Walking Fish Group Research Articles

Handling multiple objectives using k‐means clustering guided multiobjective evolutionary algorithm

AbstractMultiobjective optimization problems (MOPs) are very popular these days and have gained continuous research attention. These problems involve a minimum of two conflicting objectives that need to be optimized simultaneously. Multiobjective evolutionary algorithms (MOEAs) optimize these objectives by targeting solutions on the Pareto front. To judge the performance of different MOEAs, the convergence rate and population diversity are used. In order to create a good algorithm with a high convergence rate and diversity, the k‐means clustering algorithm is combined with a real parameter version of the improved environmental adaptation method (IEAM‐RP) in this paper. This is done as IEAM‐RP converges very fast and the k‐means clustering algorithm is used to provide diversity in decision search space. To check the performance of the proposed algorithm, it has been compared with nine state‐of‐the‐art multiobjective algorithms (MOPSOtridist, MOEA/D‐STM, MOEA/D, MOPSO, NSGA‐II, MOPSOses, MOGOA, MODA, MOALO) using standard multiobjective benchmark functions. Further, the performance of the proposed algorithm is compared against IBEA and SMS‐EMOA using walking fish group (WFG) test suit. The experimental results prove the effectiveness of the proposed algorithm.

An improved model‐based evolutionary algorithm for multi‐objective optimization

AbstractThe basic idea in the estimation of distribution algorithms is the replacement of heuristic operators with machine learning models such as regression models, clustering models, or classification models. So, recently, the model‐based evolutionary algorithms (MBEAs) have been suggested in three groups: The estimation of distribution algorithms (EDAs), surrogate assisted evolutionary algorithms, and the inversed models to map from the objective space to the decision space. In this article, a new approach, based on an inversed model of Gaussian process and random forest framework, is proposed. The main idea is applying the process of random forest variable importance with a random grouping that determines some of the best assignment of decision variables to objective functions in order to form a Gaussian process in inverse models that maps to decision space the rich solutions which are discovered from objective space. Then these inverse models through sampling the objective space generate offspring. The proposed algorithm has been tested on the benchmark test suite for evolutionary algorithms (modified Deb K, Thiele L, Laumanns M, Zitzler E (DTLZ), and Walking Fish Group (WFG)) and indicates that the proposed method is a competitive and promising approach.

An inverse model‐based multiobjective estimation of distribution algorithm using Random‐Forest variable importance methods

AbstractMost existing methods of multiobjective estimation of distributed algorithms apply the estimation of distribution of the Pareto‐solution on the decision space during the search and little work has proposed on making a regression‐model for representing the final solution set. Some inverse‐model‐based approaches were reported, such as inversed‐model of multiobjective evolutionary algorithm (IM‐MOEA), where an inverse functional mapping from Pareto‐Front to Pareto‐solution is constructed on nondominated solutions based on Gaussian process and random grouping technique. But some of the effective inverse models, during this process, may be removed. This paper proposes an inversed‐model based on random forest framework. The main idea is to apply the process of random forest variable importance that determines some of the best assignment of decision variables (xn) to objective functions (fm) for constructing Gaussian process in inversed‐models that map all nondominated solutions from the objective space to the decision space. In this work, three approaches have been used: classical permutation, Naïve testing approach, and novel permutation variable importance. The proposed algorithm has been tested on the benchmark test suite for evolutionary algorithms [modified Deb K, Thiele L, Laumanns M, Zitzler E (DTLZ) and Walking Fish Group (WFG)] and indicates that the proposed method is a competitive and promising approach.

