A parallel large-scale multiobjective evolutionary algorithm based on two-space decomposition

Many real-world multi-objective optimization problems have hundreds or even thousands of decision variables, which contrast with the current practice of multi-objective metaheuristics whose performance is typically assessed using benchmark problems with a relatively low number of decision variables (normally, no more than 30). In this paper, we propose a cooperative coevolution framework that is capable of optimizing large scale (in decision variable space) multi-objective optimization problems. We adopt a benchmark that is scalable in the number of decision variables (the ZDT test suite) and compare our proposed algorithm with respect to two state-of-the-art multi-objective evolutionary algorithms (GDE3 and NSGA-II) when using a large number of decision variables (from 200 up to 5000). The results clearly indicate that our proposed approach is effective as well as efficient for solving large scale multi-objective optimization problems.

A fast sampling based evolutionary algorithm for million-dimensional multiobjective optimization

Many real-world optimization problems are characterized by multiple conflicting objectives, which are known as multi-objective optimization problems (MOPs). In the last two decades, evolutionary algorithms have shown promising performance in solving various MOPs, and a large number of multi-objective evolutionary algorithms (MOEAs) have been proposed. In order to determine the most suitable MOEA for a specific MOP, it is usually necessary to perform experiments to compare the performance of multiple candidate MOEAs. In 2017, an evolutionary multi-objective optimization platform was proposed by us, called PlatEMO, which provides the source codes of many state-of-the-art MOEAs and helps researchers perform batch experiments on these MOEAs. In this work, we illustrate the method of using the newest version of PlatEMO to solve MOPs in applications, by means of a case study on the feature selection problem, which is an important and difficult task in machine learning and data mining. This paper details the method of adding the feature selection problem to PlatEMO, and presents the experimental results of eight MOEAs on nine datasets in feature selection.

Multi-objective evolutionary algorithms (MOEAs) of the state of the art are created with the only purpose of dealing with the number of objective functions in a multi-objective optimization problem (MOP) and treat the decision variables of a MOP as a whole. However, when dealing with MOPs with a large number of decision variables (more than 100) their efficacy decreases as the number of decision variables of the MOP increases. On the other hand, problem decomposition, in terms of decision variables, has been found to be extremely efficient and effective for solving large scale optimization problems. Nevertheless, most of the currently available approaches for large scale optimization rely on models based on cooperative coevolution or linkage learning methods that use multiple subpopulations or preliminary analysis, respectively, which is computationally expensive (in terms of function evaluations) when used within MOEAs. In this work, we study the effect of what we call operational decomposition, which is a novel framework based on coevolutionary concepts to apply MOEAs’s crossover operator without adding any extra cost. We investigate the improvements that NSGA-III can achieve when combined with our proposed coevolutionary operators. This new scheme is capable of improving efficiency of a MOEA when dealing with large scale MOPs having from 200 up to 1200 decision variables.

Reference sets generated with uniformly distributed weight vectors on a unit simplex are widely used by several multi-objective evolutionary algorithms (MOEAs). They have been employed to tackle multi-objective optimization problems (MOPs) with four or more objective functions, i.e., the so-called many-objective optimization problems. These MOEAs have shown a good performance on MOPs with regular Pareto front shapes, i.e., simplex-like shapes. However, it has been observed that in many cases, their performance degrades on MOPs with irregular Pareto front shapes. In this paper, we designed a new selection mechanism that aims to promote a Pareto front shape invariant performance of MOEAs that use weight vector-based reference sets. The newly proposed selection mechanism takes advantage of weight vector-based reference sets and seven pair-potential functions. It was embedded into the non-dominated sorting genetic algorithm III (NSGA-III) to increase its performance on MOPs with different Pareto front geometries. We use the DTLZ and DTLZ\(^{-1}\) test problems to perform an empirical study about the usage of these pair-potential functions for this selection mechanism. Our experimental results show that the pair-potential functions can enhance the distribution of solutions obtained by weight vector-based MOEAs on MOPs with irregular Pareto front shapes. Also, the proposed selection mechanism permits maintaining the good performance of these MOEAs on MOPs with regular Pareto front shapes.

Local dominance has been shown to improve significantly the overall performance of multiobjective evolutionary algorithms (MOEAs) on combinatorial optimization problems. This work proposes the control of dominance area of solutions in local dominance MOEAs to enhance Pareto selection aiming to find solutions with high convergence and diversity properties. We control the expansion or contraction of the dominance area of solutions and analyze its effects on the search performance of a local dominance MOEA using 0/1 multiobjective knapsack problems. We show that convergence of the algorithm can be significantly improved while keeping a good distribution of solutions along the whole true Pareto front by using local dominance with expansion of dominance area of solutions. We also show that by controlling the dominance area of solutions dominance can be applied within very small neighborhoods, which reduces significantly the computational cost of the local dominance MOEA

Local dominance has been shown to improve significantly the overall performance of multiobjective evolutionary algorithms (MOEAs) on combinatorial optimization problems. This work proposes the control of dominance area of solutions in local dominance MOEAs to enhance Pareto selection aiming to find solutions with high convergence and diversity properties. We control the expansion or contraction of the dominance area of solutions and analyze its effects on the search performance of a local dominance MOEA using 0/1 multiobjective knapsack problems. We show that convergence can be significantly improved while keeping a good distribution of solutions along the whole true Pareto front by using the local dominance MOEA with expansion of dominance area of solutions. We also show that dominance can be applied within very small neighborhoods by controlling the dominance area of solutions, which reduces significantly the computational cost of the local dominance MOEA.

