Benchmark Multi-objective Optimization Problems Research Articles

A space sampling based large-scale many-objective evolutionary algorithm

Large-scale multiobjective optimization problems have attracted increasing attention in both engineering applications and scientific research. Academically, large-scale multiobjective problems involve hundreds or thousands of decision variables. Due to the large decision space, the performance of traditional multiobjective evolutionary algorithms decreases dramatically when dealing with large-scale multiobjective problems, especially many-objective problems. With this in mind, a space sampling based large-scale many-objective evolutionary algorithm (LSMaOEA) is proposed in this article. Specifically, a space sampling method is developed that alternately performs upper/lower-linkage sampling and individual-linkage sampling to sample a set of individuals in the decision space. An environmental selection strategy based on nondominated sorting and reference vector association is proposed. Thus, the proposed LSMaOEA can alleviate excessively dense sampling at boundaries and improve the diversity of existing space sampling based algorithms for large-scale many-objective problems. In the experiments, the proposed algorithm is assessed by comparing it with eight state-of-the-art multi/many-objective evolutionary algorithms. The evaluation is conducted using two popular indicators across nine challenging multiobjective optimization benchmark problems with up to 2000 decision variables. The extensive experimental results consistently reveal that the proposed algorithm outperforms all the compared algorithms.

Scaling Multiobjective Evolution to Large Data With Minions: A Bayes-Informed Multitask Approach.

In an era of pervasive digitalization, the growing volume and variety of data streams poses a new challenge to the efficient running of data-driven optimization algorithms. Targeting scalable multiobjective evolution under large-instance data, this article proposes the general idea of using subsampled small-data tasks as helpful minions (i.e., auxiliary source tasks) to quickly optimize for large datasets-via an evolutionary multitasking framework. Within this framework, a novel computational resource allocation strategy is designed to enable the effective utilization of the minions while guarding against harmful negative transfers. To this end, an intertask empirical correlation measure is defined and approximated via Bayes' rule, which is then used to allocate resources online in proportion to the inferred degree of source-target correlation. In the experiments, the performance of the proposed algorithm is verified on: 1) sample average approximations of benchmark multiobjective optimization problems under uncertainty and 2) practical multiobjective hyperparameter tuning of deep neural network models. The results show that the proposed algorithm can obtain up to about 73% speedup relative to existing approaches, demonstrating its ability to efficiently tackle real-world multiobjective optimization involving evaluations on large datasets.

A multi-objective multi-tasking evolutionary algorithm based inverse mapping and adaptive transformation strategy: IM-MFEA

Multi-tasking optimization algorithm attracts much attention because the knowledge transfer between tasks enables the algorithm to process multiple related tasks simultaneously. However, negative knowledge transfer occasionally occurs, which may weaken the performance of the algorithm. To reduce the impact of negative knowledge transfer, a multi-objective multi-tasking optimization algorithm (IM-MFEA) based on inverse model mapping and an objective transformation strategy is proposed. First, correlation analysis is applied in an inverse mapping strategy to improve the accuracy of the inverse mapping model. Then, following the pattern of using the source domain solutions to assist the optimization of the target domain, the adaptive transformation strategy is used to improve the quality of the source domain solution in the objective space. These transformed solutions are reconstructed through the inverse mapping strategy. Finally, these reconstructed source domain solutions are mated with the target domain solutions to generate competitive offspring individuals for the target domain. To verify the effectiveness of the IM-MFEA, comprehensive experiments were conducted on nine multi-objective multi-factorial optimization (MFO) benchmark problems. Empirical results demonstrate that IM-MFEA is superior to other algorithms in 90% of test instances by inverted generational distance (IGD) and hypervolume (HV) value indicators.

