An improved biogeography/complex algorithm based on decomposition for many-objective optimization

MOEA/D decomposes a multi-objective optimization problem into a number of single objective optimization problems. Each single objective optimization problem is defined by a scalarizing function using a weight vector. In MOEA/D, there are several scalarizing approaches such as the weighted Tchebycheff, the weighted sum, and the PBI (penalty-based boundary intersection). However, these conventional scalarizing approaches face a difficulty to approximate a widely spread Pareto front in some problems. To enhance the spread of Pareto optimal solutions in the objective space and improve the search performance of MOEA/D especially in many-objective optimization problems, in this work we propose the inverted PBI scalarizing approach which is an extension of the conventional PBI. We use many-objective knapsack problems and WFG4 problems with 2-8 objectives, and compare the search performance of NSGA-III and four MOEA/Ds using the weighted Tchebycheff, the weighted sum, the PBI and the inverted PBI. As results, we show that MOEA/D using the inverted PBI achieves higher search performance than other algorithms in problems with many-objectives and the difficulty to obtain a widely spread Pareto front in the objective space.

Nondominated sorting plays an important role in Pareto-based multiobjective evolutionary algorithms (MOEAs). When faced with many-objective optimization problems multiobjective optimization problems (MOPs) with more than three objectives, the number of comparisons needed in nondominated sorting becomes very large. In view of this, a new corner sort is proposed in this paper. Corner sort first adopts a fast and simple method to obtain a nondominated solution from the corner solutions, and then uses the nondominated solution to ignore the solutions dominated by it to save comparisons. Obtaining the nondominated solutions requires much fewer objective comparisons in corner sort. In order to evaluate its performance, several state-of-the-art nondominated sorts are compared with our corner sort on three kinds of artificial solution sets of MOPs and the solution sets generated from MOEAs on benchmark problems. On one hand, the experiments on artificial solution sets show the performance on the solution sets with different distributions. On the other hand, the experiments on the solution sets generated from MOEAs show the influence that different sorts bring to MOEAs. The results show that corner sort performs well, especially on many-objective optimization problems. Corner sort uses fewer comparisons than others.

A modified PBI approach for multi-objective optimization with complex Pareto fronts

A reference direction and entropy based evolutionary algorithm for many-objective optimization

A collaborative decomposition-based evolutionary algorithm integrating normal and penalty-based boundary intersection methods for many-objective optimization

Evolutionary sequential transfer optimization is a paradigm that leverages search experience from solved source optimization tasks to accelerate the evolutionary search of a target task. Even though many algorithms have been developed, they mainly focus on objective-homogeneous problems, where the source and target tasks possess a similar number of objectives. In this work, we explore objective-heterogeneous problems, in which knowledge transfers across single-objective optimization problems (SOPs), multiobjective optimization problems (MOPs), and many-objective optimization problems (MaOPs). Objective-heterogeneous problems challenge the existing methods due to the diverse search and objective spaces between the source and the target task. To address this issue, we present a decision variable analysis-based transfer method that can conduct knowledge transfer across problems with the different numbers of objectives. We first separate decision variables of MOPs and MaOPs into convergence-related variables (CVs) and diversity-related variables (DVs), according to their roles while treating variables of SOPs as CVs. Then, we propose a convergence transfer module to transfer knowledge of CVs to speed up the convergence. It aligns both solutions and fitness ranks for preserving fitness rank consistency between the source and target tasks, whereby accelerating search speed. Besides, a diversity transfer module is presented to refine the distribution of DVs to maintain the population diversity. The experimental results on objective-heterogeneous test problems and a real-world case study have demonstrated the effectiveness of the proposed algorithm.

Pareto-based multi-objective evolutionary algorithms experience grand challenges in solving many-objective optimization problems due to their inability to maintain both convergence and diversity in a high-dimensional objective space. Exiting approaches usually modify the selection criteria to overcome this issue. Different from them, we propose a novel meta-objective (MeO) approach that transforms the many-objective optimization problems in which the new optimization problems become easier to solve by the Pareto-based algorithms. MeO converts a given many-objective optimization problem into a new one, which has the same Pareto optimal solutions and the number of objectives with the original one. Each meta-objective in the new problem consists of two components which measure the convergence and diversity performances of a solution, respectively. Since MeO only converts the problem formulation, it can be readily incorporated within any multi-objective evolutionary algorithms, including those non-Pareto-based ones. Particularly, it can boost the Pareto-based algorithms' ability to solve many-objective optimization problems. Due to separately evaluating the convergence and diversity performances of a solution, the traditional density-based selection criteria, for example, crowding distance, will no longer mistake a solution with poor convergence performance for a solution with low density value. By penalizing a solution in term of its convergence performance in the meta-objective space, the Pareto dominance becomes much more effective for a many-objective optimization problem. Comparative study validates the competitive performance of the proposed meta-objective approach in solving many-objective optimization problems.

