Multi-objective Test Problems Research Articles

A Multipopulation-Based Multiobjective Evolutionary Algorithm.

Multipopulation is an effective optimization component often embedded into evolutionary algorithms to solve optimization problems. In this paper, a new multipopulation-based multiobjective genetic algorithm (MOGA) is proposed, which uses a unique cross-subpopulation migration process inspired by biological processes to share information between subpopulations. Then, a Markov model of the proposed multipopulation MOGA is derived, the first of its kind, which provides an exact mathematical model for each possible population occurring simultaneously with multiple objectives. Simulation results of two multiobjective test problems with multiple subpopulations justify the derived Markov model, and show that the proposed multipopulation method can improve the optimization ability of the MOGA. Also, the proposed multipopulation method is applied to other multiobjective evolutionary algorithms (MOEAs) for evaluating its performance against the IEEE Congress on Evolutionary Computation multiobjective benchmarks. The experimental results show that a single-population MOEA can be extended to a multipopulation version, while obtaining better optimization performance.

A Scalable Test Suite for Continuous Dynamic Multiobjective Optimization.

Dynamic multiobjective optimization (DMO) has gained increasing attention in recent years. Test problems are of great importance in order to facilitate the development of advanced algorithms that can handle dynamic environments well. However, many of the existing dynamic multiobjective test problems have not been rigorously constructed and analyzed, which may induce some unexpected bias when they are used for algorithmic analysis. In this paper, some of these biases are identified after a review of widely used test problems. These include poor scalability of objectives and, more important, problematic overemphasis of static properties rather than dynamics making it difficult to draw accurate conclusion about the strengths and weaknesses of the algorithms studied. A diverse set of dynamics and features is then highlighted that a good test suite should have. We further develop a scalable continuous test suite, which includes a number of dynamics or features that have been rarely considered in literature but frequently occur in real life. It is demonstrated with empirical studies that the proposed test suite is more challenging to the DMO algorithms found in the literature. The test suite can also test algorithms in ways that existing test suites cannot.

Comparison between MOEA/D and NSGA-III on a set of novel many and multi-objective benchmark problems with challenging difficulties

Abstract Currently, evolutionary multiobjective optimization (EMO) algorithms have been successfully used to find a good approximation of many-objective optimization problems (MaOPs). To measure the performance of EMO algorithms, many benchmark multiobjective test problems have been constructed. Among them, DTLZ and WFG are two representative test suites with the scalability to the number of variables and objectives. It should be pointed out that MaOPs can be more challenging if they are involved with difficult problem features, such as objective scalability, complicated Pareto set, bias, disconnection, or degeneracy. In this paper, a set of ten new test problems with above-mentioned difficulties are constructed. Some experimental results on these test problems found by two popular EMO algorithms, i.e., MOEA/D and NSGA-III, are reported and analyzed. Moreover, the performance of these two EMO algorithms with different population sizes on these test problems are also studied.

A Review of Features and Limitations of Existing Scalable Multiobjective Test Suites

In multiobjective optimization, a scalable test problem is one that can be formulated for an arbitrary number of objectives. Scalable test problems evaluate the conceptual foundations of the so-called many-objective evolutionary algorithms. As an important class of problems, scalable test problems should contemplate a wide variety of features allowing us to evaluate and judge specific components of many-objective evolutionary algorithms. This, in fact, should promote the development of new strategies and/or methods in the design of many-objective optimization approaches. For this reason, the study of features and difficulties of this class of problems, plays a salient role in the development of many-objective approaches. As a result, a number of multiobjective scalable test problems have been proposed in recent years. In this paper, we present a review of features and limitations of existing multiobjective test problems formulated in continuous and unconstrained search spaces. We examine some features observed in some test problems which have not been properly discussed before. Additionally, we summarize a list of features and recommendations that should be considered in the design of scalable multiobjective test instances. Then, we preset a review of the state-of-the-art scalable test suites, including their features and limitations according to the recommended guidelines discussed herein. Finally, some possible paths for future research in this area are briefly discussed.

