Homework2_2011_sol

# Homework2_2011_sol - ESE 415 Optimization Assignment 2 Due...

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ESE 415 Optimization Assignment 2 Due: February 16, 2011 1. Consider f ( x ) = 2 x 2 1 + x 1 x 2 + x 2 2 + x 2 x 3 + x 2 3 - 6 x 1 - 7 x 2 - 8 x 3 + 9. i) Compute the gradient f ( x ) and Hessian H ( x ) of f . Solution: f ( x ) = 4 x 1 + x 2 - 6 x 1 + 2 x 2 + x 3 - 7 x 2 + 2 x 3 - 8 and H ( x ) = 4 1 0 1 2 1 0 1 2 ii) Evaluate f ( x ) and H ( x ) at x = 0 = 0 0 0 . Solution: f 0 0 0 = - 6 - 7 - 8 and H 0 0 0 = 4 1 0 1 2 1 0 1 2 iii) Obtain the second order approximation of f at x = 0 , i.e., the Taylor series up to the 2nd order g ( x ) = f ( x * ) + ( x - x * ) T f ( x * ) + 1 2 ( x - x * ) T 2 f ( x * )( x - x * ) , where x = 0 . Is g ( x ) identical to f ( x )? Solution: g ( x ) = f (0) + ( x - 0) T f (0) + 1 2 ( x - 0) T 2 f (0)( x - 0) , g ( x ) = 9 + ± x 1 x 2 x 3 ² - 6 - 7 - 8 + 1 2 ± x 1 x 2 x 3 ² 4 1 0 1 2 1 0 1 2 x 1 x 2 x 3 g ( x ) = 9 - 6 x 1 - 7 x 2 - 8 x 3 + 2 x 2 1 + x 2 2 + x 2 3 + x 1 x 2 + x 2 x 3 = f ( x ) 1

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2. Show the following convexity results. (a) If
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Homework2_2011_sol - ESE 415 Optimization Assignment 2 Due...

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