The box-constrained weighted maximin dispersion problem is to find a point in an $ n $-dimensional box such that the minimum of the weighted Euclidean distance from given $ m $ points is maximized. In this paper, we first propose a two-phase method to solve it. In the first phase, we adopt a block successive upper bound minimization (BSUM) algorithm framework and choose a special piecewise linear upper bound function for the weighted maximin dispersion problem. The per-iteration complexity of our algorithm is very low, since the subproblem is a one-dimensional piecewise linear minimax problem over the box constraints, or eqivalently, a two-dimensional linear programming problem which can be solved in at most $ O(m) $ time by existing algorithms. In the second phase, a useful rounding is employed to enhance the solution. Moreover, we propose another strengthened two-phase algorithm, which employs a maximum improvement successive upper-bound minimization (MISUM) algorithm instead of BSUM algorithm in the first phase. At each step, only the block that provides the maximum improvement of the upper bound function is updated. Then, it can be proved that every limit point of the iterate generated by this strengthened algorithm is a stationary point. Numerical results show that the proposed algorithms are efficient.

中文翻译：

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一种有效的低复杂度算法，用于盒约束加权极大值散点问题

框约束的加权极大值散点问题是在一个n维维的盒子中找到一个点，使得距给定的m点的加权欧几里得距离的最小值最大。在本文中，我们首先提出一种两阶段方法来解决它。在第一阶段，我们采用块连续上限最小化（BSUM）算法框架，并为加权极大值散度问题选择特殊的分段线性上限函数。我们的算法的迭代复杂度非常低，因为子问题是在框约束上的一维分段线性极大极小问题，或者等效地，二维线性规划问题最多可在$ O（m ）$现有算法的时间。在第二阶段，使用有效的舍入来增强解。此外，我们提出了另一种增强的两阶段算法，该算法在第一阶段采用最大改进的连续上限最小化（MISUM）算法代替BSUM算法。在每个步骤中，仅对提供上限功能的最大改进的块进行更新。然后，可以证明由该增强算法生成的迭代的每个极限点都是固定点。数值结果表明，该算法是有效的。可以证明，通过这种增强算法生成的迭代的每个极限点都是一个固定点。数值结果表明，该算法是有效的。可以证明，通过这种增强算法生成的迭代的每个极限点都是一个固定点。数值结果表明，该算法是有效的。