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.

In many objective optimization problems (MaOPs), more than three distinct objectives are optimized. The challenging part in MaOPs is to get the Pareto approximation (PA) with high diversity and good convergence. In Literature, in order to solve the issue of diversity and convergence in MaOPs, many approaches are proposed using different multi objective evolutionary algorithms (MOEAs). Moreover, to get better results, the researchers use the sets of reference points to differentiate the solutions and to model the search process, it further evaluates and selects the non-dominating solutions by using the reference set of solutions. Furthermore, this technique is used in some of the swarm-based evolutionary algorithms. In this paper, we have used some effective adaptations of bat algorithm with the previous mentioned approach to effectively handle the many objective problems. Moreover, we have called this algorithm as many objective bat algorithm (MaOBAT). This algorithm is a biologically inspired algorithm, which uses echolocation power of micro bats. Each bat represents a complete solution, which can be evaluated based on the problem specific fitness function and then based on the dominance relationship, non-dominated solutions are selected. In proposed MaOBAT, dominance rank is used as dominance relationship (dominance rank of a solution means by how many other solutions a solution dominated). In our proposed strategy, dynamically allocated set of reference points are used, allowing the algorithm to have good convergence and high diversity pareto fronts (PF). The experimental results show that the proposed algorithm has significant advantages over several state-of-the-art algorithms in terms of the quality of the solution.

An ACO-based Hyper-heuristic for Sequencing Many-objective Evolutionary Algorithms that Consider Different Ways to Incorporate the DM's Preferences

An adaptive penalty-based boundary intersection method for many-objective optimization problem

Many-objective optimisation problems (MaOPs) cause serious difficulties for existing multi-objective evolutionary algorithms (MOEAs). One common way to alleviate these difficulties is to use objective dimensionality reduction. Most existing objective reduction methods are time-consuming because they require MOEAs to run numerous generations. Pareto corner search evolutionary algorithm (PCSEA) was proposed in [18] to speed up objective reduction methods by only seeking corner solutions instead of whole solutions. However, the PCSEA-based objective reduction method in [18] needs to predefine a threshold to select objectives which strongly depends on problems and is not straightforward to obtain. This paper proposes a new objective dimensionality reduction method by integrating PCSEA and principal component analysis (PCA). Thanks to combining advantages of PCSEA and PCA, the proposed method not only can be efficient to eliminate redundant objectives, but also not require to define any parameter in advanced. The experimental results also show that the proposed method can perform objective reduction more successfully than the PCSEA-based objective reduction method. The results further strengthen the links between evolutionary computation and machine learning to address optimization problems.

Multi-objective optimization problems with more than three objectives are commonly referred to as many-objective optimization problems. Usually, this class of problem brings new and complex challenges to the current optimization methods, mainly maintaining the right balance between convergence and diversity. During the last years, various approaches have been proposed to solve many-objective problems. However, most existing experimental comparative studies are restricted to continuous problems. Few studies have encompassed the most recently proposed state-of-the-art approaches and made an experimental comparison applied to combinatorial optimization problems. Aiming to fill this gap, this paper presents a comparative analysis with eight algorithms covering various categories to solve a many-objective Dial-a-Ride problem. The results show that different observations can be made about the algorithms' behavior when using different test sets. Also, algorithms originally proposed to deal with problems with up to three objectives have overcome recently proposed ones.

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.

When solving multi-objective optimization problems (MOPs) with big data, traditional multi-objective evolutionary algorithms (MOEAs) meet challenges because they demand high computational costs that cannot satisfy the demands of online data processing involving optimization. The gradient heuristic optimization methods show great potential in solving large scale numerical optimization problems with acceptable computational costs. However, some intrinsic limitations make them unsuitable for searching for the Pareto fronts. It is believed that the combination of these two types of methods can deal with big MOPs with less computational cost. The main contribution of this paper is that a multi-objective memetic algorithm based on decomposition for big optimization problems (MOMA/D-BigOpt) is proposed and a gradient-based local search operator is embedded in MOMA/D-BigOpt. In the experiments, MOMA/D-BigOpt is tested on the multi-objective big optimization problems with thousands of variables. We also combine the local search operator with other widely used MOEAs to verify its effectiveness. The experimental results show that the proposed algorithm outperforms MOEAs without the gradient heuristic local search operator.