A self-organizing map based hybrid chemical reaction optimization algorithm for multiobjective optimization

Multiobjective particle swarm optimisation (MOPSO) is faced with convergence difficulties and diversity deviation, owing to combined learning orientations and premature phenomena. In MOPSO, leader selection is an important factor that can enhance the algorithm convergence rate. Inspired by this case, and aimed at balancing the convergence and diversity during the searching procedure, a self-organising map is used to construct the neighbourhood relationships among current solutions. In order to increase the population diversity, an extended chemical reaction optimisation algorithm is introduced to improve the diversity performance of the proposed algorithm. In view of the above, a self-organising map-based multiobjective hybrid particle swarm and chemical reaction optimisation algorithm (SMHPCRO) is proposed in this paper. Furthermore, the proposed algorithm is applied to 35 multiobjective test problems with all Pareto set shape and compared with 12 other multiobjective evolutionary algorithms to validate its performance. The experimental results indicate its advantages over other approaches.

An Efficient Framework Using Normalized Dominance Operator for Multi-Objective Evolutionary Algorithms

Multi-objective optimization algorithms using evolutionary optimization methods have shown strength in solving various problems using several techniques for producing uniformly distributed set of solutions. In this article, a framework is presented to solve the multi-objective optimization problem which implements a novel normalized dominance operator (ND) with the Pareto dominance concept. The proposed method has a lesser computational cost as compared to crowding-distance-based algorithms and better convergence. A parallel external elitist archive is used which enhances spread of solutions across the Pareto front. The proposed algorithm is applied to a number of benchmark multi-objective test problems with up to 10 objectives and compared with widely accepted aggregation-based techniques. Experiments produce a consistently good performance when applied to different recombination operators. Results have further been compared with other established methods to prove effective convergence and scalability.

Adaptive Epsilon dominance in decomposition-based multiobjective evolutionary algorithm

Abstract Complicated geometric shapes of Pareto fronts can cause difficulties for multiobjective evolutionary algorithms. To deal with these difficulties, efficient diversity strategies must be highly addressed in order to obtain a set of representative Pareto solutions. In decomposition-based multiobjective evolutionary algorithms, this is often done by optimizing multiple single objective subproblems defined by a set of weight vectors. For complicated Pareto fronts with extreme convexity, disconnection or degeneracy, however, it is nontrivial to set these weight vector properly. To overcome this shortcoming, we propose a new decomposition-based multiobjective evolutionary algorithm based on a hybrid weighting strategy, which optimizes both random subproblems and fixed subproblems. To maintain diversity of nondominated solutions stored in external population, a new archiving strategy based on adaptive Epsilon dominance is also suggested in our proposed algorithm. Our experimental results have showed that our proposed algorithm is superior to several other state-of-the-art multiobjective evolutionary algorithms on a set of benchmark multiobjective test problems with different challenging difficulties regarding the geometric shapes of Pareto fronts.

NSGA-II/EDA Hybrid Evolutionary Algorithm for Solving Multi-objective Economic/Emission Dispatch Problem

—In this study, a hybrid algorithm which combines the NSGA-II with a modified form of the marginal histogram model Estimation of Distribution Algorithm (EDA), herein called the NSGA-II/EDA is proposed for solving the multi-objective economic/emission power dispatch problem. The goal is to improve the convergence while preserving the diversity properties of the obtained solution set. The effect of variable interaction on the marginal histogram EDA model is reduced by performing multi-scale Principal Component Analysis on the population of solutions. Also, the concepts of non-domination and elitism have been introduced into the marginal histogram model in order for it to handle multiple objectives. Several optimization runs were carried out on the standard multi-objective test problems, including the IEEE 30- and the 118-bus test systems. Standard metrics are used to compare the performance of the developed hybrid approach with that of other multi-objective evolutionary algorithms. The effectiveness of the proposed approach in improved convergence, with good diversity is demonstrated.

PlatEMO: A MATLAB Platform for Evolutionary Multi-Objective Optimization [Educational Forum

Over the last three decades, a large number of evolutionary algorithms have been developed for solving multi-objective optimization problems. However, there lacks an upto-date and comprehensive software platform for researchers to properly benchmark existing algorithms and for practitioners to apply selected algorithms to solve their real-world problems. The demand of such a common tool becomes even more urgent, when the source code of many proposed algorithms has not been made publicly available. To address these issues, we have developed a MATLAB platform for evolutionary multi-objective optimization in this paper, called PlatEMO, which includes more than 50 multiobjective evolutionary algorithms and more than 100 multi-objective test problems, along with several widely used performance indicators. With a user-friendly graphical user interface, PlatEMO enables users to easily compare several evolutionary algorithms at one time and collect statistical results in Excel or LaTeX files. More importantly, PlatEMO is completely open source, such that users are able to develop new algorithms on the basis of it. This paper introduces the main features of PlatEMO and illustrates how to use it for performing comparative experiments, embedding new algorithms, creating new test problems, and developing performance indicators. Source code of PlatEMO is now available at: http://bimk.ahu.edu.cn/index.php?s=/Index/Software/index.html.

