Basic Concepts of Optimization
The concept of optimization is basic to much of what we do in our daily lives: a desire to do better or be the best in one field or another. In engineering we wish to produce the best possible result with the available resources. In a highly competitive modern world it is no longer sufficient to design a system whose performance of the required task is just satisfactory. It is essential to design the best system. Thus in “designing” new products in any field: aerospace, automotive, chemical, electrical, biomedical, agricultural, etc, we must use design tools which provide the desired results in a timely and economical fashion. Numerical optimization is one of the tools at our disposal.
Optimization is a very general automated design technique. In studying this technique it is important to distinguish between analysis and design. Analysis is the process of determining the response of the specified system to the certain combination of input parameters. For example, calculatin stresses in the structure as a result of certain loads. Design on the other hand, means the process of defining a system. For example, designing a structure would mean selecting specific dimensions and location of the structural members that will allow the structure to withstand the specified load.
Much of the design task in engineering is quantifiable, and so we are able to use computers to analyze alternative designs rapidly. The purpose of numerical optimization is to aid us in rationally searching among alternative designs for the best design to meet our needs.
The alternative designs of the same system differ from each other because some parameters of the system are not the same. The parameters that could be changed in the system while searching for the best design are called design variables. Although we may not always think of it this way, design process may be defined as the process of finding the minimum or maximum of some characteristic, which may be called the objective function. For the design to be acceptable it must also satisfy certain requirements. These requirements are called design constraints. Optimization automatically changes the design variables to help us find the minimum or maximum of the objective function, while satisfying all the required design constraints.
Consider example in Figure 1. One boy bets that he can locate the top of the hill while blindfolded. The other boy agrees but asks the first boy to also stay inside the fences. Translating this situation into optimization problem formulation, we see that the objective is to find the highest point on the hill. Therefore, objective function is the height achieved by the first boy with respect to his original position. The design variables are longitude and latitude – the coordinates, defining position of the boy. The constraints are that the boy has to stay inside the fences. Note here, that in general, the boy may start the search from outside the fences.
It is possible to define this physical problem mathematically, thus converting it to the engineering problem as shown in Figure 2:
Optimization is a very simple extension of the engineering problem:
Recall, that optimization automatically changes the design variables to helps us find the minimum or maximum of the objective function, while satisfying all the required constraints. The optimization process is illustrated in Figure 3. It may be broken down into the following steps:
• Find a search direction that will improve the objective while staying inside the fences;
• Search in this direction until no more improvement can be made by going in this direction;
• Repeat the process, until no search direction can be found that improves the objective.
The optimization problem formulation and the optimization process presented above are very general and can be applied to any design problem in any field. For example, if we wish to design the internal combustion engine, the objective may be to maximize the combustion efficiency. The engine may be required to provide a specific power output with an upper limit on the amount of harmful pollutants emitted into the atmosphere. These parameters will serve as constraints for optimization. The design variables that are allowed to be changed during optimization may be the compression ratio, air-fuel mixture ratio, bore and stroke, etc.
Optimization is not limited to engineering only. It is possible to optimize the financial portfolio, optimize the revenue and expenditures of a company, optimize the route of a delivery truck, optimize the chemical processing, optimize protein models, etc.
It is not debatable that optimization is useful. It has been successfully working for many years. However, at the same time it is by no means a replacement for a designer. It is also not a “push-button” tool that instantly produces the best design without human intervention. It is a process that reduces the design time, improves the design quality, and free engineers and designers for creative work by taking over tedious operations.
Optimization is the most powerful design improvement tool that is available today!