It can be reformulated and solved as a MILP using big M max xyλw � c x d 1 y s

It can be reformulated and solved as a milp using big

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It can be reformulated and solved as a MILP using big-M max x,y,λ,w ζ = c > x + d > 1 y s . t . A 1 x + B 1 y b 1 B > 2 λ = d 2 . b 2 - A 2 x - B 2 y 0 b 2 - A 2 x - B 2 y Mw λ M (1 m 2 × 1 - w ) λ 0 w ∈ { 0 , 1 } m 2 . Lizhi Wang ([email protected]) IE 534 Linear Programming November 9, 2012 8 / 28
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Difficulties with the big-M method We may not know how big is big enough. I If big-M is set too small, then it may reduce the feasible region. I If big-M is set too big, then it may cause computational problems due to rounding errors. A finite big-M may not exist. For example, when BLP is unbounded. Lizhi Wang ([email protected]) IE 534 Linear Programming November 9, 2012 9 / 28
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Solution technique: branch and bound Define the LP relaxation as max x,y,λ ζ = c > x + d > 1 y (11) s . t . A 1 x + B 1 y b 1 (12) A 2 x + B 2 y b 2 (13) B > 2 λ = d 2 (14) λ 0 . (15) If LP relaxation solution satisfies the complementarity constraints, then it’s optimal to BLP. Otherwise create two new nodes (assuming i th complementarity constraint violated) LP relaxation (11)-(15) ( A 2 x + B 2 y ) i = ( b 2 ) i LP relaxation (11)-(15) ( λ ) i = 0 Lizhi Wang ([email protected]) IE 534 Linear Programming November 9, 2012 10 / 28
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Definitions: B ( l, u ) and L ( l, u ) B ( l, u ) max x,y,λ ζ = c > x + d > 1 y s . t . A 1 x + B 1 y b 1 l A 2 x + B 2 y b 2 0 λ u B > 2 λ = d 2 0 b 2 - A 2 x - B 2 y λ 0 L ( l, u ) max x,y,λ ζ = c > x + d > 1 y s . t . A 1 x + B 1 y b 1 l A 2 x + B 2 y b 2 0 λ u B > 2 λ = d 2 Lizhi Wang ([email protected]) IE 534 Linear Programming November 9, 2012 11 / 28
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Step 0: Initialization Create the root node B ( l 1 = -∞ m 2 × 1 , u 1 = m 2 × 1 ) , which is characterized by ( l 1 , u 1 , z 1 = ) . Initialize x * = , y * = , λ * = , ζ * = -∞ , N = 1 . Go to Step 1. Lizhi Wang ([email protected]) IE 534 Linear Programming November 9, 2012 12 / 28
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Step 1: Node management For all k ∈ { 1 , ..., N } such that z k ζ * , remove node k . Update N as the number of remaining nodes. if N = 0 then if x * 6 = then 1(a) return ( x * , y * , λ * , ζ * ) is an optimal solution to the BLP (4)-(6). else 1(b) return the BLP (4)-(6) is infeasible. end else 1(c) select a node k from { 1 , ..., N } , set ( ˆ l = l k , ˆ u = u k ) , remove node k , reorder the remaining nodes from 1 to N - 1 , reduce N by 1, and go to Step 2. end Lizhi Wang ([email protected]) IE 534 Linear Programming November 9, 2012 13 / 28
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Step 2: LP relaxation Solve L ( ˆ l, ˆ u ) . if L ( ˆ l, ˆ u ) is infeasible then 2(a) go to Step 1.
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