A diversity preservation method for expensive multi-objective combinatorial optimization problems using Novel-First Tabu Search and MOEA/D

Expensive multi-objective combinatorial optimization problems have constraints in the number of objective function evaluations due to time, financial, or resource restrictions. As most combinatorial problems, they are subject to a high number of duplicated solutions. Given the fact that expensive environments limit the number of objective function evaluations, the existence of duplicated solutions heavily impacts the optimization process due to poor diversity and low convergence speed. This paper proposes the Novel-First Tabu Search, a greedy-strategy mechanism that uses Knowledge-Assisted Local Search methods to preserve the population diversity and increase the exploration and exploitation ability of MOEA/D. Experiments are conducted on constrained, unconstrained, multimodal, deceptive, linear, convex, and non-convex Pareto Front multi-objective combinatorial optimization benchmark problems. This paper also conducts an experiment on the real-world, expensive problem of Well Placement Optimization using a benchmark case based on the Namorado oil field, located in the Campos Basin, Brazil. The experimental results and performance comparison with state-of-the-art algorithms demonstrate that the proposed design significantly preserves diversity and increases convergence without violating the constraint in the number of objective function evaluations.

Multi-Objective Material Generation Algorithm (MOMGA) for Optimization Purposes

Optimization is a process of decision-making in which some iterative procedures are conducted to maximize or minimize a predefined objective function representing the overall behavior of a considered system problem. Most of the time, one specific function cannot represent the overall behavior of a system with particular levels of complexity, so the multiple objective functions should be determined for this purpose which requires an algorithm with adaptability to this situation. Multi-objective optimization is a process of decision making in which maximization or minimization of multiple objective functions is considered for reaching the acceptable levels of performance for the considered system problem. In this paper, the multi-objective version of the Material Generation Algorithm (MGA) is proposed as MOMGA, one of the recently developed metaheuristic algorithms for single-objective optimization. To evaluate the overall performance of the MOMGA, the benchmark multi-objective optimization problems of the Competitions on Evolutionary Computation (CEC) are considered alongside the real-world engineering problems. Based on the results, the MOMGA is capable of providing very acceptable results in dealing with multi-objective optimization problems.

Learnable Evolutionary Search Across Heterogeneous Problems via Kernelized Autoencoding

The design of the evolutionary algorithm with learning capability from past search experiences has attracted growing research interests in recent years. It has been demonstrated that the knowledge embedded in the past search experience can greatly speed up the evolutionary process if properly harnessed. Autoencoding evolutionary search (AEES) is a recently proposed search paradigm, which employs a single-layer denoising autoencoder to build the mapping between two problems by configuring the solutions of each problem as the input and output for the autoencoder, respectively. The learned mapping makes it possible to perform knowledge transfer across heterogeneous problem domains with diverse properties. It has shown a promising performance of learning and transferring the knowledge from past search experiences to facilitate the evolutionary search on a variety of optimization problems. However, despite the success enjoyed by AEES, the linear autoencoding model cannot capture the nonlinear relationship between the solution sets used in the mapping construction. Taking this cue, in this article, we devise a kernelized autoencoder to construct the mapping in a reproducing kernel Hilbert space (RKHS), where the nonlinearity among problem solutions can be captured easily. Importantly, the proposed kernelized autoencoding method also holds a closed-form solution which will not bring much computational burden in the evolutionary search. Furthermore, a kernelized autoencoding evolutionary-search (KAES) paradigm is proposed that adaptively selects the linear and kernelized autoencoding along the search process in pursuit of effective knowledge transfer across problem domains. To validate the efficacy of the proposed KAES, comprehensive empirical studies on both benchmark multiobjective optimization problems as well as real-world vehicle crashworthiness design problem are presented.

