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class_8 - CS205 Class 8 Covered In Class 1 3 4 5 6 Reading...

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CS205 – Class 8 Covered In Class : 1, 3, 4, 5, 6 Reading : Heath Chapter 6 1. Optimization – given an objective function f , find relative maxima or minima. Note that since max f = min f it is enough to only consider minima. a. We’ll start with scalar functions f of one variable for now. b. unconstained – any x n R is acceptable c. constrained – minimize f on a subset S R d. usually find local minima, since global minima are hard to find i. one option is to find many local minima and compare them to find a global minimum e. Not equivalent to solving for f(x) = 0. There might exist no such x or the minimum may be attained somewhere f(x) < 0. f. poorly conditioned since '( ) 0 fx at a minimum, i.e. locally flat (similar to a multiple root) – error tolerance should be more like as opposed to g. given a critical point where '( ) 0 , we can use the sign of the second derivative to determine whether we have a local minimum, a local maximum, or an inflection point i. if ''( ) 0 , concave up, minimum ii. if ''( ) 0 , concave down, maximum iii. otherwise when the second derivative vanishes, we have an inflection point, i.e. neither a minimum nor a maximum -2 0 2 -2 0 2 -15 -10 -5 0 -2 0 2 -2 0 2 5 10 15 -5 -2.5 0 2.5 5 -5 -2.5 0 2.5 5 -20 0 20 Local maxima Local minima Saddle h. unimodal [ , *] ax is monotonically decreasing and [ *, ] xb is monotonically increasing * x is the minimum – most schemes need a unimodal interval in order to converge 2. golden section search - Using the magic number ( 5 1)/2
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