Lecture10-2003 - Lecture X Finding the Minimum Using...

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1 Lecture X Finding the Minimum Using Newton’s Method I. Finding the Univariate Minimum (Algorithm1.ma) A. A Sufficiently Complex Function 1. It is obvious that finding the minimum of a simple univariate quadratic function is trivial given the rules we discussed in the preceding section. For example U x x x ( ) = + 5 4 2 2 has a minimum determined by its first derivative U x x x x ( ) = = = 10 4 0 2 5 . In addition, straightforward transformations such as e x x 5 4 2 2 + offer little additional complexity [ ] U x x U x x x ( ) ( ) = = = 10 4 0 2 5 . 2. One possibility is the rational function f x x x x ( ) = + + 5 4 2 10 2 . f(x) 0 5 10 15 20 25 30 35 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 However, these functions typically have bizarre discontinuities. For example, plotting the above function over a broader range indicates that
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AEB 6533 – Static and Dynamic Optimization Lecture 10 Professor Charles B. Moss 2 -208.2 -800 -600 -400 -200 0 200 400 600 -19 -16 -13 -9 -6 -3 0 3 6 9 12 15 18 If we restrict our attention to the range of x’s such that x is greater than -10 the problem becomes more tractable.
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