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Lower bounds for nonlinear assignment problems using many body interactions.

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Title: Lower bounds for nonlinear assignment problems using many body interactions.
Authors: Ramachandran, Bala1, Pekny, J.F.1 pekny@ecn.purdue.edu
Source: European Journal of Operational Research. 02/16/98, Vol. 105 Issue 1, p202-215. 14p. 3 Charts.
Subjects: Nonlinear assignment problems, Many-body problem, Quadratic assignment problem, Linear programming, Integer programming
Abstract: This paper concerns lower bounding techniques for the general α-adic assignment problem. The nonlinear objective function is linearized by the introduction of additional variables and constraints, thus yielding a mixed integer linear programming formulation of the problem. The concept of many body interactions is introduced to strengthen this formulation and incorporated in a modified formulation obtained by lifting the original representation to a higher dimensional space. This process involves two steps — (i) addition of new variables and constraints and (ii) incorporation of the new variables in the objective function. If this lifting process is repeated β times on an α-adic assignment problem along with the incorporation of higher order interactions, it results in the mixed-integer formulation of an equivalent (α + β)-adic assignment problem. The incorporation of many body interactions in the higher dimensional formulation improves its degeneracy properties and is also critical to the derivation of decomposition methods for the solution of these large scale mathematical programs in the higher dimensional space. It is shown that a lower bound to the optimal solution of the corresponding linear programming relaxation can be obtained by dualizing a subset of constraints in this formulation and solving O(N[sup 2(α+β-1)]) linear assignment problems, whose coefficients depend on the dual values. Moreover, it is proved that the optimal solution to the LP relaxation is obtained if we use the optimal duals for the solution of the linear assignment problems. This concept of many body interactions could be applied in designing algorithms for the solution of formulations obtained by lifting general MILP's. We illustrate all these concepts on the quadratic assignment problem. With these decomposition bounds, we have found the provably optimal solutions of two unsolved QAP's of size 32 and have also improved upon existing lower bounds for other QAP's. [ABSTRACT FROM AUTHOR]
Copyright of European Journal of Operational Research is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Lower bounds for nonlinear assignment problems using many body interactions.
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  Data: <searchLink fieldCode="AR" term="%22Ramachandran%2C+Bala%22">Ramachandran, Bala</searchLink><relatesTo>1</relatesTo><br /><searchLink fieldCode="AR" term="%22Pekny%2C+J%2EF%2E%22">Pekny, J.F.</searchLink><relatesTo>1</relatesTo><i> pekny@ecn.purdue.edu</i>
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  Data: <searchLink fieldCode="JN" term="%22European+Journal+of+Operational+Research%22">European Journal of Operational Research</searchLink>. 02/16/98, Vol. 105 Issue 1, p202-215. 14p. 3 Charts.
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  Data: <searchLink fieldCode="DE" term="%22Nonlinear+assignment+problems%22">Nonlinear assignment problems</searchLink><br /><searchLink fieldCode="DE" term="%22Many-body+problem%22">Many-body problem</searchLink><br /><searchLink fieldCode="DE" term="%22Quadratic+assignment+problem%22">Quadratic assignment problem</searchLink><br /><searchLink fieldCode="DE" term="%22Linear+programming%22">Linear programming</searchLink><br /><searchLink fieldCode="DE" term="%22Integer+programming%22">Integer programming</searchLink>
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  Data: This paper concerns lower bounding techniques for the general α-adic assignment problem. The nonlinear objective function is linearized by the introduction of additional variables and constraints, thus yielding a mixed integer linear programming formulation of the problem. The concept of many body interactions is introduced to strengthen this formulation and incorporated in a modified formulation obtained by lifting the original representation to a higher dimensional space. This process involves two steps — (i) addition of new variables and constraints and (ii) incorporation of the new variables in the objective function. If this lifting process is repeated β times on an α-adic assignment problem along with the incorporation of higher order interactions, it results in the mixed-integer formulation of an equivalent (α + β)-adic assignment problem. The incorporation of many body interactions in the higher dimensional formulation improves its degeneracy properties and is also critical to the derivation of decomposition methods for the solution of these large scale mathematical programs in the higher dimensional space. It is shown that a lower bound to the optimal solution of the corresponding linear programming relaxation can be obtained by dualizing a subset of constraints in this formulation and solving O(N[sup 2(α+β-1)]) linear assignment problems, whose coefficients depend on the dual values. Moreover, it is proved that the optimal solution to the LP relaxation is obtained if we use the optimal duals for the solution of the linear assignment problems. This concept of many body interactions could be applied in designing algorithms for the solution of formulations obtained by lifting general MILP's. We illustrate all these concepts on the quadratic assignment problem. With these decomposition bounds, we have found the provably optimal solutions of two unsolved QAP's of size 32 and have also improved upon existing lower bounds for other QAP's. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Data: <i>Copyright of European Journal of Operational Research is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – Type: doi
        Value: 10.1016/S0377-2217(97)00038-6
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      – Code: eng
        Text: English
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        PageCount: 14
        StartPage: 202
    Subjects:
      – SubjectFull: Nonlinear assignment problems
        Type: general
      – SubjectFull: Many-body problem
        Type: general
      – SubjectFull: Quadratic assignment problem
        Type: general
      – SubjectFull: Linear programming
        Type: general
      – SubjectFull: Integer programming
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      – TitleFull: Lower bounds for nonlinear assignment problems using many body interactions.
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            – D: 16
              M: 02
              Text: 02/16/98
              Type: published
              Y: 1998
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