet simplified manner which is both highly accessible to and useful for a wide audience." - David F. Thompson, Graduate Program Director, University of Cincinnati "I have used several optmization books over the past 10 years to support my various graduate optimization courses. Of all the books that I have used, I prefer Dr. Arora’s Introduction to Optimum Design, 2nd Ed…The strength of this book lies in his attention to detail using numeric exercises to demonstrate the numerical processes used in the various optimization methods. I particularly like his choice of nomenclature throughout the book, as it conforms to the standard symbols and function names used in classical optimization literature. The application exercises presented cover a broad range in technologies, which makes it a good textbook for any engineering discipline." - Tom R. Mincer, California State University "...this book is well written and covers just about every topic that one needs to know about the optimum design process. It includes a good balance of theory and application. The book will therefore be appealing to all users." - Practice Periodical On Structural Design and Construction - ASCE, Nov. 2005

Table of Contents


1 Introduction to Design Optimization
1.1 The Design Process
1.2 Engineering Design versus EngineeringAnalysis
1.3 Conventional versus Optimum Design Process
1.4 Optimum Design versus Optimal Control
1.5 Basic Terminology and Notation

2 Optimum Design Problem Formulation
2.1 The Problem Formulation Process
2.2 Design of a Can
2.3 Insulated Spherical Tank Design
2.4 Sawmill Operation
2.5 Design of a Two-Bar Bracket
2.6 Design of a Cabinet
2.6.1 Formulation 1 for Cabinet Design
2.6.2 Formulation 2 for Cabinet Design
2.6.3 Formulation 3 for Cabinet Design
2.7 Minimum-Weight Tubular Column Design
2.7.1 Formulation 1 for Column Design
2.7.2 Formulation 2 for Column Design
2.8 Minimum-Cost Cylindrical Tank Design
2.9 Design of Coil Springs
2.10 Minimum-Weight Design of a SymmetricmThree-Bar Truss
2.11 A General Mathematical Model for Optimum Design
Exercises for Chapter 2

3 Graphical Optimization and Basic Concepts
3.1 Graphical Solution Process
3.2 Use of Mathematica for Graphical Optimization
3.3 Use of MATLAB for Graphical Optimization
3.4 Design Problem with Multiple Solutions
3.5 Problem with Unbounded Solution
3.6 Infeasible Problem
3.7 Graphical Solution for the Minimum-Weight Tubular Column
3.8 Graphical Solution for a Beam Design Problem
Exercises for Chapter 3 83

4 Optimum Design Concepts: Optimality Conditions
4.1 Definitions of Global and Local Minima
4.2 Review of Some Basic Calculus Concepts
4.3 Concept of Necessary and Sufficient Conditions
4.4 Optimality Conditions: Unconstrained Problem
4.5 Necessary Conditions: Equality-Constrained Problem
4.6 Necessary Conditions for a General Constrained Problem
4.7 Postoptimality Analysis: The Physical Meaning of Lagrange Multipliers
4.8 Global Optimality
4.9 Engineering Design Examples

5 More on Optimum Design Concepts: Optimality Conditions
5.1 Alternate Form of KKT Necessary Conditions
5.2 Irregular Points
5.3 Second-Order Conditions for Constrained Optimization
5.4 Second-Order Conditions for Rectangular Beam Design Problem
5.5 Duality in Nonlinear Programming
Exercises for Chapter 5


6 Optimum Design with Excel Solver
6.1 Introduction to Numerical Methods for Optimum Design
6.2 Excel Solver: An Introduction
6.3 Excel Solver for Unconstrained Optimization Problems
6.4 Excel Solver for Linear Programming Problems
6.5 Excel Solver for Nonlinear Programming: Optimum Design of Springs
6.6 Optimum Design of Plate Girders Using Excel Solver
6.7 Optimum Design of Tension Members
6.8 Optimum Design of Compression Members
6.9 Optimum Design of Members for Flexure
6.10 Optimum Design of Telecommunication Poles
Exercises for Chapter 6

7 Optimum Design with MATLAB
7.1 Introduction to the Optimization Toolbox
7.2 Unconstrained Optimum Design Problems
7.3 Constrained Optimum Design Problems
7.4 Optimum Design Examples with MATLAB
Exercises for Chapter 7

8 Linear Programming Methods for Optimum Design
8.1 Linear Functions
8.2 Definition of a Standard Linear Programming Problem
8.3 Basic Concepts Related to Linear Programming Problems
8.4 Calculation of Basic Solutions
8.5 The Simplex Method
8.6 The Two-Phase Simplex Method-Artificial Variables
8.7 Postoptimality Analysis
Exercises for Chapter 8

9 More on Linear Programming Methods
for Optimum Design 377
9.1 Derivation of the Simplex Method
9.2 An Alternate Simplex Method
9.3 Duality in Linear Programming
9.4 KKT Conditions for the LP Problem
9.5 Quadratic Programming Problems
Exercises for Chapter 9