A multi-objective DIRECT algorithm for ship hull optimization

The paper is concerned with black-box nonlinear constrained multi-objective optimization problems. Our interest is the definition of a multi-objective deterministic partition-based algorithm. The main target of the proposed algorithm is the solution of a real ship hull optimization problem. To this purpose and in pursuit of an efficient method, we develop an hybrid algorithm by coupling a multi-objective DIRECT-type algorithm with an efficient derivative-free local algorithm. The results obtained on a set of “hard” nonlinear constrained multi-objective test problems show viability of the proposed approach. Results on a hull-form optimization of a high-speed catamaran (sailing in head waves in the North Pacific Ocean) are also presented. In order to consider a real ocean environment, stochastic sea state and speed are taken into account. The problem is formulated as a multi-objective optimization aimed at (i) the reduction of the expected value of the mean total resistance in irregular head waves, at variable speed and (ii) the increase of the ship operability, with respect to a set of motion-related constraints. We show that the hybrid method performs well also on this industrial problem.

Grasshopper optimization algorithm for multi-objective optimization problems

This work proposes a new multi-objective algorithm inspired from the navigation of grass hopper swarms in nature. A mathematical model is first employed to model the interaction of individuals in the swam including attraction force, repulsion force, and comfort zone. A mechanism is then proposed to use the model in approximating the global optimum in a single-objective search space. Afterwards, an archive and target selection technique are integrated to the algorithm to estimate the Pareto optimal front for multi-objective problems. To benchmark the performance of the algorithm proposed, a set of diverse standard multi-objective test problems is utilized. The results are compared with the most well-regarded and recent algorithms in the literature of evolutionary multi-objective optimization using three performance indicators quantitatively and graphs qualitatively. The results show that the proposed algorithm is able to provide very competitive results in terms of accuracy of obtained Pareto optimal solutions and their distribution.

On the use of two reference points in decomposition based multiobjective evolutionary algorithms

Decomposition based multiobjective evolutionary algorithms approximate the Pareto front of a multiobjective optimization problem by optimizing a set of subproblems in a collaborative manner. Often, each subproblem is associated with a direction vector and a reference point. The settings of these parameters have a very critical impact on convergence and diversity of the algorithm. Some work has been done to study how to set and adjust direction vectors to enhance algorithm performance for particular problems. In contrast, little effort has been made to study how to use reference points for controlling diversity in decomposition based algorithms. In this paper, we first study the impact of the reference point setting on selection in decomposition based algorithms. To balance the diversity and convergence, a new variant of the multiobjective evolutionary algorithm based on decomposition with both the ideal point and the nadir point is then proposed. This new variant also employs an improved global replacement strategy for performance enhancement. Comparison of our proposed algorithm with some other state-of-the-art algorithms is conducted on a set of multiobjective test problems. Experimental results show that our proposed algorithm is promising.

Late parallelization and feedback approaches for distributed computation of evolutionary multi-objective optimization algorithms

Distributing of the multi-objective evolutionary algorithms into various computational devices in a parallel fashion is a method for speeding up the execution time of the algorithms. When the processors are increased in number, the gain from parallelization decreases. For this reason, the aim of the parallelization method is not only to decrease the overall algorithm execution time, but also to obtain a higher gain from the use of parallel processors. Therefore, in this study two new parallelization approaches are proposed and discussed to increase the benefit from the computational devices. The proposed schemes are named as late parallelization and feedback approaches. These proposed approaches are based on four-layered structure, and each layer (i) has the $$2^i$$ number of computational units where population at each unit is divided into two equal subpopulations at ordered layers. Consequently, the sum of the members at every layer equals to each other. Of these layer-based cognate schemes, in late parallelization, the layers are executed sequentially beginning with the first layer. Until the last layer is executed, a delay time is defined such that layers are executed more than one except the last layer. However, at feedback approach, the order of the layers is different and the last layer is connected with the front layer. So, the population is not only divided into subpopulations but also they are gathered with the in-osculation of the subpopulations. Finally, the performances of these approaches are evaluated on multi-objective test problems with respect to the statistical properties of the number of function evaluations.

