hw2 - method (and has be re-discovered many times): Input:...

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22M:174 Optimization Techniques Homework 2 — due Friday Feb 3 1. Show using Taylor series that a function f : R n R is convex if and only if the Hessian matrix 2 f ( x ) is positive semi-definite for all x . 2. Implement a simple steepest descent algorithm with fixed step size param- eter s . The algorithm can be represented in pseudo-code as follows: Input: f , f , x 0 , step parameter s > 0 , tolerance ε > 0 , maximum iteration count k max Output: solution estimate x x x 0 ; k 0 while k f ( x ) k ≥ ε k k max x x - s f ( x ) print k , x , f ( x ) k k + 1 end while Try this out on the function f ( x , y ) = x 4 - x 2 y + 2 y 2 - 2 y with x 0 = ( x 0 , y 0 ) = ( 0 , 0 ) . Use s = 1 , 0 . 1 , 0 . 01 , 0 . 001 and set k max = 100. Comment on the behavior you see. Also try it out on the function f ( x , y ) = ( x - 1 ) 2 + 100 ( y - x 2 ) 2 with x 0 = ( x 0 , y 0 ) = ( 2 , 0 ) for the same values of s . Do not just hand in your output, which will probably be very long. Edit it down to something easy to read and understand. You may need a text editor (for example, gedit ). You might also use the diary command in Matlab to say the values of k , x and f ( x ) printed out. 3. The Goldstein/Armijo line search algorithm is a very well-known line search
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Unformatted text preview: method (and has be re-discovered many times): Input: f , f , x , > , < c 1 < 1 , search direction p Output: where f ( x + s p ) f ( x ) + c 1 p T f ( x ) while f ( x + p ) > f ( x ) + c 1 p T f ( x ) / 2 end while 1 Show that this algorithm terminates in nite time provided p T f ( x ) < and f is at least once continuously differentiable. 4. Many programmers implementing a line search algorithm like the Gold-stein/Armijo line search might use the condition while f ( x + p ) f ( x ) ... If we use x + p with p =- f ( x ) for the new value of x after each line search, this method might not converge to a critical point. Can you explain why? 2...
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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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hw2 - method (and has be re-discovered many times): Input:...

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