hw2-ans - 22M:174 Optimization Techniques Homework 2...

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Unformatted text preview: 22M:174 Optimization Techniques Homework 2 answers 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 . Using Taylor series with integral 2nd order remainder, f ( y ) = f ( x )+ ∇ f ( x ) T ( y- x )+ Z 1 ( 1- s )( y- x ) T ∇ 2 f ( x + s ( y- x ))( y- x ) ds . If ∇ 2 f ( z ) is positive semi-definite for all z , then ( y- x ) T ∇ 2 f ( x + s ( y- x ))( y- x ) ≥ 0 for all x , y , and s . So then f ( y ) ≥ f ( x )+ ∇ f ( x ) T ( y- x ) for all y and x , and thus f is convex. On the other hand, if f is convex then f ( y ) ≥ f ( x )+ ∇ f ( x ) T ( y- x ) for all x and y . Thus R 1 ( 1- s )( y- x ) T ∇ 2 f ( x + s ( y- x ))( y- x ) ds ≥ 0 for all x and y . Putting y = x + α d for d and α 6 = 0, we have α 2 R 1 ( 1- s ) d ∇ 2 f ( x + s α d ) d ds ≥ 0. Since α 2 > 0, we can divide by α 2 to get R 1 ( 1- s ) d T ∇ 2 f ( x + s α d ) d ds ≥ 0. Taking α → 0 we get R 1 ( 1- s ) d T ∇ 2 f ( x ) d ds = 1 2 d T ∇ 2 f...
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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-ans - 22M:174 Optimization Techniques Homework 2...

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