MOEA/D-ARA+SBX: A new multi-objective evolutionary algorithm based on decomposition with artificial raindrop algorithm and simulated binary crossover

In the field of optimization computation, there has been a growing interest in applying intelligent algorithms to solve multi-objective optimization problems (MOPs). This paper focuses mainly on the multi-objective evolutionary algorithm based on decomposition, MOEA/D for short, which offers a practical general algorithmic framework of evolutionary multi-objective optimization, and has been achieved great success for a wide range of MOPs. Like most other algorithms, however, MOEA/D has its limitations, which are reflected in three aspects: the problem of balancing diversity and convergence, non-uniform distribution of the Pareto front (PF), and weak convergence of the algorithm. To alleviate these limitations, a new combination of the artificial raindrop algorithm (ARA) and a simulated binary crossover (SBX) operator is first integrated into the framework of MOEA/D to balance the convergence and diversity. Thus, our proposed approach is called MOEA/D with ARA and SBX (MOEA/D-ARA+SBX). On the other hand, the raindrop pool in ARA is further extended to an external elitist archive, which retains only non-dominated solutions and discards all others. In addition, the k-nearest neighbors approach is introduced to prune away redundant non-dominated solutions. In such a way, a Pareto approximate subset with good distribution to the true PF may be achieved. Based on the relevant mathematical theory and some assumptions, it is proven that MOEA/D-ARA+SBX can converge to the true PF with probability one. For performance evaluation and comparison purposes, the proposed approach was applied to 44 multi-objective test problems with all types of Pareto set shape, and compared with 16 other versions of MOEA/D. The experimental results indicate its advantages over other approaches.

An elitist non-dominated sorting bat algorithm NSBAT-II for multi-objective optimization of phthalic anhydride reactor

Nature inspired meta-heuristic algorithms are an integral part of modern optimization techniques. One such algorithm is bat algorithm which is inspired from echolocation behavior of bats and has been successfully applied to non-linear single-objective optimization problems. In this paper, a multi-objective extension of bat algorithm is proposed using the concepts of Pareto non-dominance and elitism. The novel algorithm is tested using thirty multi-objective test problems. The performance is measured using metrics namely, hyper-volume ratio, generational distance and spacing. The newly developed algorithm is then applied to a real-world multi-objective optimization problem of a phthalic anhydride reactor. It shows faster convergence for test problems as well as the industrial optimization problem than two popular nature inspired meta-heuristic algorithms, i.e. multi-objective non-dominated sorting particle swarm optimization and real-coded elitist non-dominated sorting genetic algorithm.

Biased Multiobjective Optimization and Decomposition Algorithm.

The bias feature is a major factor that makes a multiobjective optimization problem (MOP) difficult for multiobjective evolutionary algorithms (MOEAs). To deal with this problem feature, an algorithm should carefully balance between exploration and exploitation. The decomposition-based MOEA decomposes an MOP into a number of single objective subproblems and solves them in a collaborative manner. Single objective optimizers can be easily used in this algorithm framework. Covariance matrix adaptation evolution strategy (CMA-ES) has proven to be able to strike good balance between the exploration and the exploitation of search space. This paper proposes a scheme to use both differential evolution (DE) and covariance matrix adaptation in the MOEA based on decomposition. In this scheme, single objective optimization problems are clustered into several groups. To reduce the computational overhead, only one subproblem from each group is selected to optimize by CMA-ES while other subproblems are optimized by DE. When an evolution strategy procedure meets some stopping criteria, it will be reinitialized and used for solving another subproblem in the same group. A set of new multiobjective test problems with bias features are constructed in this paper. Extensive experimental studies show that our proposed algorithm is suitable for dealing with problems with biases.

