Homework 04 - IEOR E4007 G. Iyengar March 30th, 2009...

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IEOR E4007 G. Iyengar March 30th, 2009 Homework # 4 Due: April 12th, 2009 1. Problem on unconstrained optimization For each of the following optimization problems either verify that the given x is a stationary point or find a direction d that locally improves at x . (a) max 10 x 2 1 + 12 ln( x 2 ), x = (1 , 2) (b) max x 1 x 2 - 10 x 1 + 4 x 2 , x = ( - 4 , 10) 2. Local optimality For each of the following functions f , classify the specified x as a definitely a local maximum, possibly local maximum, definitely local minimum, possibly local minimum or none of the above. (a) f ( x ) = - x 2 1 - 6 x 1 x 2 - 9 x 2 2 , x = ( - 3 , 1) (b) f ( x ) = 12 x 2 - x 2 1 + 3 x 1 x 2 - 3 x 2 2 , x = (12 , 8) (c) f ( x ) = 6 x 1 + ln( x 1 ) + x 2 2 , x = (1 , 2) (d) f ( x ) = 4 x 2 1 + 3 /x 2 - 8 x 1 + 3 x 2 , x = (1 , 1) 3. Problem on recognizing convex functions/sets Determine whether each of the following is a convex program (a) min { x 1 + x 2 : x 1 x 2 9 , | x 1 | ≤ 5 , | x 2 | ≤ 5 } (b) max { 62 x 1
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Homework 04 - IEOR E4007 G. Iyengar March 30th, 2009...

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