3D fast convex-hull-based evolutionary multiobjective optimization algorithm

The receiver operating characteristic (ROC) and detection error tradeoff (DET) curves have been widely used in the machine learning community to analyze the performance of classifiers. The area (or volume) under the convex hull has been used as a scalar indicator for the performance of a set of classifiers in ROC and DET space. Recently, 3D convex-hull-based evolutionary multiobjective optimization algorithm (3DCH-EMOA) has been proposed to maximize the volume of convex hull for binary classification combined with parsimony and three-way classification problems. However, 3DCH-EMOA revealed high consumption of computational resources due to redundant convex hull calculations and a frequent execution of nondominated sorting. In this paper, we introduce incremental convex hull calculation and a fast replacement for non-dominated sorting. While achieving the same high quality results, the computational effort of 3DCH-EMOA can be reduced by orders of magnitude. The average time complexity of 3DCH-EMOA in each generation is reduced from O(n2logn) to O(nlogn) per iteration, where n is the population size. Six test function problems are used to test the performance of the newly proposed method, and the algorithms are compared to several state-of-the-art algorithms, including NSGA-III, RVEA, etc., which were not compared to 3DCH-EMOA before. Experimental results show that the new version of the algorithm (3DFCH-EMOA) can speed up 3DCH-EMOA for about 30 times for a typical population size of 300 without reducing the performance of the method. Besides, the proposed algorithm is applied for neural networks pruning, and several UCI datasets are used to test the performance.