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± ² 7 OPTIMIZATION 46 7 OPTIMIZATION The engineer is continually faced with non-trivial decisions, and discerning the best among alternatives is one of the most useful and general tasks that one can master. Optimization exists because in nearly every endeavor, one is faced with tradeoffs. Here are some examples: Contributing to savings versus achieving enjoyment from purchases made now; Buying an expensive bicycle from one of many manufacturers - you are faced with choices on accessories, weight, style, warranty, performance, reputation, and so on; Writing a very simple piece of code that can solves a particular problem versus devel- oping a more professional and general-use product; Size of the column to support a roof load; How fast to drive on the highway; Design of strength bulkheads inside an airplane wing assembly The field of optimization is very broad and rich, with literally hundreds of different classes of problems, and many more methods of solution. Central to the subject is the concept of the parameter space denoted as X , which describes the region where specific decisions x may lie. For instance, acceptable models of a product off the shelf might be simply indexed as x i . x can also be a vector of specific or continuous variables, or a mixture of the two. Also critical is the concept of a cost f ( x ) that is associated with a particular parameter set x .W e c a n say that f will be minimized at the optimal set of parameters x : f ( x ) = min f ( x ) . x±X We will develop in this section some methods for continuous parameters and others for discrete parameters. will consider some concepts also from planning and multi-objective optimization, e.g., the case where there is more than one cost function. 7.1 Single-Dimension Continuous Optimization Consider the case of only one parameter, and one cost function. When the function is known and is continuous - as in many engineering applications - a very reasonable first method to try is to zero the derivative. In particular, df ( x ) =0 . dx x = x The user has to be aware even in this first problem that there can exist multiple points with zero derivative. These are any locations in X where f ( x ) is flat, and indeed these could be
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7 OPTIMIZATION 47 at local minima and at local and global maxima. Another important point to note is that if X is a finite domain, then there may be no location where the function is flat. In this case, the solution could lie along the boundary of X , or take the minimum within X . In the figure below, points A and C are local maxima, E is the global maxima, B and D are local minima, and F is the global minimum shown. However, the solution domain X does not admit F, so the best solution would be B. In all the cases shown, however, we have at the maxima and minima f ± ( x ) = 0. Furthermore, at maxima f ±± ( x ) < 0, and at minima f ±± ( x ) > 0.
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This note was uploaded on 11/29/2011 for the course CIVIL 1.00 taught by Professor Georgekocur during the Spring '05 term at MIT.

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