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

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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.
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Unformatted text preview: method (and has be re-discovered many times): Input: f , ∇ f , x , α > , < c 1 < 1 , search direction p Output: α where f ( x + s p ) ≤ f ( x ) + c 1 α p T ∇ f ( x ) α ← α while f ( x + α p ) > f ( x ) + c 1 α p T ∇ f ( x ) α ← α / 2 end while 1 Show that this algorithm terminates in finite 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 + α p ) ≥ f ( x ) ... If we use x + α 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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