Hindrances for robust multi-objective test problems

Despite the significant number of benchmark problems for evolutionary multi-objective optimisation algorithms, there are few in the field of robust multi-objective optimisation. This paper investigates the characteristics of the existing robust multi-objective test problems and identifies the current gaps in the literature. It is observed that the majority of the current test problems suffer from simplicity, so five hindrances are introduced to resolve this issue: bias towards non-robust regions, deceptive global non-robust fronts, multiple non-robust fronts (multi-modal search space), non-improving (flat) search spaces, and different shapes for both robust and non-robust Pareto optimal fronts. A set of 12 test functions are proposed by the combination of hindrances as challenging test beds for robust multi-objective algorithms. The paper also considers the comparison of five robust multi-objective algorithms on the proposed test problems. The results show that the proposed test functions are able to provide very challenging test beds for effectively comparing robust multi-objective optimisation algorithms. Note that the source codes of the proposed test functions are publicly available at www.alimirjalili.com/RO.html.

Stopping criteria for MAPLS-AW, a hybrid multi-objective evolutionary algorithm

Evolutionary algorithms are widely used to solve multi-objective optimization problems effectively by performing global search over the solution space to find better solutions. Hybrid evolutionary algorithms have been introduced to enhance the quality of solutions obtained. One such hybrid algorithm is memetic algorithm with preferential local search using adaptive weights (MAPLS-AW) (Bhuvana and Aravindan in Soft Comput, doi:10.1007/s00500-015-1593-9, 2015). MAPLS-AW, a variant of NSGA-II algorithm, recognizes the elite solutions of the population and preferences are given to them for local search during the evolution. This paper proposes a termination scheme derived from the features of MAPLS-AW. The objective of the proposed scheme is to detect convergence of population without compromising quality of solutions generated by MAPLS-AW. The proposed termination scheme consists of five stopping measures, among which two are newly proposed in this paper to predict the convergence of the population. Experimental study has been carried out to analyze the performance of the proposed termination scheme and to compare with existing termination schemes. Several constrained and unconstrained multi-objective benchmark test problems are used for this comparison. Additionally, a real-time application economic emission and load dispatch has also been used to check the performance of the proposed scheme. The results show that the proposed scheme identifies convergence of population much earlier than the existing stopping schemes without compromising the quality of solutions.

Shifted robust multi-objective test problems

In 2013 Gaspar-Cunha et al. proposed a set of novel robust multi-objective benchmark functions to increase the difficulty of the current test problems and effectively mimic the characteristics of real search spaces. Despite the merits of the proposed benchmark problems, it is observed that the robust Pareto optimal fronts are located on the boundaries of the search space, which may result in the infeasibility of solutions obtained in case of perturbations along the negative side of the second parameter. This paper modifies the proposed test functions by Gaspar-Cunha et al. to mimic real problems better and allow the parameters to be fluctuated by any degree of perturbations. In fact, the robust fronts are shifted to the centre of the search space, so that any degree of uncertainties can be considered. The paper considers theoretical and experimental analysis of both set of test functions as well.

Novel frameworks for creating robust multi-objective benchmark problems

Robust optimization deals with considering different types of uncertainties during the optimization process in order to obtain reliable solutions, a critical issue when solving real problems. Multiple objectives are another vital aspect of real problems that should be considered during optimization. In order to benchmark the performance of different meta-heuristics, test problems are essential, as the literature shows. Despite the significant number of studies in developing multi-objective test problems, there is currently neither study on the suitability of the current robust multi-objective benchmark problems, nor standard frameworks to create them. This motivates our attempts to investigate the features of the current robust test problems and propose three novel frameworks to generate various robust multi-objective test problems with alterable parameters. As case studies, Robust Multi-Objective Particle Swarm Optimization (RMOPSO), Robust Non-dominated Sorting Genetic Algorithm (RNSGA-II), Robust Multiobjective Evolutionary Algorithm Based on Decomposition (RMOEA/D), Robust Two Local Best Multi-objective Particle Swarm Optimization (R2LB-MOPSO), and Robust Decomposition-Based Multi-objective Evolutionary Algorithm with an Ensemble of Neighborhood Sizes (RENS-MOEA/D) are benchmarked on the proposed test problems. The results show that the proposed frameworks are able to generate robust multi-objective test problems with different adjustable characteristics and levels of difficulty. In addition, the results show that the test problems generated by the proposed frameworks can provide very challenging test beds for effectively benchmarking the performance of robust meta-heuristics.