Optimization Concepts and Applications in Engineering

Contents
Preface
1 Preliminary Concepts
	1.1 Introduction
	1.2 Historical Sketch
	1.3 The Nonlinear Programming Problem
	1.4 Optimization Problem Modeling
	1.5 Graphical Solution of One- and Two-Variable Problems
	1.6 Existence of a Minimum and a Maximum: Weierstrass Theorem
	1.7 Quadratic Forms and Positive Definite Matrices
	1.8 Cn Continuity of a Function
	1.9 Gradient Vector, Hessian Matrix, and their Numerical Evaluation using Divided Differences
	1.10 Taylor’s Theorem, Linear and Quadratic Approximations
	1.11 Miscellaneous Topics
	Chapter 1 Problems
	Chapter 1 Computer Programs
	Bibliography
	A Few Leads to Optimization Software
	Some Optimization Conferences
	Some Optimization Specific Journals
2 One-Dimensional Unconstrained Minimization
	2.1 Introduction
	2.2 Theory Related to Single-Variable (Univariate) Minimization
	2.3 Unimodality and Bracketing the Minimum
	2.4 Fibonacci Method
	2.5 Golden Section Search Method
	2.6 Polynomial-Based Methods
	2.7 Optimization of Non-unimodal Functions
	2.8 Using MATLAB
	2.9 Zero of a Function
	Chapter 2 Problems
	Chapter 2 Computer Programs
	Bibliography
3 Unconstrained Optimization
	3.1 Introduction
	3.2 Necessary and Sufficient Conditions for Optimality
	3.3 Convexity
	3.4 Basic Concepts: Starting Design, Direction Vector, and Step Size
	3.5 The Steepest Descent Method
	3.6 The Conjugate Gradient Method
	3.7 Newton’s Method
	3.8 Quasi-Newton Methods
	3.9 Approximate Line Search
	3.10 Using MATLAB
	Chapter 3 Problems
	Chapter 3 Computer Programs
	Bibliography
4 Linear Programming
	4.1 Introduction
	4.2 Linear Programming Problem
	4.3 Problem Illustrating Modeling, Solution, Solution Interpretation, and Lagrange Multipliers
	4.4 Problem Modeling
	4.5 Geometric Concepts: Hyperplanes, Half-spaces, Polytopes, Extreme Points
	4.6 Standard Form of an LP
	4.7 The Simplex Method – Starting with LE (≤) Constraints
	4.8 Treatment of GE and EQ Constraints
	4.9 Revised Simplex Method
	4.10 Duality in Linear Programming
	4.11 The Dual Simplex Method
	4.12 Criss-Cross Method
	4.13 Sensitivity Analysis
	4.14 Interior Approach
	4.15 Quadratic Programming (QP) and the Linear Complementary Problem (LCP)
	Chapter 4 Problems
	Chapter 4 Computer Programs
	Bibliography
5 Constrained Minimization
	5.1 Introduction
	5.2 Graphical Solution of Two-Variable Problems
	5.3 Use of Excel SOLVER and MATLAB
	5.4 Formulation of Problems in Standard NLP Form
	5.5 Necessary Conditions for Optimality
	5.6 Sufficient Conditions for Optimality
	5.7 Convexity
	5.8 Sensitivity of Optimum Solution to Problem Parameters
	5.9 Rosen’s Gradient Projection Method for Linear Constraints
	5.10 Zoutendijk’s Method of Feasible Directions (Nonlinear Constraints)
	5.11 The Generalized Reduced Gradient Method (Nonlinear Constraints)
	5.12 Sequential Quadratic Programming (SQP)
	5.13 Features and Capabilities of Methods Presented in this Chapter
	Chapter 5 Problems
	Chapter 5 Computer Programs
	Bibliography
6 Penalty Functions, Duality, and Geometric Programming
	6.1 Introduction
	6.2 Exterior Penalty Functions
	6.3 Interior Penalty Functions
	6.4 Extended Interior Penalty Functions
	6.5 Duality
	6.6 The Augmented Lagrangian Method
	6.7 Geometric Programming
	Chapter 6 Problems
	Chapter 6 Computer Programs
	Bibliography
7 Direct Search Methods for Nonlinear Optimization
	7.1 Introduction
	7.2 Cyclic Coordinate Search
	7.3 Hooke and Jeeves Pattern Search Method
	7.4 Rosenbrock’s Method
	7.5 Powell’s Method of Conjugate Directions
	7.6 Nelder and Mead Simplex Method
	7.7 Simulated Annealing (SA)
	7.8 Genetic Algorithm (GA)
	7.9 Differential Evolution (DE)
	7.10 Box’s Complex Method for Constrained Problems
	7.11 Transformation of Variables to Remove some Bounds and Constraints
	Chapter 7 Problems
	Chapter 7 Computer Programs
	Bibliography
8 Multiobjective Optimization
	8.1 Introduction
	8.2 Concept of Pareto Optimality
	8.3 Significance and Generation of the Entire Pareto Surface
	8.4 Methods to Identify a Single Best Compromise Solution
	Chapter 8 Problems
	Chapter 8 Computer Programs
	Bibliography
9 Integer and Discrete Programming
	9.1 Introduction
	9.2 Zero–One Programming
	9.3 Branch and Bound Algorithm for Mixed Integers (LP-based)
	9.4 Gomory Cut Method
	9.5 Farkas’ Method for Discrete Nonlinear Monotone Structural Problems
	9.6 Genetic Algorithm for Discrete Programming
	Chapter 9 Problems
	Chapter 9 Computer Programs
	Bibliography
10 Dynamic Programming
	10.1 Introduction
	10.2 The Dynamic Programming Problem and Approach
	10.3 Problem Modeling and Computer Implementation
	Chapter 10 Problems
	Chapter 10 Computer Programs
	Bibliography
11 Optimization Applications for Transportation, Assignment, and Network Problems
	11.1 Introduction
	11.2 Transportation Problem
	11.3 Assignment Problems
	11.4 Network Problems
	Chapter 11 Problems
	Chapter 11 Computer Programs
	Bibliography
12 Finite Element and Simulation-Based Optimization
	12.1 Introduction
	12.2 Derivative Calculations
	12.3 Sizing (i.e. Parameter) Optimization via Optimality Criteria and Nonlinear Programming Methods
	12.4 Topology Optimization of Continuum Structures
	12.5 Shape Optimization
	12.6 Optimization With Dynamic Response
	Chapter 12 Problems
	Chapter 12 Computer Programs
	Bibliography
Index