10 Numerical Methods for Unconstrained Optimum Design
10.1 Gradient-Based and Direct Search Methods
10.2 General Concepts: Gradient-Based Methods
10.3 Descent Direction and Convergence of Algorithms
10.4 Step Size Determination: Basic Ideas
10.5 Numerical Methods to Compute Step Size
10.6 Search Direction Determination: The Steepest-Descent Method
10.7 Search Direction Determination: The Conjugate Gradient Method
10.8 Other Conjugate Gradient Methods
Exercises for Chapter 10

11 More on Numerical Methods for Unconstrained Optimum Design
11.1 More on Step Size Determination
11.2 More on the Steepest-Descent Method
11.3 Scaling of Design Variables
11.4 Search Direction Determination: Newton’s Method
11.5 Search Direction Determination: Quasi-Newton Methods
11.6 Engineering Applications of Unconstrained Methods
11.7 Solutions to Constrained Problems Using Unconstrained Optimization Methods
11.8 Rate of Convergence of Algorithms
11.9 Direct Search Methods
Exercises for Chapter 11

12 Numerical Methods for Constrained Optimum Design
12.1 Basic Concepts Related to Numerical Methods
12.2 Linearization of the Constrained Problem
12.3 The Sequential Linear Programming Algorithm
12.4 Sequential Quadratic Programming
12.5 Search Direction Calculation: The QP Subproblem
12.6 The Step Size Calculation Subproblem
12.7 The Constrained Steepest-Descent Method
Exercises for Chapter 12

13 More on Numerical Methods for Constrained Optimum Design
13.1 Potential Constraint Strategy
13.2 Inexact Step Size Calculation
13.3 Bound-Constrained Optimization
13.4 Sequential Quadratic Programming: SQP Methods
13.5 Other Numerical Optimization Methods
13.6 Solution to the Quadratic Programming Subproblem
Exercises for Chapter 13

14 Practical Applications of Optimization
14.1 Formulation of Practical Design Optimization Problems
14.2 Gradient Evaluation of Implicit Functions
14.3 Issues in Practical Design Optimization
14.4 Use of General-Purpose Software
14.5 Optimum Design of Two-Member Frame with Out-of-Plane Loads
14.6 Optimum Design of a Three-Bar Structure for Multiple Performance Requirements
14.7 Optimal Control of Systems by Nonlinear Programming
14.8 Alternative Formulations for Structural Optimization Problems
14.9 Alternative Formulations for Time-Dependent Problems
Exercises for Chapter 14


15 Discrete Variable Optimum Design Concepts and Methods
15.1 Basic Concepts and Definitions
15.2 Branch-and-Bound Methods
15.3 Integer Programming
15.4 Sequential Linearization Methods
15.5 Simulated Annealing
15.6 Dynamic Rounding-Off Method
15.7 Neighborhood Search Method
15.8 Methods for Linked Discrete Variables
15.9 Selection of a Method
15.10 Adaptive Numerical Method for Discrete Variable Optimization
Exercises for Chapter 15

16 Genetic Algorithms for Optimum Design
16.1 Basic Concepts and Definitions
16.2 Fundamentals of Genetic Algorithms
16.3 Genetic Algorithm for Sequencing-Type Problems
16.4 Applications 653
Exercises for Chapter 16

17 Multi-objective Optimum Design Concepts and Methods
17.1 Problem Definition
17.2 Terminology and Basic Concepts
17.3 Multi-objective Genetic Algorithms
17.4 Weighted Sum Method
17.5 Weighted Min-Max Method
17.6 Weighted Global Criterion Method
17.7 Lexicographic Method
17.8 Bounded Objective Function Method
17.9 Goal Programming
17.10 Selection of Methods
Exercises for Chapter 17

18 Global Optimization Concepts and Methods
18.1 Basic Concepts of Solution Methods
18.2 Overview of Deterministic Methods
18.3 Overview of Stochastic Methods
18.4 Two Local-Global Stochastic Methods
18.5 Numerical Performance of Methods
Exercises for Chapter 18

19 Nature-Inspired Search Methods
19.1 Differential Evolution Algorithm
19.2 Ant Colony Optimization
19.3 Particle Swarm Optimization
Exercises for Chapter 19

20 Additional Topics on Optimum Design
20.1 Meta-Models for Design Optimization
20.2 Design of Experiments for Response Surface Generation
20.3 Discrete Design with Orthogonal Arrays
20.4 Robust Design Approach
20.5 Reliability-Based Design Optimization-Design under Uncertainty

Appendix A: Vector and Matrix Algebra
A.1 Definition of Matrices
A.2 Types of Matrices and Their Operations
A.3 Solving n Linear Equations in n Unknowns
A.4 Solution to m Linear Equations in n Unknowns
A.5 Concepts Related to a Set of Vectors
A.6 Eigenvalues and Eigenvectors
A.7 Norm and Condition Number of a Matrix
Exercises for Appendix A

Appendix B: Sample Computer Programs
B.1 Equal Interval Search
B.2 Golden Section Search
B.3 Steepest-Descent Method
B.4 Modified Newton’s Method


Answers to Selected Exercises