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This textbook is for readers new or returning to the practice of optimization whose interest in the subject may relate to a wide range of products and processes. Rooted in the idea of “minimum principles,” the book introduces the reader to the analytical tools needed to apply optimization practices to an array of single- and multi-variable problems. While comprehensive and rigorous, the treatment requires no more than a basic understanding of technical math and how to display mathematical results visually. It presents a group of simple, robust methods and illustrates their use in clearly-defined examples. Distinct from the majority of optimization books on the market intended for a mathematically sophisticated audience who might want to develop their own new methods of optimization or do research in the field, this volume fills the void in instructional material for those who need to understand the basic ideas.

The text emerged from a set of applications-driven lecture notes used in optimization courses the author has taught for over 25 years. The book is class-tested and refined based on student feedback, devoid of unnecessary abstraction, and ideal for students and practitioners from across the spectrum of engineering disciplines. It provides context through practical examples and sections describing commercial application of optimization ideas, such as how containerized freight and changing sea routes have been used to continually reduce the cost of moving freight across oceans. It also features 2D and 3D plots and an appendix illustrating the most widely used MATLAB optimization functions.

### Chapter 1. Optimization: The Big Idea

Optimization, no matter how it is applied, is built around a small number of basic ideas. The tools of optimization maximize or minimize some quantity called an objective function. The things that can be changed to make this happen are the design variables. Restrictions on values of the design variables are constraints. With these three ideas in mind, it is easy to notice examples of optimization all around you. Modern optimization usually requires mathematical calculations, but people began optimizing products and processes long before computers and modern analytical methods were available.
Mark French

### Chapter 2. Getting Started in Optimization: Problems of a Single Variable

One measure of the size of an optimization problem is the number of design variables – more design variables imply a larger and, thus, more challenging optimization problem. It’s helpful to start with problems of a single variable in order to become comfortable with the overall ideas in optimization before moving on to problems with more variables. It helps, also, that methods which work for any number of variables sometimes include a step that uses single variable methods.
Mark French

### Chapter 3. Minimum Principles: Optimization in the Fabric of the Universe

Minimum principles are routinely used in designing products and processes. However, they lie at the heart of many physical laws. Nature seems to be parsimonious, often acting in a way that minimizes effort in some way. This grand idea underlies topics as varied as evolution and structural mechanics. Optimization seems to be woven into the fabric of existence.
Mark French

### Chapter 4. Problems with More than One Variable

Almost all useful optimization problems have more than one design variable, so it’s important to understand methods that work for any number of variables. Conveniently, some of these methods are just extensions of single-variable methods. The methods in this chapter are not hard to implement and some equivalent functions are available in MATLAB. We will focus on two variable problems since they can be represented using surface or contour plots. However, methods that work for two design variables generally work for any larger number of design variables as well.
Mark French

### Chapter 5. Constraints: Placing Limits on the Solution

Most practical optimization problems include limits, called constraints, on the values that the design variables can take. For example, constraints may describe allowable limits on the stresses in a structure or the maximum speed of a ship moving cargo across an ocean. This chapter presents the basic concepts and a method for constrained optimization that is a simple extension of familiar unconstrained methods.
Mark French

### Chapter 6. General Conditions for Solving Optimization Problems: Karush-Kuhn-Tucker Conditions

There is a formal set of mathematical conditions that describe the minima of constrained or unconstrained optimization problems. Called the Karush-Kuhn-Tucker (KKT) conditions, they can sometimes be used to solve directly for minimum points in design space. This chapter shows a series of examples on how the KKT conditions can be used to find the minima of unconstrained and constrained optimization problems.
Mark French

### Chapter 7. Discrete Variables

Most optimization methods assume that the design variables are continuous and can take any value that satisfies the constraints. It is generally true of the design variables used to describe many problems, but there is a class of problems in which the design variables can only take discrete values. An example might be the number of engines to power a ship or the number of trucks needed to support a product delivery organization. Since the conventional ideas of slope and curvature don’t generally work with discrete variables, there are methods specifically for problems involving discrete variables. To date, discrete methods are generally capable of finding approximate solutions.
Mark French

### Chapter 8. Aerospace Applications

It is a general rule that the closer one is to the limits of what is possible, the more engineering is needed. Since the aerospace world often works near this limit, airplanes and spacecraft have seen some of the important applications of optimization. For example, the payload of a large rocket may comprise only a few percent of the takeoff weight. Thus any decrease in the weight of the structure may disproportionately increase the allowable payload.
Mark French

### Chapter 9. Structural Optimization

One of the earliest applications of optimization to engineering design was that of designing structures for minimum weight. While widely applied, it is particularly useful for aerospace structures, where the penalty for dragging unneeded weight into the sky is often quite high but where even minor structural failures can be catastrophic. The methods used for structural optimization are not unique, but the problem formulations are. A particular application of aerospace structural optimization is that of designing aeroelastically scaled wind tunnel models. These represent a type of inverse problem in which the mass and stiffness properties are known beforehand and the designer must create a structure that both fits inside the required aerodynamic envelope and has those properties.
Mark French

### Chapter 10. Multiobjective Optimization

There are often problems in which it isn’t possible to identify a single objective function. For example, in a mechanical system, it might not be possible to isolate either cost or weight as the objective. Or, for a package delivery organization, it might not be possible to choose between delivery time or delivery cost as the objective function. These problems are part of a class called multiobjective problems – problems with more than one objective function. There is an entire body of literature devoted to multiobjective problems, and some of the basic ones are presented here.
Mark French