One of the relevant research topics to which Chris Floudas contributed was quadratically constrained quadratic programming (QCQP). This paper considers one of the simplest hard cases of QCQP, the two trust region subproblem (TTRS). In this case, one needs to minimize a quadratic function constrained by the intersection of two ellipsoids. The Lagrangian dual of the TTRS is a semidefinite program (SDP) and this result has been extensively used to solve the problem efficiently. We focus on numerical aspects of branch-and-bound solvers with three goals in mind. We provide (i) a detailed analysis of the ability of state-of-the-art solvers to complete the global search for a solution, (ii) a quantitative approach for measuring the cluster effect on each solver and (iii) a comparison between the branch-and-bound and the SDP approaches. We perform the numerical experiments on a set of 212 challenging problems provided by Kurt Anstreicher. Our findings indicate that SDP relaxations and branch-and-bound have orthogonal difficulties, thus pointing to a possible benefit of a combined method. The following solvers were selected for the experiments: Antigone 1.1, Baron 16.12.7, Lindo Global 10.0, Couenne 0.5 and SCIP 3.2.

中文翻译：

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关于两个信任区域子问题的全局优化求解器的计算研究。

克里斯·弗洛达斯（Chris Floudas）贡献的相关研究主题之一是二次约束二次规划（QCQP）。本文考虑了QCQP最简单的困难案例之一，两个信任区域子问题（TTRS）。在这种情况下，需要最小化由两个椭圆体的交点约束的二次函数。TTRS的拉格朗日对偶是一个半定程序（SDP），并且该结果已广泛用于有效解决问题。我们着眼于三个目标，着眼于分支定界求解器的数值方面。我们提供（i）最先进的求解器完成全局搜索解决方案能力的详细分析，（ii）衡量每个求解器的簇效应的定量方法，以及（iii）分支定界和SDP方法。我们对Kurt Anstreicher提供的212个具有挑战性的问题进行了数值实验。我们的发现表明，SDP弛豫和分支定界具有正交困难，因此指出了组合方法的可能好处。实验选择了以下求解器：Antigone 1.1，Baron 16.12.7，Lindo Global 10.0，Couenne 0.5和SCIP 3.2。