计算Pareto前沿表示的改进超盒算法
An improved hyperboxing algorithm for calculating a Pareto front representation
摘要：
在求解具有多个目标函数的最优化问题时，经常会遇到一个或多个目标函数为非凸函数，或者不能很容易地证明所有目标函数的凸性。在这种情况下，需要一个计算非支配集的表示的通用算法。一种合适的方法在于所谓的超装箱算法，其特征在于将目标空间分裂成轴平行的超矩形。因此，只有非支配性的财产被用来减少所谓的搜索区域。在文献中，这样的算法已经显示出相对于计算的代表点的数量提供了非常好的帕累托前沿覆盖。然而，该算法的计算成本是禁止的问题，超过五个目标。在本文中，我们提出了算法的进步，提高了算法的性能，使其适用于多达9个目标的问题。我们说明了一组测试问题的性能增益和质量的表示。我们还将改进的算法应用到放射治疗计划领域的一个实际问题中。
英文摘要：
When solving optimization problems with multiple objective functions we are often faced with the situation that one or several objective functions are non-convex or that we can not easily show the convexity of all functions involved. In this case a general algorithm for computing a representation of the nondominated set is required. A suitable approach consists in a so-called hyperboxing algorithm that is characterized by splitting the objective space into axis-parallel hyperrectangles. Thereby, only the property of nondominance is exploited for reducing the so-called search region. In the literature such an algorithm has already shown to provide a very good coverage of the Pareto front relative to the number of representation points calculated. However, the computational cost for the algorithm was prohibitive for problems with more than five objectives. In this paper, we present algorithmic advances that improve the performance of the algorithm and make it applicable to problems with up to nine objectives. We illustrate the performance gain and the quality of the representation for a set of test problems. We also apply the improved algorithm to a real world problem in the field of radiotherapy planning.
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