Nonlinear Optimization

Models and Applications

William P. Fox

• Publisher: Chapman and Hall/CRC
• Year: 2020
• Print ISBN: 9780367444150
• eBook ISBN: 9781000196962
• Edition: 1
• Topic: Mathematics
• Subtopic: Operations
• Index: Mathematics

Optimization is the act of obtaining the "best" result under given circumstances. In design, construction, and maintenance of any engineering system, engineers must make technological and managerial decisions to minimize either the effort or cost required or to maximize benefits. There is no single method available for solving all optimization problems efficiently. Several optimization methods have been developed for different types of problems. The optimum-seeking methods are mathematical programming techniques (specifically, nonlinear programming techniques).

Nonlinear Optimization: Models and Applications presents the concepts in several ways to foster understanding. Geometric interpretation: is used to re-enforce the concepts and to foster understanding of the mathematical procedures. The student sees that many problems can be analyzed, and approximate solutions found before analytical solutions techniques are applied. Numerical approximations: early on, the student is exposed to numerical techniques. These numerical procedures are algorithmic and iterative. Worksheets are provided in Excel, MATLAB®, and Maple™ to facilitate the procedure. Algorithms: all algorithms are provided with a step-by-step format. Examples follow the summary to illustrate its use and application.

Nonlinear Optimization: Models and Applications:

• 
  
  

• Emphasizes process and interpretation throughout

• 
  
  

• Presents a general classification of optimization problems

• 
  
  

• Addresses situations that lead to models illustrating many types of optimization problems

• 
  
  

• Emphasizes model formulations

• 
  
  

• Addresses a special class of problems that can be solved using only elementary calculus

• 
  
  

• Emphasizes model solution and model sensitivity analysis

About the author:

William P. Fox is an emeritus professor in the Department of Defense Analysis at the Naval Postgraduate School. He received his Ph.D. at Clemson University and has taught at the United States Military Academy and at Francis Marion University where he was the chair of mathematics. He has written many publications, including over 20 books and over 150 journal articles. Currently, he is an adjunct professor in the Department of Mathematics at the College of William and Mary. He is the emeritus director of both the High School Mathematical Contest in Modeling and the Mathematical Contest in Modeling.

Chapter 1

Introduction to Optimization Models

1.1 Introduction

Optimization is the act of obtaining the “best” result under given circumstances. In design, construction, and maintenance of any engineering system, engineers have to take many technological and managerial decisions at several stages. The ultimate goal of all such decisions is to minimize either the effort or cost required and maximize benefits. Since the effort required and the benefits desired in any practical situation are expressed as a function of certain decision variables, optimization can be defined as the process of finding the conditions that give the maximum or the minimum value of the function. As seen in Figure 1.1, the point x corresponds to the maximum value of f(x). There is no single method available for solving all optimization problems efficiently. Hence, a number of optimization methods have been developed for solving different types of optimization problems.

The optimum-seeking methods are also known as mathematical programming techniques (specifically, nonlinear programming techniques) and are generally studied as part of operations research or applied mathematics. Operations research is a branch of mathematics that is concerned with the application of scientific methods and techniques to decision-making problems, and with establishing the best or optimal solutions. Table 1.1 lists various mathematical techniques used in the areas of operations research. As operations research is ever evolving, the list is always growing.

Table 1.1 Some Operations Research Methods

	Methods of Operations Research
Mathematical Programming	Stochastic Techniques	Statistical Methods
Calculus	Decision Theory	Regression Analysis
Calculus of Variations	Markov Processes	Cluster Analysis
Linear Progra...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface: Nonlinear Optimization—Models and Applications
• Acknowledgments
• Author
• 1 Introduction to Optimization Models
• 2 Review of Differential Calculus
• 3 Single-Variable Unconstrained Optimization
• 4 Numerical Search Techniques in Single-Variable Optimization
• 5 Review of Multivariable Differential Calculus
• 6 Models Using Unconstrained Optimization: Maximization and Minimization with Several Variables
• 7 Multivariate Optimization Search Techniques
• 8 Optimization with Equality Constraints
• 9 Inequality Constraints: Necessary/Sufficient Kuhn–Tucker Conditions (KTC)
• 10 Specialized Nonlinear Optimization Methods
• 11 Dynamic Programming
• 12 Data Analysis with Regression Models, Advanced Regression Models, and Machine Learning through Optimization
• Answers to Selected Problems
• Index