In real-world multi-objective problems, each objective has a totally different scale. However, some frequently-used multi-objective evolutionary algorithms (MOEAs) have no objective space normalization mechanisms. The effect of objective space normalization on the performance of decomposition-based MOEAs (e.g., MOEA/D and NSGA-III) has already been examined for artificial test problems (e.g., DTLZ and WFG) in the literature. In this paper, we examine its practical usefulness for real-world multi-objective problems using various MOEAs. Our experimental results clearly show that objective space normalization is needed not only in decomposition-based MOEAs but also in hypervolume-based MOEAs. We also explain why objective space normalization is needed in these two types of MOEAs.

In this paper, we propose a framework to accelerate the computational efficiency of evolutionary algorithms on large-scale multiobjective optimization. The main idea is to track the Pareto optimal set (PS) directly via problem reformulation. To begin with, the algorithm obtains a set of reference directions in the decision space and associates them with a set of weight variables for locating the PS. Afterwards, the original large-scale multiobjective optimization problem is reformulated into a low-dimensional single-objective optimization problem. In the reformulated problem, the decision space is reconstructed by the weight variables and the objective space is reduced by an indicator function. Thanks to the low dimensionality of the weight variables and reduced objective space, a set of quasi-optimal solutions can be obtained efficiently. Finally, a multiobjective evolutionary algorithm is used to spread the quasi-optimal solutions over the approximate Pareto optimal front evenly. Experiments have been conducted on a variety of large-scale multiobjective problems with up to 5000 decision variables. Four different types of representative algorithms are embedded into the proposed framework and compared with their original versions, respectively. Furthermore, the proposed framework has been compared with two state-of-the-art algorithms for large-scale multiobjective optimization. The experimental results have demonstrated the significant improvement benefited from the framework in terms of its performance and computational efficiency in large-scale multiobjective optimization.

This paper presents a random-based dynamic grouping strategy (RDG) for cooperative coevolution to deal with large scale multi-objective optimization problems (MOPs) by decomposing the whole dimension into several groups of variables with an equal size. First, a decomposer pool containing different group sizes is designed. Then, a group size is dynamically selected with probability in the evolution process. The probability of each group size in the pool is computed based on the historical performance measured by C-metric, a common metric in multi-objective optimization. Under the selected group size, random grouping is executed to decompose the whole dimension into groups. Through this, both the group size and the group components are dynamic. Finally, combining RDG with a traditional and famous multi-objective evolutionary algorithm (MOEA) named MOEA/D, we develop MOEA/D-RDG to cope with large scale MOPs. The efficacy of the proposed MOEA/D-RDG is verified on two sets of MOPs (UF1-UF10 and WFG1-WFG9) through comparing with two MOEA/D variants.

Multioperator search strategy for evolutionary multiobjective optimization

Under certain smoothness assumptions, the Pareto set of a continuous multiobjective optimization problem is a piecewise continuous manifold in the decision space, which can be derived from the Karush-Kuhn-Tucker condition. Despite that a number of multiobjective evolutionary algorithms (MOEAs) have been proposed, their performance on multiobjective optimization problems with complicated Pareto sets (MOP-cPS) is still unsatisfying. In this article, we adopt the concept of manifold and propose a manifold learning-inspired mating strategy to enhance the diversity maintenance in MOEAs for solving MOP-cPS efficiently. In the proposed strategy, all of the individuals are first clustered into different manifolds according to their distribution in the objective space, and then the mating reproduction is restricted among individuals in the same manifold. Moreover, we embed the proposed mating strategy in three representative MOEAs and compare the embedded MOEAs with their original versions using the assortative genetic operators on a variety of MOP-cPS. The experimental results demonstrate the significant performance improvements benefitting from the proposed mating restriction strategy.

Many multi-objective evolutionary algorithms (MOEAs) have been successful in approximating the Pareto Front. However, well-distributed solutions in the objective and decision spaces are still required in many real-life applications. In this paper, a novel MOEA is proposed to this problem. Distinct from other MOEAs, the proposed algorithm suggests a framework, which includes two crowding estimation methods, multiple selection methods for mating and search strategies for variation, to improve the MOEA' s searching ability, and the diversity of its solutions. The algorithm emphasizes the importance of using the decision space and the objective space diversities. The objective space crowding and decision space crowding distances are designed using different ideas. To produce new individuals, three different types of mating selections and their respective search strategies are constructed for the main population and the two sparse populations, with the help of the two crowding measurements. Finally, based on the experimental tests on 17 unconstrained multi-objective optimization problems, the proposed algorithm is demonstrated to have better results compared to several state-of-the-art MOEAs. A detailed analysis on the effectiveness and robustness of the framework is also presented.

This study proposes a novel recombination operator, called hybrid crossover operator (HX), which is performed in gene level of chromosome to enhance the optimization performance of multi-objective evolutionary algorithms (MOEAs). The proposed HX operator combines the advantages of simulated binary crossover with local search ability and differential evolution with strong global search capability. When HX is embedded into two state-of-the-art MOEAs, i.e., NSGA-II and MOEA/D-DE, the experimental results validate the improvement of HX when compared to the original counterpart.

Towards Software Product Lines Optimization Using Evolutionary Algorithms