In this paper, we study the maximum diversity problem (MDP) which is equivalent to the quadratic unconstrained binary optimization (QUBO) problem with cardinality constraint. The MDP aims to select a subset of elements with given cardinality such that the sum of pairwise distances between any two elements in the selected subset is maximized. For solving this computationally challenging problem, we propose a two-phase tabu search based evolutionary algorithm (TPTS/EA), which integrates several distinguishing features to ensure the diversity and the quality of the evolution, such as a two-phase tabu search algorithm which consists of a dynamic candidate list (DCL) strategy-based traditional tabu search in the first phase and a solution-based tabu search procedure to refine the search in the second phase, and two path-relinking based recombination operators to generate new offspring solutions. Tested on three sets of totally 140 public instances in the literature, the study demonstrates the efficacy of the proposed TPTS/EA algorithm in terms of both solution quality and computational efficiency. Specifically, our proposed TPTS/EA algorithm is able to improve the previous best known results for 2 instances, while matching the previous best-known solutions for 130 instances. We also provide experimental evidences to highlight the beneficial effect of several important components in our TPTS/EA algorithm.

在本文中，我们研究了最大分集问题（MDP），该问题等同于具有基数约束的二次无约束二进制优化（QUBO）问题。MDP旨在选择具有给定基数的元素子集，以使所选子集中的任意两个元素之间的成对距离之和最大。为了解决这一计算难题，我们提出了一种基于两阶段禁忌搜索的进化算法（TPTS / EA），该算法融合了几个明显的特征以确保进化的多样性和质量，例如两阶段禁忌搜索算法由第一阶段的基于动态候选者列表（DCL）策略的传统禁忌搜索和第二阶段的基于解决方案的禁忌搜索过程优化搜索组成，和两个基于路径重新链接的重组算子以生成新的后代解决方案。在文献中对三组共140个公共实例进行了测试，该研究证明了所提出的TPTS / EA算法在解决方案质量和计算效率方面的功效。具体而言，我们提出的TPTS / EA算法能够改善2个实例的先前最著名的结果，同时匹配130个实例的先前最著名的解决方案。我们还提供实验证据，以突出我们TPTS / EA算法中几个重要组件的有益效果。具体而言，我们提出的TPTS / EA算法能够改善2个实例的先前最著名的结果，同时匹配130个实例的先前最著名的解决方案。我们还提供实验证据，以突出我们TPTS / EA算法中几个重要组件的有益效果。具体而言，我们提出的TPTS / EA算法能够改善2个实例的先前最著名的结果，同时匹配130个实例的先前最著名的解决方案。我们还提供实验证据，以突出我们TPTS / EA算法中几个重要组件的有益效果。