一种计算Pareto近似集超体积指标的新算法
Novel algorithm to calculate hypervolume indicator of Pareto approximation set
摘要：
超体积指标是比较多目标优化器生成的Pareto近似集的一种普遍接受的质量度量。用于计算$d$维空间中的$n$点的最佳已知算法具有具有特殊数据结构的$O(n^{d/2})$的运行时间。本文提出了一种递归的顶点分裂算法，用于计算$d>2$维中一组$n$非可比点的超体积指标。它分裂出多个不能被分裂参考点支配的子超长方体。特别地，仔细选择分裂参考点以使子超长方体中的点的数量最小化。复杂度分析表明，该算法在最坏情况下达到了$O((\frac{d}{2})^n)$时间复杂度和$O(dn^2)$空间复杂度。
英文摘要：
Hypervolume indicator is a commonly accepted quality measure for comparing Pareto approximation set generated by multi-objective optimizers. The best known algorithm to calculate it for $n$ points in $d$-dimensional space has a run time of $O(n^{d/2})$ with special data structures. This paper presents a recursive, vertex-splitting algorithm for calculating the hypervolume indicator of a set of $n$ non-comparable points in $d>2$ dimensions. It splits out multiple child hyper-cuboids which can not be dominated by a splitting reference point. In special, the splitting reference point is carefully chosen to minimize the number of points in the child hyper-cuboids. The complexity analysis shows that the proposed algorithm achieves $O((\frac{d}{2})^n)$ time and $O(dn^2)$ space complexity in the worst case.
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