Scalable and customizable benchmark problems for many-objective optimization

Solving many-objective problems (MaOPs) is still a significant challenge in the multi-objective optimization (MOO) field. One way to measure algorithm performance is through the use of benchmark functions (also called test functions or test suites), which are artificial problems with a well-defined mathematical formulation, known solutions and a variety of features and difficulties. In this paper we propose a parameterized generator of scalable and customizable benchmark problems for MaOPs. It is able to generate problems that reproduce features present in other benchmarks and also problems with some new features. We propose here the concept of generative benchmarking, in which one can generate an infinite number of MOO problems, by varying parameters that control specific features that the problem should have: scalability in the number of variables and objectives, bias, deceptiveness, multimodality, robust and non-robust solutions, shape of the Pareto front, and constraints. The proposed Generalized Position-Distance (GPD) tunable benchmark generator uses the position-distance paradigm, a basic approach to building test functions, used in other benchmarks such as Deb, Thiele, Laumanns and Zitzler (DTLZ), Walking Fish Group (WFG) and others. It includes scalable problems in any number of variables and objectives and it presents Pareto fronts with different characteristics. The resulting functions are easy to understand and visualize, easy to implement, fast to compute and their Pareto optimal solutions are known.

A Novel Multi-Objective Five-Elements Cycle Optimization Algorithm

Inspired by the mechanism of generation and restriction among five elements in Chinese traditional culture, we present a novel Multi-Objective Five-Elements Cycle Optimization algorithm (MOFECO). During the optimization process of MOFECO, we use individuals to represent the elements. At each iteration, we first divide the population into several cycles, each of which contains several individuals. Secondly, for every individual in each cycle, we judge whether to update it according to the force exerted on it by other individuals in the cycle. In the case of an update, a local or global update is selected by a dynamically adjustable probability P s ; otherwise, the individual is retained. Next, we perform combined mutation operations on the updated individuals, so that a new population contains both the reserved and updated individuals for the selection operation. Finally, the fast non-dominated sorting method is adopted on the current population to obtain an optimal Pareto solution set. The parameters’ comparison of MOFECO is given by an experiment and also the performance of MOFECO is compared with three classic evolutionary algorithms Non-dominated Sorting Genetic Algorithm II (NSGA-II), Multi-Objective Particle Swarm Optimization algorithm (MOPSO), Pareto Envelope-based Selection Algorithm II (PESA-II) and two latest algorithms Knee point-driven Evolutionary Algorithm (KnEA) and Non-dominated Sorting and Local Search (NSLS) on solving test function sets Zitzler et al’s Test suite (ZDT), Deb et al’s Test suite (DTLZ), Walking Fish Group (WFG) and Many objective Function (MaF). The experimental results indicate that the proposed MOFECO can approach the true Pareto-optimal front with both better diversity and convergence compared to the five other algorithms.

Multi-type multi-objective imaging scheduling method based on improved NSGA-III for satellite formation system

This paper studies the imaging scheduling issue for China’s L-band differential interferometric synthetic aperture radar (InSAR) satellite formation which can work in different observation modes to satisfy different imaging requests. Previous imaging scheduling mainly focuses on maximizing the total revenue of selected targets but ignores that different types of targets actually have different demands. Therefore, a constraint satisfaction model which classifies targets into three types according to targets’ size and number of observations is established, meanwhile, three objective functions are considered to satisfy demands of different types of targets. Furthermore, an imaging scheduling method based on improved non-dominated sorting genetic algorithm III (NSGA-III) is proposed to obtain a set of well-converged and well-diversified non-dominating solutions. New Niche-Preservation operation with the same penalty value for convergence and diversity performance is adopted in NSGA-III. Numerical comparison simulations on walking fish group (WFG) test suits and three imaging scheduling instances of different size show the superiority of the proposed methodology.

An Optimization Framework of Multiobjective Artificial Bee Colony Algorithm Based on the MOEA Framework.

The artificial bee colony (ABC) algorithm has become one of the popular optimization metaheuristics and has been proven to perform better than many state-of-the-art algorithms for dealing with complex multiobjective optimization problems. However, the multiobjective artificial bee colony (MOABC) algorithm has not been integrated into the common multiobjective optimization frameworks which provide the integrated environments for understanding, reusing, implementation, and comparison of multiobjective algorithms. Therefore, a unified, flexible, configurable, and user-friendly MOABC algorithm framework is presented which combines a multiobjective ABC algorithm named RMOABC and the multiobjective evolution algorithms (MOEA) framework in this paper. The multiobjective optimization framework aims at the development, experimentation, and study of metaheuristics for solving multiobjective optimization problems. The framework was tested on the Walking Fish Group test suite, and a many-objective water resource planning problem was utilized for verification and application. The experiment's results showed the framework can deal with practical multiobjective optimization problems more effectively and flexibly, can provide comprehensive and reliable parameters sets, and can complete reference, comparison, and analysis tasks among multiple optimization algorithms.

