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hw11 - ¯ x for this NLP as described in Karush-Kuhn-Tucker...

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CO250 I NTRODUCTION TO O PTIMIZATION - HW 11 This assignment covers material in Chapters 6 and 7. This assignment will not be graded, but you are expected to know the material covered. Solutions are posted on UW-ACE. Exercise 1. Consider the following LP, which we will call (P): max 2 x 1 + x 2 subject to x 1 x 2 2 x 1 + x 2 2 x 1 2 x 2 3 (a) Sketch the feasible region of (P) in the ( x 1 , x 2 ) -plane, indicate all extreme points, and at each extreme point show the cone of tight constraints. (b) Express c = ( 2 , 1) T as an element of the cone of tight constraints at the point (0 , 2) T . This gives a pair ( y 1 , y 2 ) . (c) Using (b), find an appropriate solution to the dual of (P) and show how this demonstrates that (0 , 2) T is optimal for (P). (Hint: use the idea in the proof of Theorem 35, the last theorem in Chapter 6.) Exercise 2. Consider the following NLP: a) min 2 x 1 2 x 2 s.t. x 2 1 x 2 0 (1) x 1 + x 2 2 0 (2) x 1 + 1 4 0 (3) and the vector ¯ x := (1 , 1) T . Write down the optimality conditions for
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Unformatted text preview: ¯ x for this NLP as described in Karush-Kuhn-Tucker Theorem. Using these conditions and the theorem, prove that ¯ x is optimal. b) Suppose that we replace the objective function by, min − x 1 + αx 2 . Indicate for what values of α is ¯ x going to be optimal for (NLP). Exercise 3 (15 marks) . 1 2 a) Let g 1 , g 2 , g 3 : ℜ → ℜ be defined by g 1 ( x ) := − x, g 2 ( x ) := 2 , g 3 ( x ) := x. Plot these functions on ℜ 2 . Identify on your plot, the function ˆ g defined by, ˆ g ( x ) := max { g 1 ( x ) , g 2 ( x ) , g 3 ( x ) } . Prove that ˆ g is a convex function. b) Suppose g 1 , g 2 , . . . , g m : ℜ n → ℜ are given convex functions. Define the function ˆ g : ℜ n → ℜ where ˆ g ( x ) := max { g 1 ( x ) , g 2 ( x ) , . . . , g m ( x ) } . Prove that ˆ g is a convex function....
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