Homework Solutions 04

# Homework Solutions 04 - IEOR E4007 G Iyengar Nov 5th 2008...

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Unformatted text preview: IEOR E4007 G. Iyengar Nov. 5th, 2008 Homework # 4 Due: Nov. 19th 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) The gradient ∇ f ( x ) at x = (1 , 2) is given by ∇ f ( x ) = parenleftbigg 20 x 1 12 /x 2 parenrightbigg = parenleftbigg 20 6 parenrightbigg Since the gradient is not equal to zero, the pt. (1 , 2) is not a stationary pt. Since this is a maximization problem, the direction Δ x = ∇ f ( x ) improves the objective. (b) max x 1 x 2- 10 x 1 + 4 x 2 , x = (- 4 , 10) The gradient at x = (- 4 , 10) is given by ∇ f ( x ) = parenleftbigg x 2- 10 x 1 + 4 parenrightbigg = , i.e. x = (- 4 , 10) is a stationary pt. 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) The gradient at x = (- 3 , 1) is given by ∇ f ( x ) = parenleftbigg- 2 x 1- 6 x 2- 6 x 1- 18 x 2 parenrightbigg = So a stationary pt. The Hessian at x is ∇ 2 f ( x ) = bracketleftbigg- 2- 6- 6- 18 bracketrightbigg and its eigenvalues are { ,- 20 } . Thus, the pt. is possibly a local maximum. 1 (b) f ( x ) = 12 x 2- x 2 1 + 3 x 1 x 2- 3 x 2 2 , x = (12 , 8) The gradient at x = (12 , 8) is given by ∇ f ( x ) = parenleftbigg- 2 x 1 + 3 x 2 12 + 3 x 1- 6 x 2 parenrightbigg = So a stationary pt. The Hessian at x is ∇ 2 f ( x ) = bracketleftbigg- 2 3 3- 6 bracketrightbigg and its eigenvalues are {- . 3944 ,- 7 . 6056 } . Therefore, the pt. is definitely a local maximum. (c) f ( x ) = 6 x 1 + ln( x 1 ) + x 2 2 , x = (1 , 2) The gradient at x = (1 , 3) is given by ∇ f ( x ) = parenleftbigg 6 + 1 /x 1 2 x 2 parenrightbigg = parenleftbigg 7 6 parenrightbigg Not a stationary pt. (d) f ( x ) = 4 x 2 1 + 3 /x 2- 8 x 1 + 3 x 2 , x = (1 , 1) The gradient at x = (1 , 1) is given by ∇ f ( x ) = parenleftbigg 8 x 1- 8- 3 x 2 2 + 3 parenrightbigg = . A stationary pt. The Hessian at x is ∇ 2 f ( x ) = bracketleftbigg 8 0 0 6 bracketrightbigg and its eigenvalues are { 6 , 8 } . Therefore, definitely a local minimum. 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 } The constraint x 1 x 2 ≤ 9 is non-convex. Consequently, this problem is unlikely to be convex. Consider two points x = (5 , 9 / 5) prime , y = (9 / 5 , 5) prime and θ = 1 / 2. Then z = 0 . 5 x + 0 . 5 y = 17 / 5(1 , 1) prime violates the constraint (17 / 5) 2 = 11 . 56 > 9. Thus, the feasible region is not convex and the optimization problem is not convex.the feasible region is not convex and the optimization problem is not convex....
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Homework Solutions 04 - IEOR E4007 G Iyengar Nov 5th 2008...

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