REVIEW OF THE MULTI-OBJECTIVE SWARM INTELLIGENCE OPTIMIZATION ALGORITHMS

Multi-objective swarm intelligence (MOSI) metaheuristics were proposed to solve multi-objective optimization problems (MOPs) that consists of two or more conflict objectives, in which improving an objective leads to the degradation of the other. The MOSI algorithms are based on the integration of single objective algorithms and multi-objective optimization (MOO) approach. The MOO approaches include scalarization, Pareto dominance, decomposition and indicator-based. In this paper, the status of MOO research and state-of-the-art MOSI algorithms namely, multi-objective particle swarm, artificial bee colony, firefly algorithm, bat algorithm, gravitational search algorithm, grey wolf optimizer, bacterial foraging and moth-flame optimization algorithms have been reviewed. These reviewed algorithms were mainly developed to solve continuous MOPs. The review is based on how the algorithms deal with objective functions using MOO approaches, the benchmark MOPs used in the evaluation and performance metrics. Furthermore, it describes the advantages and disadvantages of each MOO approach and provides some possible future research directions in this area. The results show that several MOO approaches have not been used in most of the proposed MOSI algorithms. Integrating other different MOO approaches may help in developing more effective optimization algorithms, especially in solving complex MOPs. Furthermore, most of the MOSI algorithms have been evaluated using MOPs with two objectives, which clarifies open issues in this research area.

Runtime Analysis of Somatic Contiguous Hypermutation Operators in MOEA/D Framework

Somatic contiguous hypermutation (CHM) operators are important variation operators in artificial immune systems. The few existing theoretical studies are only concerned with understanding the optimization behavior of CHM operators on solving single-objective optimization problems. The MOEA/D framework is one of the most popular strategies for solving multi-objective optimization problems (MOPs). In this paper, we present a runtime analysis of using two CHM operators in MOEA/D framework for solving five benchmark MOPs, including four bi-objective and one many-objective problems. Our analyses show that the expected runtimes of CHM operators on the four bi-objective problems are better than or as good as that of the well-studied standard bit mutation operator. Moreover, using CHM operators in MOEA/D framework can improve the best known upper bound on the many-objective problem by a factor of n. This paper provides insight into understanding the optimization behavior of CHM operators in the well-known MOEA/D framework, and indicates that using the CHM operator in MOEA/D framework is a promising method for handling MOPs.

A reference point-based evolutionary algorithm for approximating regions of interest in multiobjective problems

Most evolutionary multiobjective optimization algorithms are designed to approximate the entire Pareto front. During the last decade, a series of preference-based evolutionary algorithms have been developed, where a part of the Pareto front is approximated by incorporating the preferences of a Decision Maker. However, only a few such algorithms are able to obtain well-distributed solutions covering the complete “region of interest” that is determined by a reference point. In this paper, a preference-based evolutionary algorithm for approximating the region of interest is proposed. It is based on the state-of-the-art genetic algorithm NSGA-II and the CHIM approach introduced in the NBI method which is used to obtain uniformly distributed solutions in the region of interest. The efficiency of the proposed algorithm has been experimentally evaluated and compared to other state-of-the-art multiobjective preference-based evolutionary algorithms by solving a set of multiobjective optimization benchmark problems. It has been shown that the incorporation of the Decision Maker’s preferences and the CHIM approach into the NSGA-II algorithm allows approximating the whole region of interest accurately while maintaining a good distribution of the obtained solutions.

A repository of real-world datasets for data-driven evolutionary multiobjective optimization

Many real-world optimization applications have more than one objective, which are modeled as multiobjective optimization problems. Generally, those complex objective functions are approximated by expensive simulations rather than cheap analytic functions, which have been formulated as data-driven multiobjective optimization problems. The high computational costs of those problems pose great challenges to existing evolutionary multiobjective optimization algorithms. Unfortunately, there have not been any benchmark problems reflecting those challenges yet. Therefore, we carefully select seven benchmark multiobjective optimization problems from real-world applications, aiming to promote the research on data-driven evolutionary multiobjective optimization by suggesting a set of benchmark problems extracted from various real-world optimization applications.

