Fast Greedy Subset Selection from Large Candidate Solution Sets in Evolutionary Multi-objective Optimization

Subset selection is an interesting and important topic in the field of evolutionary multi-objective optimization (EMO). Especially, in an EMO algorithm with an unbounded external archive, subset selection is an essential post-processing procedure to select a pre-specified number of solutions as the final result. In this paper, we discuss the efficiency of greedy subset selection for the hypervolume, IGD and IGD+ indicators. Greedy algorithms usually efficiently handle subset selection. However, when a large number of solutions are given (e.g., subset selection from tens of thousands of solutions in an unbounded external archive), they often become time-consuming. Our idea is to use the submodular property, which is known for the hypervolume indicator, to improve their efficiency. First, we prove that the IGD and IGD+ indicators are also submodular. Next, based on the submodular property, we propose an efficient greedy inclusion algorithm for each indicator. Then, we demonstrate through computational experiments that the proposed algorithms are much faster than the standard greedy subset selection algorithms.

Evolutionary Multi-Objective Optimization Algorithm Framework with Three Solution Sets

It is assumed in the evolutionary multi-objective optimization (EMO) community that a final solution is selected by a decision maker from a non-dominated solution set obtained by an EMO algorithm. The number of solutions to be presented to the decision maker can be totally different. In some cases, the decision maker may want to examine only a few representative solutions from which a final solution is selected. In other cases, a large number of non-dominated solutions may be needed to visualize the Pareto front. In this paper, we suggest the use of a general EMO framework with three solution sets to handle various situations with respect to the required number of solutions. The three solution sets are the main population of an EMO algorithm, an external archive to store promising solutions, and a final solution set which is presented to the decision maker. The final solution set is selected from the archive. Thus the population size and the archive size can be arbitrarily specified as long as the archive size is not smaller than the required number of solutions. The final population is not necessarily to be a good solution set since it is not presented to the decision maker. Through computational experiments, we show the advantages of this framework over the standard final population and final archive frameworks. We also discuss how to select a final solution set and how to explain the reason for the selection, which is the first attempt towards an explainable EMO framework.

Decomposition-Based Multi-Objective Evolutionary Algorithm Design under Two Algorithm Frameworks

The development of efficient and effective evolutionary multi-objective optimization (EMO) algorithms has been an active research topic in the evolutionary computation community. Over the years, many EMO algorithms have been proposed. The existing EMO algorithms are mainly developed based on the final population framework. In the final population framework, the final population of an EMO algorithm is presented to the decision maker. Thus, it is required that the final population produced by an EMO algorithm is a good solution set. Recently, the use of solution selection framework was suggested for the design of EMO algorithms. This framework has an unbounded external archive to store all the examined solutions. A pre-specified number of solutions are selected from the archive as the final solutions presented to the decision maker. When the solution selection framework is used, EMO algorithms can be designed in a more flexible manner since the final population is not necessarily to be a good solution set. In this paper, we examine the design of MOEA/D under these two frameworks. We use an offline genetic algorithm-based hyper-heuristic method to find the optimal configuration of MOEA/D in each framework. The DTLZ and WFG test suites and their minus versions are used in our experiments. The experimental results suggest the possibility that a more flexible, robust and high-performance MOEA/D algorithm can be obtained when the solution selection framework is used.

Algorithm Configurations of MOEA/D with an Unbounded External Archive

In the evolutionary multi-objective optimization (EMO) community, it is usually assumed that the final population is presented to the decision maker as the result of the execution of an EMO algorithm. Recently, an unbounded external archive was used to evaluate the performance of EMO algorithms in some studies where a pre-specified number of solutions are selected from all the examined non-dominated solutions. In this framework, which is referred to as the solution selection framework, the final population does not have to be a good solution set. Thus, the solution selection framework offers higher flexibility to the design of EMO algorithms than the final population framework. In this paper, we examine the design of MOEA/D under these two frameworks. First, we show that the performance of MOEA/D is improved by linearly changing the reference point specification during its execution through computational experiments with various combinations of initial and final specifications. Robust and high performance of the solution selection framework is observed. Then, we examine the use of a genetic algorithm-based offline hyper-heuristic method to find the best configuration of MOEA/D in each framework. Finally, we further discuss solution selection after the execution of an EMO algorithm in the solution selection framework.

Lazy Greedy Hypervolume Subset Selection from Large Candidate Solution Sets

Subset selection is a popular topic in recent years and a number of subset selection methods have been proposed. Among those methods, hypervolume subset selection is widely used. Greedy hypervolume subset selection algorithms can achieve good approximations to the optimal subset. However, when the candidate set is large (e.g., an unbounded external archive with a large number of solutions), the algorithm is very time-consuming. In this paper, we propose a new lazy greedy algorithm exploiting the submodular property of the hypervolume indicator. The core idea is to avoid unnecessary hypervolume contribution calculation when finding the solution with the largest contribution. Experimental results show that the proposed algorithm is hundreds of times faster than the original greedy inclusion algorithm and several times faster than the fastest known greedy inclusion algorithm on many test problems.

Solution Subset Selection for Final Decision Making in Evolutionary Multi-Objective Optimization

In general, a multi-objective optimization problem does not have a single optimal solution but a set of Pareto optimal solutions, which forms the Pareto front in the objective space. Various evolutionary algorithms have been proposed to approximate the Pareto front using a pre-specified number of solutions. Hundreds of solutions are obtained by their single run. The selection of a single final solution from the obtained solutions is assumed to be done by a human decision maker. However, in many cases, the decision maker does not want to examine hundreds of solutions. Thus, it is needed to select a small subset of the obtained solutions. In this paper, we discuss subset selection from a viewpoint of the final decision making. First we briefly explain existing subset selection studies. Next we formulate an expected loss function for subset selection. We also show that the formulated function is the same as the IGD plus indicator. Then we report experimental results where the proposed approach is compared with other indicator-based subset selection methods.

Effects of Discretization of Decision and Objective Spaces on the Performance of Evolutionary Multiobjective Optimization Algorithms

Recently, the discretization of decision and objective spaces has been discussed in the literature. In some studies, it is shown that the decision space discretization improves the performance of evolutionary multi-objective optimization (EMO) algorithms on continuous multi-objective test problems. In other studies, it is shown that the objective space discretization improves the performance on combinatorial multi-objective problems. However, the effect of the simultaneous discretization of both spaces has not been examined in the literature. In this paper, we examine the effects of the decision space discretization, objective space discretization and simultaneous discretization on the performance of NSGA-II through computational experiments on the DTLZ and WFG problems. Using various settings about the number of decision variables and the number of objectives, our experiments are performed on four types of problems: standard problems, large-scale problems, many-objective problems, and large-scale many-objective problems. We show that the decision space discretization has a positive effect for large-scale problems and the objective space discretization has a positive effect for many-objective problems. We also show the discretization of both spaces is useful for large-scale many-objective problems.