Improved genetic algorithm based on particle swarm optimization-inspired reference point placement

ABSTRACTThis article investigates the use of optimal reference point placement to improve performance of non-dominated sorting genetic algorithm (NSGA). Placement of reference points for many-objective optimization is inspired by wheel and Von Neumann topologies of Particle Swarm Optimization (PSO). Results obtained show that the pattern of reference point placement determines performance efficiency of NSGA. The better-performing wheel topology (called wheel reference point genetic algorithm (wRPGA), is compared to three other many-objective evolutionary algorithms: knee-driven evolutionary algorithm (KnEA), non-dominated sorting genetic algorithm III (NSGAIII) and multi-objective evolutionary algorithm based on dominance and decomposition (MOEAD/D). The selected many-objective benchmark problems are Walking Fish Group 2 (WFG2) and Deb-Thiele-Laumanns-Zitzler 2 (DTLZ2). It is also tested on a 3-objective cost function for a hypothetical model of a stand-alone microgrid. Through the simulations, the wheel configuration performed 88.9% better than the Von Neumann configuration. The wheel topology also achieved better performance with respect to inverted generational distance (IGD) compared to KnEA, NSGAIII and MOEAD/D for 7 out of 15 IEEE Congress on Evolutionary Computation (CEC) 2017 benchmark problems. wRPGA gave a good approximation of the Pareto front for the 3-objective model representing the hypothetical microgrid.

Multi-Objective Optimization Algorithm Based on Sperm Fertilization Procedure (MOSFP)

In this paper, we propose an extended multi-objective version of single objective optimization algorithm called sperm swarm optimization algorithm. The proposed multi-objective optimization algorithm based on sperm fertilization procedure (MOSFP) operates based on Pareto dominance and a crowding factor, that crowd and filter out the list of the best sperms (global best values). We divide the sperm swarm into three equal parts, after that, different types of turbulence (mutation) operators are applied on these parts, such as uniform mutation, non-uniform mutation, and without any mutation. Our algorithm is compared against three well-known algorithms in the field of optimization. These algorithms are NSGA-II, SPEA2, and OMOPSO. These algorithms are compared using a very popular benchmark function suites called Zitzler-Deb-Thiele (ZDT) and Walking-Fish-Group (WFG). We also adopt three quality metrics to compare the convergence, accuracy, and diversity of these algorithms, including, inverted generational distance (IGD), spread (SP), and epsilon (∈). The experimental results show that the performance of the proposed MOSFP is highly competitive, which outperformed OMOPSO in solving problems such as ZDT3, WFG5, and WFG8. In addition, the proposed MOSFP outperformed both of NSGA-II or SPEA2 algorithms in solving all the problems.

Impact of selection methods on the diversity of many-objective Pareto set approximations

Selection methods are a key component of all multi-objective and, consequently, many-objective optimisation evolutionary algorithms. They must perform two main tasks simultaneously. First of all, they must select individuals that are as close as possible to the Pareto optimal front (convergence). Second, but not less important, they must help the evolutionary approach to provide a diverse population. In this paper, we carry out a comprehensive analysis of state-of-the-art selection methods with different features aimed to determine the impact that this component has on the diversity preserved by well-known multi-objective optimisers when dealing with many-objective problems. The algorithms considered herein, which incorporate Pareto-based and indicator-based selection schemes, are analysed through their application to the Walking Fish Group (WFG) test suite taking into account an increasing number of objective functions. Algorithmic approaches are assessed via a set of performance indicators specifically proposed for measuring the diversity of a solution set, such as the Diversity Measure and the Diversity Comparison Indicator. Hypervolume, which measures convergence in addition to diversity, is also used for comparison purposes. The experimental evaluation points out that the reference-point-based selection scheme of the Non-dominated Sorting Genetic Algorithm III (NSGA-III) and a modified version of the Non-dominated Sorting Genetic Algorithm II (NSGA-II), where the crowding distance is replaced by the Euclidean distance, yield the best results.