Analysis of semi-asynchronous multi-objective evolutionary algorithm with different asynchronies

This paper proposes a novel master–slave parallel evolutionary algorithm (EA) approach with different asynchrony and provides its detailed analyses on multi-objective optimization problems. We express the proposed EA with different asynchrony as a semi-asynchronous EA. A semi-asynchronous EA generates new solutions whenever evaluations of the predefined number of solutions complete, unlike a conventional synchronous EA waits for evaluations of all solutions to generate the next population. To establish a semi-asynchronous EA, this paper introduces an asynchrony parameter that is used to decide how many solutions are waited to generate new solutions. We conduct an experiment to verify the effectiveness of the proposed semi-asynchronous EA on benchmark problems with the several variances of the evaluation time. In the experiment, we apply a semi-asynchronous EA to NSGA-II and NSGA-III, which are well-known multi-objective EAs. The semi-asynchronous NSGA-IIs and the semi-asynchronous NSGA-IIIs with different asynchronies are compared on multi-objective optimization benchmark problems. The experimental result reveals that semi-asynchronous approaches with an appropriate asynchrony have possibility to outperform the asynchronous and the synchronous ones. Additionally, detailed analysis reveals that an appropriate asynchrony may vary not only depends on a target problem but also depends on the degree of the evolution process.

Computing budget allocation in multi-objective evolutionary algorithms for stochastic problems

Abstract Multi-objective stochastic problems are important problems in practice and are often solved through multi-objective evolutionary algorithms. Researchers have developed different noise handling techniques to improve the efficiency and accuracy of such algorithms, primarily by integrating these methods into the evaluation or environmental selection steps of the algorithms. In this work, a combination of studies that compare integration of different computing budget allocation methods into either the evaluation or the environmental selection steps are conducted. These comparisons are performed on stochastic problems derived from benchmark multi-objective optimization problems and consider varying levels of noise. The algorithms are compared in terms of both proximity to and coverage of the true Pareto-optimal front and sufficient studies are performed to allow statistically significant conclusions to be drawn. It is shown that integrating computing budget allocation methods into the environmental selection step is better than integration within the evaluation step.

Subset simulation for multi-objective optimization

Subset simulation is an efficient Monte Carlo technique originally developed for structural reliability problems, and further modified to solve single-objective optimization problems based on the idea that an extreme event (optimization problem) can be considered as a rare event (reliability problem). In this paper subset simulation is extended to solve multi-objective optimization problems by taking advantages of Markov Chain Monte Carlo and a simple evolutionary strategy. In the optimization process, a non-dominated sorting algorithm is introduced to judge the priority of each sample and handle the constraints. To improve the diversification of samples, a reordering strategy is proposed. A Pareto set can be generated after limited iterations by combining the two sorting algorithms together. Eight numerical multi-objective optimization benchmark problems are solved to demonstrate the efficiency and robustness of the proposed algorithm. A parametric study on the sample size in a simulation level and the proportion of seed samples is performed to investigate the performance of the proposed algorithm. Comparisons are made with three existing algorithms. Finally, the proposed algorithm is applied to the conceptual design optimization of a civil jet.

The elitist non-dominated sorting genetic algorithm with inheritance (i-NSGA-II) and its jumping gene adaptations for multi-objective optimization

Like elitism, parent inheritance plays an important role to decide the quality of offspring and it is believed that the parents with high intelligence quotient (IQ) like to produce children with high IQ. Inspiring this concept, the improved pool of an initial random population involving the best set of chromosomes are incorporated in the framework of multi-objective optimization genetic algorithm. The effects of parent inheritance in the elitist non-dominated sorting genetic algorithm (called, i-NSGA-II) on the speed of convergence to the global Pareto-optimal front is compared with the binary coded NSGA-II using different benchmark multi-objective optimization problems. The parent inheritance is also incorporated in several jumping gene (JG) adapted NSGA-II algorithms. The efficacy of inheritance in NSGA-II and its several JG adaptations is tested by quantifying several indicators, namely, generational distance, spacing and hyper-volume ratio using different benchmark multi-objective optimization problems from the literature. The inclusion of the inheritance operator improves the speed of convergence to global Pareto-optimal front significantly with a minimum number of generations over existing NSGA-II and several JG adapted NSGA-II algorithms. The effectiveness of the proposed operator is further established by solving real-life robust multi-objective optimization problems involving the drilling of oil-well and synthesis of sal oil biodiesel.

