By Staff of Aero Publishers

Approximately 60 illustrations (4 pages in color). specified plane development images and lots of of the opposite photos are full-page. nice profile drawings.

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**Example text**

10 3) The cost and constraint functions are usually implicit as well as explicit functions of the design variables. These functions, however, can be calculated using various analysis procedures once a design is specified. Their derivatives can also be evaluated using special procedures. This is commonly known as design sensitivity analysis, which is also discussed elsewhere in this volume. Basic Definitions and Existence of Solution Before presenting various concepts related to design optimization, it is prudent to define what is meant by a minimum point for the function f(X) and to discuss the existence of the optimum solution for the problem P.

Purchased from American Institute of Aeronautics and Astronautics MATHEMATICAL PROGRAMMING METHODS l 33 ) (11) This search direction is used in Eq. (9) to update the design. It is surprising how often this method is used today and presented as an "advanced" algorithm. However, it must be stated that this method is one of the worst available and should never be used. The steepest descent method will seldom converge reliably to a solution in even the simplest of nonlinear minimization tasks. Furthermore, it is almost trivial to modify this method to make it efficient, although this simple modification is still not considered to be the best algorithm available.

Usually the answer to this question is that the solution is only a local minimum. The global solution for the problem can be found by either some exhaustive search of the constraint set or by showing the problem to be convex. Both procedures can require extensive computations. If the problem is convex, then any local minimum is also a global minimum, and the KKTfirstorder conditions are necessary as well as sufficient. The question of convexity of a problem is briefly addressed here. If the cost function f(X) is convex over the convex constraint set 5, then the problem P is called a convex programming problem.