HW4Solution

# HW4Solution - EAS6939 Homework#4 1 Consider the following...

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EAS6939 Homework #4 1. Consider the following design optimization problem: 22 12 1 11 3 32 1 Minimize ( ) 4 4 Subject to ( ) 0 () 0 ( 1 ) 0 fx x x gx x = +− + =− ≤ = −− ≤ x x x x (i) Find the optimum point graphically (ii) Show that the optimum point does not satisfy K-T condition. Explain (i) As shown in the figure, (1, 0) is the optimum point and f = 1 at the optimum point. x * 1.0 g 3 ( x *) g 2 ( x *) f ( x *) x 1 x 2 f =1 f =4 (ii) The Lagrangian function for the problem can be defined as 2 2 1 1 1 1 2 3 2 1 3 44( )( )[( 1) ] Lx x x x s x λλ λ =+− ++−+ + −+ + −− + The K-T condition is 3 3 1 223 2 2 21 3 24 3 ( 0 20 0 0 (1 ) 0 0, 1,2,3 ii xx x xs s si = −+ = = = = == At x = (1, 0) since g 1 is inactive, and g 2 and g 3 are active, the slack variables should be 123 0, 0, 0 ss === The first equation in the K-T condition can’t be satisfied by substituting into these values. As is clear from the figure, the gradients of two active constraints are not independent: [0, -1] and [0, 1]. In the mathematical term, x * is not a regular point of the feasible domain. The K-T

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HW4Solution - EAS6939 Homework#4 1 Consider the following...

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