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chapt4 - Section 4.1 Minimization of Functions of One...

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Section 4.1: Minimization of Functions of One Variable Unconstrained Optimization 4 In this chapter we study mathematical programming techniques that are commonly used to extremize nonlinear functions of single and multiple ( n ) design variables subject to no constraints. Although most structural optimization problems involve constraints that bound the design space, study of the methods of unconstrained op- timization is important for several reasons. First of all, if the design is at a stage where no constraints are active then the process of determining a search direction and travel distance for minimizing the objective function involves an unconstrained func- tion minimization algorithm. Of course in such a case one has constantly to watch for constraint violations during the move in design space. Secondly, a constrained optimization problem can be cast as an unconstrained minimization problem even if the constraints are active. The penalty function and multiplier methods discussed in Chapter 5 are examples of such indirect methods that transform the constrained min- imization problem into an equivalent unconstrained problem. Finally, unconstrained minimization strategies are becoming increasingly popular as techniques suitable for linear and nonlinear structural analysis problems (see Kamat and Hayduk[1]) which involve solution of a system of linear or nonlinear equations. The solution of such systems may be posed as finding the minimum of the potential energy of the system or the minimum of the residuals of the equations in a least squared sense. 4.1 Minimization of Functions of One Variable In most structural design problems the objective is to minimize a function with many design variables, but the study of minimization of functions of a single de- sign variable is important for several reasons. First, some of the theoretical and numerical aspects of minimization of functions of n variables can be best illustrated, especially graphically, in a one dimensional space. Secondly, most methods for un- constrained minimization of functions f ( x ) of n variables rely on sequential one- dimensional minimization of the function along a set of prescribed directions, s k , in the multi-dimensional design space R n . That is, for a given design point x 0 and a specified search direction at that point s 0 , all points located along that direction can be expressed in terms of a single variable α by x = x 0 + α s 0 , (4 . 1 . 1) 115
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Chapter 4: Unconstrained Optimization where α is usually referred to as the step length. The function f ( x ) to be minimized can, therefore, be expressed as f ( x ) = f ( x 0 + α s 0 ) = f ( α ) . (4 . 1 . 2) Thus, the minimization problem reduces to finding the value α * that minimizes the function, f ( α ). In fact, one of the simplest methods used in minimizing functions of n variables is to seek the minimum of the objective function by changing only one variable at a time, while keeping all other variables fixed, and performing a one- dimensional minimization along each of the coordinate directions of an n -dimensional design space. This procedure is called the
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