Semidefinite Programming Relaxations For The Quadratic Assignment Problem

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Semidefinite Programming Relaxations For The Quadratic Assignment Problem (1998)

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by Qing Zhao , Stefan E. Karisch , Franz Rendl , Henry Wolkowicz

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81 - 22 self

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@MISC{Zhao98semidefiniteprogramming,
    author = {Qing Zhao and Stefan E. Karisch and Franz Rendl and Henry Wolkowicz},
    title = {Semidefinite Programming Relaxations For The Quadratic Assignment Problem},
    year = {1998}
}

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Abstract

Semidefinite programming (SDP) relaxations for the quadratic assignment problem (QAP) are derived using the dual of the (homogenized) Lagrangian dual of appropriate equivalent representations of QAP. These relaxations result in the interesting, special, case where only the dual problem of the SDP relaxation has strict interior, i.e. the Slater constraint qualification always fails for the primal problem. Although there is no duality gap in theory, this indicates that the relaxation cannot be solved in a numerically stable way. By exploring the geometrical structure of the relaxation, we are able to find projected SDP relaxations. These new relaxations, and their duals, satisfy the Slater constraint qualification, and so can be solved numerically using primal-dual interior-point methods. For one of our models, a preconditioned conjugate gradient method is used for solving the large linear systems which arise when finding the Newton direction. The preconditioner is found by exploiting th...

Keyphrases

quadratic assignment problem semidefinite programming relaxation slater constraint qualification projected sdp relaxation preconditioned conjugate gradient method large linear system primal problem semidefinite programming dual problem new relaxation appropriate equivalent representation sdp relaxation relaxation cannot stable way geometrical structure duality gap lagrangian dual newton direction primal-dual interior-point method

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