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lec16_10182006

# lec16_10182006 - 10.34 Numerical Methods Applied to...

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10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #16: Unconstrained Optimization. Unconstrained Optimization min x f(x ) x [k+1] x guess Require: x [k] f ( x [k+1] )<f(x [k] ) Which direction to move? Move Downhill Æ “Steepest Descent” very robust but poor convergence at the end x Figure 1. Diagram of steepest descent approach to global minimum. Unless you start on the center line, you will zigzag inefficiently going down contour lines is easy with this method f approx (x ) = f (x [k] ) + f | x [ k] ·(x –x [k] ) + ½(x –x [k] ) T B ·(x – x [k] ) (x – x [k] ) = p = - f (|| f || 2 / f T B f ) {Cauchy} B must be positive definite and not singular. p cauchy (- f /||f ||) Δ Å max step size allowed p steepest descent = p cauchy t a k e m i n | | p || Look at Figures 5.5 and 5.6 in BEERS Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

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Figure 2. An example of poor scaling. 10.34, Numerical Methods Applied to Chemical Engineering
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lec16_10182006 - 10.34 Numerical Methods Applied to...

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