二次指派问题半定松弛的中心化ADMM
Centering ADMM for the Semidefinite Relaxation of the QAP
AI总结：
提出Centering ADMM方法求解QAP的半定松弛，结合内点法中心化步骤与标准ADMM，证明其全局收敛性且数值实验验证了对部分实例的有效性。
摘要：
我们提出了一种求解二次指派问题（QAP）半定（SD）松弛的新方法，称为Centering ADMM。Centering ADMM是一种交替方向乘子法（ADMM），结合了内点法中使用的中心化步骤。Centering ADMM的第一阶段通过将障碍函数项纳入目标函数来更新迭代点，使其趋近中心路径，如同内点法一样。如果当前迭代点充分接近中心路径且障碍参数值充分小，该方法将切换至标准版本的ADMM。我们证明了Centering ADMM（不采用动态更新惩罚参数）具有全局收敛性质。为观察中心化步骤的效果，我们对QAPLIB中实例的SD松弛问题进行了数值实验。结果表明，中心化步骤对某些类别的实例非常有效。
英文摘要：
We propose a new method for solving the semidefinite (SD) relaxation of the quadratic assignment problem (QAP), called Centering ADMM. Centering ADMM is an alternating direction method of multipliers (ADMM) combining the centering steps used in the interior-point method. The first stage of Centering ADMM updates the iterate so that it approaches the central path by incorporating a barrier function term into the objective function, as in the interior-point method. If the current iterate is sufficiently close to the central path with a sufficiently small value of the barrier parameter, the method switches to the standard version of ADMM. We show that Centering ADMM (not employing a dynamic update of the penalty parameter) has global convergence properties. To observe the effect of the centering steps, we conducted numerical experiments with SD relaxation problems of instances in QAPLIB. The results demonstrate that the centering steps are quite efficient for some classes of instances.
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