We consider the Boolean Quadratic Programming problem with Generalized Upper Bound constraints (BQP-GUB). This problem belongs to the NP-hard complexity class of problems and is a generalization of the Quadratic Semi-Assignment Problem (QSAP), which has applications in different areas such as production planning and image segmentation. Non-exact methods for solving BQP-GUB instances are commonly based on exploring solution neighborhoods, which consist of flip moves that choose an even number of binary variables and flip their values to the complementary value (from 1 to 0 or from 0 to 1). In order to find fairly good solutions, these methods must evaluate a large number of flip moves, and that can be time consuming. The best-known formulae used to evaluate flip moves for BQP-GUB take $O(\mathit{nr})$ time, where n is the number of variables and r is the number of flips. In this paper, we seek to improve the processing time of evaluating flip moves. We extend the results in the literature and prove two closed-form formulae for evaluating flip moves in $O(\mathit{mr})$ and $O(r^2)$ times, where m is the number of variables with values equal to one. Additionally, for experimental purposes, we prove a reduction from QSAP to BQP-GUB. We report on computational experiments with local search and Iterated Tabu Search algorithms using our formulae and those in the literature. Our results show that the implementations using one of our formulae achieved the best performance in all of the experiments.

我们考虑具有广义上界约束（BQP-GUB）的布尔二次规划问题。此问题属于NP-hard复杂性问题类别，是二次半分配问题（QSAP）的推广，它在生产计划和图像分割等不同领域中都有应用。解决BQP-GUB的非精确方法实例通常基于探索解决方案邻域，包括选择偶数个二进制变量并将其值翻转为互补值（从1到0或从0到1）的翻转移动。为了找到相当好的解决方案，这些方法必须评估大量的翻转动作，这可能很耗时。最有名的公式来评估倒装移动BQP-GUB起飞$Ø（\mathit{r}）$时间，其中n是变量数，r是翻转数。在本文中，我们力求缩短评估翻转动作的处理时间。我们将结果扩展到文献中，并证明了两个闭合形式的公式，用于评估$Ø（\mathit{先生}）$ 和 $Ø（{\mathrm{[R}}^{\mathrm{2个}}）$次，其中m是值等于1的变量的数量。此外，出于实验目的，我们证明了从QSAP减少到BQP-GUB的方法。我们报告使用我们的公式和文献中的局部搜索和迭代禁忌搜索算法进行的计算实验。我们的结果表明，使用我们的公式之一的实现在所有实验中均实现了最佳性能。