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9 Lagrangian Relaxation (1)

# 9 Lagrangian Relaxation (1) - IEE 598 9(II.3.6 Lagrangian...

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IEE 598 - 9 (II.3.6) Lagrangian Relaxation Muhong Zhang DEPARTMENT OF INDUSTRIAL ENGINEERING Mar. 04, 2011

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Lagrangian Relaxation Recall that for LP relaxation, we dropped the integrality constraint. For an optimization problem z IP = max { cx : Ax b , Dx d , x Z n + } , by dropping some “complicating” constraints Dx d , we get a relaxation of the original IP. However, such relaxation might be weak. We might try to penalizing the violation of dropping constraints. Lagrangian relaxation: LR ( λ ) : z LR ( λ ) = max { cx - λ ( Dx - d ) : Ax b , x Z n + } Proposition: Problem LR ( λ ) is a relaxation of the IP problem for all λ 0. Corollary: z IP z LR ( λ ) for all λ 0. Zhang IEE 598 OPT II 2 / 19
Lagrangian Relaxation Lagrangian relaxation gives an upper bound of the IP problem. To determine the best λ , we solve the following Lagrangian Dual problem: z LD = min λ 0 z LR ( λ ) . Corollary: z IP z LD . Proposition: Let x ( λ ) be an optimal solution for LR ( λ ) . If (1) Dx ( λ ) d and (ii) λ ( d - Dx ) = 0, then x ( λ ) is optimal for the IP problem. Zhang IEE 598 OPT II 3 / 19

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Uncapacited facility location problem Variables: x j : indicator of build facility j , j N ; y ij : proportion of demand i to be meet by facility j , i I , j N . Formulation: max i I j N c ij y ij - j N f j x j s.t. j N y ij = 1 i I y ij x j i I , j N y R mn + , x B n Let Q = { ( x , y ) B n × R mn + : y ij x j , i I , j N } . LR ( λ ) : Zhang IEE 598 OPT II 4 / 19
Lagrangian Dual 1 How to solve the Lagrangian Dual problem? 2 How strong is the z LD ? 3 When is z LD = z IP ? Zhang IEE 598 OPT II 5 / 19

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The Strength of the Lagrangian Dual LR ( λ ) : z LR ( λ ) = max { cx + λ ( d - Dx ) : Ax b , x Z n + } Let Q = { x Z n + : Ax b } and S = { x Q
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9 Lagrangian Relaxation (1) - IEE 598 9(II.3.6 Lagrangian...

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