Celis-Dennis-Tapia问题的快速界
Sharp and fast bounds for the Celis-Dennis-Tapia problem
摘要：
在Celis-Dennis-Tapia（CDT）问题中，一个二次函数在由两个严格凸二次约束定义的区域上被最小化.在本文中，我们重新推导出一个必要的和最优性条件的精确性的对偶拉格朗日界（相当于在这种情况下的Shor松弛界）。从这样的条件出发，我们建议通过增加一个或两个线性切割拉格朗日松弛来加强对偶拉格朗日约束。这样的切割是从两个约束之一的支撑超平面获得的。因此，它们对于原问题是多余的，但对于拉格朗日松弛却不是。计算实验表明，新的界限是有效的，并需要有限的计算时间。特别是，提出的边界之一能够解决[Burer，Anstreicher，2013]中提出的CDT问题的212个困难实例中的所有实例。
英文摘要：
In the Celis-Dennis-Tapia (CDT) problem a quadratic function is minimized over a region defined by two strictly convex quadratic constraints. In this paper we re-derive a necessary and optimality condition for the exactness of the dual Lagrangian bound (equivalent to the Shor relaxation bound in this case). Starting from such condition we propose to strengthen the dual Lagrangian bound by adding one or two linear cuts to the Lagrangian relaxation. Such cuts are obtained from supporting hyperplanes of one of the two constraints. Thus, they are redundant for the original problem but they are not for the Lagrangian relaxation. The computational experiments show that the new bounds are effective and require limited computing times. In particular, one of the proposed bounds is able to solve all but one of the 212 hard instances of the CDT problem presented in [Burer, Anstreicher, 2013].
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