The Quadratic Assignment Problem formulated as an integer program:

Minimize $$\sum_{i=1}^n\sum_{j=1}^nc_{ij} x_{ij} + \sum_{i=1}^n\sum_{j=1}^n\sum_{k=1}^n\sum_{l=1}^n c_{ijkl}x_{ij}x_{kl}$$ subject to $$\sum_{i=1}^n x_{ij} = 1 \quad \forall j=1,\ldots,n,$$ $$\sum_{j=1}^n x_{ij} = 1 \quad \forall i=1,\ldots,n,$$ $$x_{ij}\in{0,1}\quad \forall i,j=1,\ldots,n.$$

A simple and natural linearization:

Minimize $$\sum_{i=1}^n\sum_{j=1}^nc_{ij} x_{ij} + \sum_{i=1}^n\sum_{j=1}^n\sum_{k=1}^n\sum_{l=1}^n c_{ijkl}y_{ijkl}$$ subject to $$\sum_{i=1}^n x_{ij} = 1 \quad \forall j=1,\ldots,n,$$ $$\sum_{j=1}^n x_{ij} = 1 \quad \forall i=1,\ldots,n,$$ $$y_{ijkl} \le x_{ij} \quad \forall i,j,k,l=1,\ldots,n,$$ $$ x_{ij} + x_{kl} \le 1+y_{ijkl} \quad \forall i,j,k,l=1,\ldots,n,$$ $$ y_{ijkl} = y_{klij} \quad \forall i,j,k,l=1,\ldots,n,$$ $$x_{ij},y_{ijkl}\in{0,1}\quad \forall i,j,k,l=1,\ldots,n.$$

My question is: Has this linearization already been studied in the literature? I did not see it in any papers (this one for example).

asked 02 Dec '15, 04:23

f10w's gravatar image

f10w
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accept rate: 0%

edited 02 Dec '15, 04:29

@Rob Pratt: Your comment answers my question. Please put it as an answer so that I can accept it. Thanks.

(09 Dec '15, 14:18) f10w

Liberti, L. (2007), "Compact Linearization for Binary Quadratic Problems," 4OR: A Quarterly Journal of Operations Research, 5, 231–245.

Attributes the "usual linearization" to Fortet (1960).

link

answered 03 Dec '15, 15:36

Rob%20Pratt's gravatar image

Rob Pratt
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accept rate: 28%

link

answered 03 Dec '15, 03:20

Erling_MOSEK's gravatar image

Erling_MOSEK
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accept rate: 3%

Thanks but I don't see the considered linearization being mentioned in that paper.

(09 Dec '15, 14:20) f10w

I once saw this paper where they list several linerizations. I do not recall if the one you propose is one of them, though.

link

answered 04 Dec '15, 02:50

Sune's gravatar image

Sune
958413
accept rate: 20%

Thanks but the considered linearization is not in the list.

(09 Dec '15, 14:20) f10w
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Asked: 02 Dec '15, 04:23

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Last updated: 09 Dec '15, 18:51

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