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Unformatted text preview: ams/econ 11b supplementary notes ucsc Optimization in several variables I, first and second order conditions c 2010, Yonatan Katznelson This note covers the basics of (unconstrained) optimization, and attempts to explain things like • why critical points are critical, and • why the second derivative test works. The key definitions and facts are summarized in neat little boxes, (like this one). 1. Local extreme values and critical points. 1.1 Local extreme values. The value z = f ( x ,y ) is called a local minimum value of the function if there is a number δ > 0 such that f ( x,y ) ≥ f ( x ,y ) (1) for all points ( x,y ) satisfying  x x  < δ and  y y  < δ . Likewise, z 1 = f ( x 1 ,y 1 ) is a local maximum value if f ( x,y ) ≤ f ( x 1 ,y 1 ) (2) for all points ( x,y ) satisfying  x x 1  < δ and  y y 1  < δ , for some number δ > 0. † The local maximum and minimum values of a function are also called local extreme values , when the distinction between maximum and minimum is not important. More generally, ˜ w = f (˜ x 1 ,..., ˜ x n ) is a local maximum (or minimum) value if and only if there is some number δ > 0 such that f ( x 1 ,...,x n ) ≤ f (˜ x 1 ,..., ˜ x n ) (or f ( x 1 ,...,x n ) ≥ f (˜ x 1 ,..., ˜ x n ) , respectively) for all points ( x 1 ,...,x n ) satisfying  x j ˜ x j  < δ , for j = 1 , 2 ,...,n . 1.2 First order conditions. By studying the first order Taylor approximation of the function z = f ( x,y ), we can identify the points ( x ,y ) where local extreme values might occur. Recall that the first order approximation can be written f ( x,y ) f ( x ,y ) ≈ f x ( x ,y )( x x ) + f y ( x ,y )( y y ) , (3) where the approximation is accurate for all points ( x,y ) sufficiently close to ( x ,y ), i.e., for all points satisfying  x x  < δ and  y y  < δ for some positive number δ . ‡ † The condition ‘  x x 1  < δ and  y y 1  < δ , for some number δ > 0’ is a more precise way of saying ‘sufficiently close to ( x 1 ,y 1 )’. ‡ For our purposes in this note, the precise value of δ is not important. The important thing is that δ exists. 1 Suppose that f x ( x ,y ) > 0, then by choosing y = y and x 1 > x , the approximation (1.3) tells us that f ( x 1 ,y ) f ( x ,y ) ≈ f x ( x ,y )( x 1 x ) > , as long as x 1 is sufficiently close to x . This means that f ( x ,y ) cannot possibly be a local maximum value, since there are points arbitrarily close to ( x ,y ) where the function takes larger values. Still assuming that f x ( x ,y ) > 0 and staying with y = y , but choosing x 1 < x , we see that f ( x 1 ,y ) f ( x ,y ) ≈ f x ( x ,y )( x 1 x ) < , as long as x 1 is sufficiently close to x . This implies that f ( x ,y ) cannot possibly be a local minimum value, since there are points arbitrarily close to ( x ,y ) where the function takes smaller values....
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 Spring '08
 BINICI

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