NM6_optim_f01 - CGN 3421 - Computer Methods Gurley...

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CGN 3421 - Computer Methods Gurley Numerical Methods Lecture 6 - Optimization page 86 of 94 Numerical Methods Lecture 6 - Optimization Topics: numerical optimization - Newton again - Random search - Golden Section Search Optimization - motivation What? Locating where some function reaches a maximum or minimum Find x where or Why? For example: A function represents the cost of a project minimize the cost of the project When? When an exact solution is not available or a big pain Example 1: Given: , Find x where y is minimum analytical solution: ==> Example 2: Given: , Find x where y is minimum Depends on the range we’re interested in. .. Having lots of local maxima and minima means having lots of zero slope cases. An exact solution would be a big pain. .. H Z () OCZ ± HZ OKP ± [ ²³ Z ´ Ō µ ± F[ FZ ----- ·µ Z ´ Ō ¸ ±± Z ´ ± [Z UKP ´ ·¹² Z ² Ō UKP Ō GZR ±
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CGN 3421 - Computer Methods Gurley Numerical Methods Lecture 6 - Optimization page 87 of 94 Single variable - Newton Recall the Newton method for finding a root of an equation , where We can use a similar approach to find a min or max of The min / max occurs where the slope is zero So if we find the root of the derivative, we find the max / min location Find roots of using Newton = Example: Newton Method find the maximum of this function we’ll need these for x = [0,3] OR we can use central difference to replace the analytical derivatives. Z M ± ² Z M HZ M () H Ŏ Z M ---------------- Ō ³ H Ŏ Z M FH Z FZ ------------ ³ H Z H Ŏ Z IZ ´ ³³ Z M ± ² Z M M I ' Z M -------------- Ō ³ Z M H Ŏ Z H ŎŎ Z --------------- Ō to find where the max occurs. ..
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This note was uploaded on 06/10/2011 for the course CGN 3421 taught by Professor Long during the Spring '08 term at University of Florida.

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NM6_optim_f01 - CGN 3421 - Computer Methods Gurley...

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