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Homework2_2011

# Homework2_2011 - f x 1 ∥ x 3 − x 1 ∥ ≥ f x 2 − f...

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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. (a) Compute the gradient f ( x ) and Hessian H ( x ) of f . (b) Evaluate f ( x ) and H ( x ) at x = 0 = 0 0 0 . (c) 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 )? 2. Show the following convexity results. (a) If f : R n R is a convex function and x = Ay + b , where A R n × n and b R n , then g ( y ) = f ( Ay + b ) is convex. (b) Let C R n be a convex set and let f : C R be twice continuously differentiable over C . If C = R n and f is convex, then H ( x ) 0, x C (Hint: show this by contradiction and use the continuity of 2 f ). (c) Show that if f 1 and f 2 are convex functions then their pointwise maximum f , defined by f ( x ) = max { f 1 ( x ) , f 2 ( x ) } , with dom f = dom f 1 dom f 2 , is also convex, where dom f denotes the domain of f . (d) Let x 1 , x 2 , x 3 R n such that x 2 ( x 1 , x 3 ). Show that if f is convex in R n then f ( x 3 )

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Unformatted text preview: f ( x 1 ) ∥ x 3 − x 1 ∥ ≥ f ( x 2 ) − f ( x 1 ) ∥ x 2 − x 1 ∥ . 3. Let C ⊂ R n be a convex set and let f ( x ) = h ( g ( x )), where g : C → R and h : R → R . (a) Let f ( x ) = e g ( x ) . If g ( x ) is convex, then f ( x ) is convex. True or false? Why? (b) Let f ( x ) = 1 g ( x ) . If g is convex and positive, then f is convex. True or false? Why? 1 4. (a) Show that x 4 + 3 y 4 ≤ √ ln ( e x 2 4 + 3 4 e y 2 ) for all positive number x and y . (Hint: The desired inequality follows from the convexity of an appropriate function.) (b) Show that ± x 2 + y 3 + z 12 + w 12 ² 4 ≤ 1 2 x 4 + 1 3 y 4 + 1 12 z 4 + 1 12 w 4 with equality if and only if x = y = z = w . 2...
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