Benchmarking Multi- and Many-Objective Evolutionary Algorithms Under Two Optimization Scenarios

Recently, a large number of multi-objective evolutionary algorithms (MOEAs) for many-objective optimization problems have been proposed in the evolutionary computation community. However, an exhaustive benchmarking study has never been performed. As a result, the performance of the MOEAs has not been well understood yet. Moreover, in almost all previous studies, the performance of the MOEAs was evaluated based on nondominated solutions in the final population at the end of the search. Such traditional benchmarking methodology has several critical issues. In this paper, we exhaustively investigate the anytime performance of 21 MOEAs using an unbounded external archive (UEA), which stores all nondominated solutions found during the search process. Each MOEA is evaluated under two optimization scenarios called UEA and reduced UEA in addition to the standard final population scenario. These two scenarios are more practical in real-world applications than the final population scenario. Experimental results obtained under the two scenarios are significantly different from the previously reported results under the final population scenario. For example, results on the Walking Fish Group test problems with up to six objectives indicate that some recently proposed MOEAs are outperformed by some classical MOEAs. We also analyze the reason why some classical MOEAs work well under the UEA and the reduced UEA scenarios.

Pareto Fronts of Many-Objective Degenerate Test Problems

In general, an ${M}$ -objective continuous optimization problem has an ( $ {M-1}$ )-dimensional Pareto front in the objective space. If its dimension is smaller than ( $ {M-1}$ ), it is called a degenerate Pareto front. Deb–Thiele–Laumanns–Zitzler (DTLZ)5 and Walking Fish Group (WFG)3 have often been used as many-objective continuous test problems with degenerate Pareto fronts. However, it was noted that DTLZ5 has a nondegenerate part of the Pareto front. Constraints have been proposed to remove the nondegenerate part. In this letter, first we show that WFG3 also has a nondegenerate part. Then, we derive constraints to remove the nondegenerate part. Finally, we show that the existence of the nondegenerate part makes WFG3 an interesting test problem through computational experiments.

A coevolutionary approach to many objective optimization based on a novel ranking method

Ranking solutions of the population in an evolutionary algorithm that solves a many objective optimization problem is a challenging task which has been vastly studied in recent years. Loss in the hypervolume of the population when a solution is omitted could be a good measure for ranking solutions but calculating this value for high dimensional problems is not tractable. In the selection operator of evolutionary algorithms, the actual hypervolume values are not important, with only a relative knowledge of these values we can select best solutions. In this paper we have proposed a novel method for approximating the ranking induced by the hypervolume indicator. This method is compared to the similar methods and its efficiency and performance is proved via proper experiments. The method is tested using benchmark test problems from the Walking Fish Group problem family which are scalable both in number of variables and objectives.

A clustering-ranking method for many-objective optimization

In evolutionary multi-objective optimization, balancing convergence and diversity remains a challenge and especially for many-objective (three or more objectives) optimization problems (MaOPs). To improve convergence and diversity for MaOPs, we propose a new approach: clustering-ranking evolutionary algorithm (crEA), where the two procedures (clustering and ranking) are implemented sequentially. Clustering incorporates the recently proposed non-dominated sorting genetic algorithm III (NSGA-III), using a series of reference lines as the cluster centroid. The solutions are ranked according to the fitness value, which is considered to be the degree of closeness to the true Pareto front. An environmental selection operation is performed on every cluster to promote both convergence and diversity. The proposed algorithm has been tested extensively on nine widely used benchmark problems from the walking fish group (WFG) as well as combinatorial travelling salesman problem (TSP). An extensive comparison with six state-of-the-art algorithms indicates that the proposed crEA is capable of finding a better approximated and distributed solution set.

