hw4solutions

hw4solutions - ORF 307 Homework 4 Solutions Exercise 1. The...

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ORF 307 Homework 4 Solutions Exercise 1. The Lagrangian of this problem is: L ( x, λ 1 , λ 2 , ν ) = c T x + λ T 1 Ax - λ T 2 x + ν T ( Dx - f ) with λ 1 , λ 2 0 The Lagrange dual function is: g ( λ 1 , λ 2 , ν ) = inf x±D L ( x, λ 1 , λ 2 , ν ) = inf x±D c T x + λ T 1 Ax - λ T 2 x + ν T ( Dx - f ) = inf x±D ( c T + λ T 1 A - λ T 2 + ν T D ) x - ν T f = - ν T f if c + A T λ 1 - λ 2 + D T ν = 0, = -∞ otherwise Thus the dual problem is: max g ( λ 1 , λ 2 , ν ) s.t. λ 1 , λ 2 0 This is equivalent to: max - ν T f s.t. c + A T λ 1 - λ 2 + D T ν = 0 λ 1 , λ 2 0 Note that λ 2 is like a surplus variable so this problem is equivalent to: max - ν T f s.t. A T λ 1 + D T ν ≥ - c λ 1 0 Exercise 2. The Lagrangian of this problem is: L ( x, λ, ν ) = c T x - λ T x + ν T (1 T x - 1) with λ 0 The Lagrange dual function is: 1
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g ( λ, ν ) = inf x±D L ( x, λ, ν ) = inf x±D c T x -
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hw4solutions - ORF 307 Homework 4 Solutions Exercise 1. The...

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