Contents
Preface
1
xvii
Introduction to Optimization
1
1.1
Introduction
1.2
Historical Development
1.3
Engineering Applications of Optimization
1.4
Statement of an Optimization Problem
1.5
1
3
1.4.1
Design Vector
1.4.2
Design Constraints
7
1.4.3
Constraint Surface
8
1.4.4
Objective Function
9
1.4.5
Objective Function Surfaces
5
6
6
9
Classification of Optimization Problems
14
1.5.1
Classification Based on the Existence of Constraints
1.5.2
Classification Based on the Nature of the Design Variables
1.5.3
Classification Based on the Physical Structure of the Problem
16
1.5.4
Classification Based on the Nature of the Equations Involved
19
1.5.5
Classification Based on the Permissible Values of the Design Variables
1.5.6
Classification Based on the Deterministic Nature of the Variables
1.5.7
Classification Based on the Separability of the Functions
1.5.8
Classification Based on the Number of Objective Functions
1.6
Optimization Techniques
1.7
Engineering Optimization Literature
1.8
Solution of Optimization Problems Using MATLAB
References and Bibliography
Review Questions
Problems
2
15
28
29
30
32
35
35
36
39
45
46
Classical Optimization Techniques
63
2.1
Introduction
2.2
Single-Variable Optimization
2.3
Multivariable Optimization with No Constraints
2.4
14
63
2.3.1
Semidefinite Case
2.3.2
Saddle Point
63
68
73
73 Optimization with Equality Constraints
Multivariable
2.4.1
Solution by Direct Substitution
2.4.2
Solution by the Method of Constrained Variation
2.4.3
Solution by the Method of Lagrange Multipliers
75
76
77
85
vii
viii
Contents
2.5
2.6
Multivariable Optimization with Inequality Constraints
2.5.1
Kuhn-Tucker Conditions
2.5.2
Constraint Qualification
Review Questions
Problems
3
98
98
104
Convex Programming Problem
References and Bibliography
93
105
105
106
Linear Programming I: Simplex Method
119
3.1
Introduction
3.2
Applications of Linear Programming
119
3.3
Standard Form of a Linear Programming Problem
3.4
Geometry of Linear Programming Problems
3.5
Definitions and Theorems
3.6
Solution of a System of Linear Simultaneous Equations
3.7
Pivotal Reduction of a General System of Equations
3.8
Motivation of the Simplex Method
3.9
Simplex Algorithm
120
127
138
140
3.9.1
Identifying an Optimal Point
3.9.2
Improving a Nonoptimal Basic Feasible Solution
Two Phases of the Simplex Method
150
3.11
MATLAB Solution of LP Problems
156
References and Bibliography
Review Questions
Problems
141
158
158
160
Linear Programming II: Additional Topics and Extensions
4.1
Introduction
4.2
Revised Simplex Method
4.3
Duality in Linear Programming
177
177
177
192
4.3.1
Symmetric Primal-Dual Relations
4.3.2
General Primal-Dual Relations
4.3.3
Primal-Dual Relations When the Primal Is in Standard Form
4.3.4
Duality Theorems
192
4.3.5
Dual Simplex Method
193
195
195
200
4.4
Decomposition Principle
4.5
Sensitivity or Postoptimality Analysis
4.6
133
135
139
3.10
4
122
124
207
4.5.1
Changes in the Right-Hand-Side Constants bi
4.5.2
Changes in the Cost Coefficients cj
4.5.3
Addition of New Variables
4.5.4
Changes in the Constraint Coefficients aij
4.5.5
Addition of Constraints
Transportation Problem
220
208
212
214
218
215
193
Contents
4.7
Karmarkar's Interior Method
222
4.7.1
Statement of the Problem
4.7.2
Conversion of an LP Problem into the Required Form
4.7.3
Algorithm
Quadratic Programming
4.9
MATLAB Solutions
229
235
237
References and Bibliography
Problems
5
239
239
Nonlinear Programming I: One-Dimensional Minimization Methods
5.1
Introduction
5.2
Unimodal Function
5.3
253
254
254
Unrestricted Search
5.3.1
Search with Fixed Step Size
5.3.2
Search with Accelerated Step Size
5.4
Exhaustive Search
5.5
Dichotomous Search
5.6
Interval Halving Method
5.7
Fibonacci Method
5.8
263
5.9
Golden Section
Method
Comparison
of Elimination
Methods
254
255
256
257
260
267
INTERPOLATION METHODS
Quadratic Interpolation Method
5.11
Cubic Interpolation Method
5.12
Direct Root Methods
273
280
286
5.12.1
Newton Method
5.12.2
Quasi-Newton Method
5.12.3
Secant Method
Practical Considerations
271
271
5.10
5.14
248
248
ELIMINATION METHODS
5.13
224
226
4.8
Review Questions
223
286
288
290
293
5.13.1
How to Make the Methods Efficient and More Reliable
293
5.13.2
Implementation in Multivariable Optimization Problems
293
5.13.3
Comparison of Methods
294
MATLAB Solution of One-Dimensional Minimization Problems
References and Bibliography
Review Questions
Problems
295
296
295
294
ix