This paper aims to answer an open question recently posed in the literature, that is to find a fast exact method for solving the max-sum diversity problem, a nonconcave quadratic binary maximization problem. We show that, for Euclidean max-sum diversity problems (EMSDP), the distance matrix defining the quadratic term is always conditionally negative definite. This interesting property ensures that the cutting plane method is exact for (EMSDP), even in the absence of concavity. As such, the cutting plane method, which is primarily designed for concave maximisation problems, converges to the optimal solution of (EMDSP). The method was evaluated on several standard benchmark test sets, where it was shown to outperform other exact solution methods for (EMSDP), and is capable of solving two-coordinate problems of up to eighty-five thousand variables.

本文旨在回答文献中最近提出的一个悬而未决的问题，即找到一种快速精确的方法来解决最大和多样性问题，即非凹二次二元最大化问题。我们证明，对于欧几里得最大和分集问题（EMSDP），定义二次项的距离矩阵始终是有条件负定的。这个有趣的属性确保了即使没有凹面，切割平面方法对于 (EMSDP) 也是精确的。因此，主要针对凹最大化问题设计的割平面方法收敛于（EMDSP）的最优解。该方法在多个标准基准测试集上进行了评估，结果表明该方法优于其他精确求解方法 (EMSDP)，并且能够解决多达八万五千个变量的两坐标问题。