An Efficient Non-dominated Sorting Method for Evolutionary Algorithms

Abstract

We present a new non-dominated sorting algorithm to generate the non-dominated fronts in multi-objective optimization with evolutionary algorithms, particularly the NSGA-II. The non-dominated sorting algorithm used by NSGA-II has a time complexity of O(MN2) in generating non-dominated fronts in one generation (iteration) for a population size N and M objective functions. Since generating non-dominated fronts takes the majority of total computational time (excluding the cost of fitness evaluations) of NSGA-II, making this algorithm faster will significantly improve the overall efficiency of NSGA-II and other genetic algorithms using non-dominated sorting. The new non-dominated sorting algorithm proposed in this study reduces the number of redundant comparisons existing in the algorithm of NSGA-II by recording the dominance information among solutions from their first comparisons. By utilizing a new data structure called the dominance tree and the divide-and-conquer mechanism, the new algorithm is faster than NSGA-II for different numbers of objective functions. Although the number of solution comparisons by the proposed algorithm is close to that of NSGA-II when the number of objectives becomes large, the total computational time shows that the proposed algorithm still has better efficiency because of the adoption of the dominance tree structure and the divide-and-conquer mechanism.

An Efficient Approach to Nondominated Sorting for Evolutionary Multiobjective Optimization

“…Actually, this sorting method will likely consume more runtime in simulation due to its recursive nature. In addition, as Clymont and Keedwell [12] and Fang et al [13] pointed out, Jensen's sorting algorithm is not applicable in many cases, for instance, when strong-dominance [14] or ϵ-dominance [15] is used in comparison, or when the population contains duplicate solutions.…”

“…There are a few other non-dominated sorting approaches inspired by different ideas, such as the nondominated rank sort of the omni-optimizer [16], better non-dominated sort [17], immune recognition based algorithm [18], quick sort [19], sorting based algorithm [20], and divide-and-conquer based non-dominated sorting algorithm [13]. Most of these approaches are effective in dealing with MOPs that have a small number of objectives, however, their efficiency often seriously degrades as the number of objectives increases.…”

A Decision Variable Clustering-Based Evolutionary Algorithm for Large-Scale Many-Objective Optimization

“…It is worth noting that, there are also some other treestructure based non-dominated sorting approaches. For example, the sorting algorithm proposed in [73] stores the non-dominated solutions in a M -d tree (M is the number of objectives), and adds new solution via inserting and deleting nodes from the tree; another sorting algorithm proposed in [74] uses a binary tree to store the dominance relationships between solutions, by adjusting the binary tree the solutions in each front can be identified; and the recently proposed M-front [54] keeps an Kd tree to perform approximate nearest neighbor search for determining whether a new solution is dominated by the existing non-dominated solutions. Compared to these existing tree-structure based approaches, the proposed T-ENS holds a considerably less computational complexity as it benefits from the framework of EN-S [55], which allows candidate solutions to be inserted into the tree one by one without adjusting the structure of it.…”

Effectiveness and efficiency of non-dominated sorting for evolutionary multi- and many-objective optimization

“…The eight methods under consideration include fast non-dominated sort (FNS) [7], generalized Jensen's sort [54], divide-and-conquer-based sort [53], deductive sort [28], corner sort [30], M-front [55], ENS-SS [29], and T-ENS [23].…”