hw1 - k ( x ) = exp (-x T x ) is not. 4. Plot the function...

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22M:270 Optimization Techniques Homework 1 — due Friday Jan 27 1. Find all the critical points of the following function f ( x , y ) = x 4 - x 2 y + 2 y 2 - 2 y . Determine if they are local minima, local maxima, or saddle points, by look- ing at the Hessian matrices at the critical points. Evaluate the function at the critical points. Can you determine if the function is coercive (that is, f ( x , y ) if p x 2 + y 2 )? Do you think that you have found the global minimum? Explain why. 2. Repeat Q1 for the following function f ( x , y ) = x 3 + 2 xy + y 2 . 3. Suppose that g : R n R is convex and f : R R is convex and increasing. Show that f g : R n R is convex. Use this to show that h ( x ) = exp ( x T x ) is convex; show as a counterexample that
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Unformatted text preview: k ( x ) = exp (-x T x ) is not. 4. Plot the function f ( x ) = cos ( 3 x )-. 5sin ( 5 x )+ . 2cos ( 10 x- / 4 ) . Esti-mate how many critical points it has on the interval [ , 2 ] . Now consider g : R 20 R given by g ( x ) = f ( x 1 ) f ( x 2 ) f ( x 20 ) . Show that g ( x ) = is equivalent to f ( x i ) = 0 for all i . From this, estimate how many critical points g has in the cube [ , 2 ] 20 . Clearly there is a potential to have many (irrelevant) local minima. Can you suggest ways of handling problems like this? 1...
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This note was uploaded on 04/01/2012 for the course 22M 174 taught by Professor Davidstewart during the Spring '12 term at University of Iowa.

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