Novel search scheme for multi-objective evolutionary algorithms to obtain well-approximated and widely spread Pareto solutions

In multi-objective optimization, the quality of Pareto-optimal solutions is evaluated by the efficiency of the optimal front (proximity), uniformity, and spread. This paper introduces a novel search scheme for multi-objective evolutionary algorithms (MOEAs), whose solutions demonstrate improved proximity and spread metrics. Our proposed scheme comprises two search phases with different search objectives. The first phase uses a reference-point-based approach to improve proximity; the second phase adopts a distributed-cooperation scheme (DC-scheme) to broaden the range of solutions. We experimentally investigate the effectiveness of our proposed scheme on the walking fish group (WFG) test suite of scalable multi-objective problems. Finally, we show the applicability of the proposed scheme to various types of MOEAs.

Choice function based hyper-heuristics for multi-objective optimization

A selection hyper-heuristic is a high level search methodology which operates over a fixed set of low level heuristics. During the iterative search process, a heuristic is selected and applied to a candidate solution in hand, producing a new solution which is then accepted or rejected at each step. Selection hyper-heuristics have been increasingly, and successfully, applied to single-objective optimization problems, while work on multi-objective selection hyper-heuristics is limited. This work presents one of the initial studies on selection hyper-heuristics combining a choice function heuristic selection methodology with great deluge and late acceptance as non-deterministic move acceptance methods for multi-objective optimization. A well-known hypervolume metric is integrated into the move acceptance methods to enable the approaches to deal with multi-objective problems. The performance of the proposed hyper-heuristics is investigated on the Walking Fish Group test suite which is a common benchmark for multi-objective optimization. Additionally, they are applied to the vehicle crashworthiness design problem as a real-world multi-objective problem. The experimental results demonstrate the effectiveness of the non-deterministic move acceptance, particularly great deluge when used as a component of a choice function based selection hyper-heuristic.

Multiobjective Estimation of Distribution Algorithm Based on Joint Modeling of Objectives and Variables

This paper proposes a new multiobjective estimation of distribution algorithm (EDA) based on joint probabilistic modeling of objectives and variables. This EDA uses the multidimensional Bayesian network as its probabilistic model. In this way, it can capture the dependencies between objectives, variables and objectives, as well as the dependencies learned between variables in other Bayesian network-based EDAs. This model leads to a problem decomposition that helps the proposed algorithm find better tradeoff solutions to the multiobjective problem. In addition to Pareto set approximation, the algorithm is also able to estimate the structure of the multiobjective problem. To apply the algorithm to many-objective problems, the algorithm includes four different ranking methods proposed in the literature for this purpose. The algorithm is first applied to the set of walking fish group problems, and its optimization performance is compared with a standard multiobjective evolutionary algorithm and another competitive multiobjective EDA. The experimental results show that on several of these problems, and for different objective space dimensions, the proposed algorithm performs significantly better and on some others achieves comparable results when compared with the other two algorithms. The algorithm is then tested on the set of CEC09 problems, where the results show that multiobjective optimization based on joint model estimation is able to obtain considerably better fronts for some of the problems compared with the search based on conventional genetic operators in the state-of-the-art multiobjective evolutionary algorithms.

A multi-objective hyper-heuristic based on choice function

Hyper-heuristics are emerging methodologies that perform a search over the space of heuristics in an attempt to solve difficult computational optimization problems. We present a learning selection choice function based hyper-heuristic to solve multi-objective optimization problems. This high level approach controls and combines the strengths of three well-known multi-objective evolutionary algorithms (i.e. NSGAII, SPEA2 and MOGA), utilizing them as the low level heuristics. The performance of the proposed learning hyper-heuristic is investigated on the Walking Fish Group test suite which is a common benchmark for multi-objective optimization. Additionally, the proposed hyper-heuristic is applied to the vehicle crashworthiness design problem as a real-world multi-objective problem. The experimental results demonstrate the effectiveness of the hyper-heuristic approach when compared to the performance of each low level heuristic run on its own, as well as being compared to other approaches including an adaptive multi-method search, namely AMALGAM.