Duality evolution: an efficient approach to constraint handling in multi-objective particle swarm optimization

This paper proposes an efficient approach for constraint handling in multi-objective particle swarm optimization. The particles population is divided into two non-overlapping populations, named infeasible population and feasible population. The evolution process in each population is done independent of the other one. The infeasible particles are evolved in the constraint space toward feasibility. During evolution process, if an infeasible particle becomes a feasible one, it migrates to feasible population. In a parallel process, the particles in feasible population are evolved in the objective space toward Pareto optimality. At each generation of multi-objective particle swarm optimization, a leader should be assigned to each particle to move toward it. In the proposed method, a different leader selection algorithm is proposed for each population. For feasible population, the leader is selected using a priority-based method in three levels and for infeasible population, a leader replacement method integrated by an elitism-based method is proposed. The proposed approach is tested on several constrained multi-objective optimization benchmark problems, and its results are compared with two popular state-of-the-art constraint handling multi-objective algorithms. The experimental results indicate that the proposed algorithm is highly competitive in solving the benchmark problems.

A multiagent, dynamic rank-driven multi-deme architecture for real-valued multiobjective optimization

Multiobjective real parameter optimization is a challenging problem in majority of engineering applications. This paper presents a creative multiagent and dynamic multi-deme architecture based on a novel collaboration mechanism for the solution of multiobjective real-parameter optimization problems. The proposed architecture comprises a number of multiobjective metaheuristic agents that act on subsets of a population based on a cyclic assignment order. The proposed multiagent architecture works iteratively in sessions including two consecutive phases: in the first phase, a population of solutions is divided into subpopulations based on dominance ranks of its elements. In the second phase, each multiobjective metaheuristic is assigned to work on a subpopulation based on a cyclic or round-robin order. Hence, each metaheuristic operates on a different-rank subpopulation in subsequent sessions, where a session starts with a new assignment of metaheuristics and ends when termination criteria for the session are satisfied. Individual agents have their local archives of non-dominated solutions extracted in a session, while there is a global archive keeping all non-dominated solutions found so far. At the end of each session, all subpopulations are combined into one global population to be used for the initialization of the next session. Similarly, all local archives are merged with the global archive to get the set of all non-dominated solutions found by all metaheuristics through working on subsets of different rank-levels. This way, the metaheuristics cooperate with each other by sharing their search experiences through collecting them in a common population and a common global archive. The proposed multiagent system is experimentally evaluated using the well-known CEC2009 multiobjective optimization benchmark problems set. Analysis of the experimental results demonstrated that the proposed architecture achieves better performance compared to majority of its state-of-the-art competitors in almost all problem instances.

A preference-based multi-objective evolutionary algorithm R-NSGA-II with stochastic local search

Incorporation of a decision maker’s preferences into multi-objective evolutionary algorithms has become a relevant trend during the last decade, and several preference-based evolutionary algorithms have been proposed in the literature. Our research is focused on improvement of a well-known preference-based evolutionary algorithm R-NSGA-II by incorporating a local search strategy based on a single agent stochastic approach. The proposed memetic algorithm has been experimentally evaluated by solving a set of well-known multi-objective optimization benchmark problems. It has been experimentally shown that incorporation of the local search strategy has a positive impact to the quality of the algorithm in the sense of the precision and distribution evenness of approximation.

Universal partially evolved parallelization of MOEA/D for multi-objective optimization on message-passing clusters

This paper presents a universal partially evolved parallelization of the multi-objective evolutionary algorithm based on decomposition (MOEA/D) for multi-objective optimization on message-passing clusters to reduce its computation time. The partially evolved MOEA/D (peMOEA/D) is suitable not only for the bi-objective space, but also for higher dimensional objective spaces by using a partially evolved island model. This model improves the algorithm universality and population diversity by keeping a subpopulation equal in size to the entire population, but only evolving a partial (equal to a partition size) subpopulation on each separate processor in a cluster. Furthermore, a nearest-neighbor partitioning approach, hybrid migration policy and adaptive neighbor-based topology are adopted in the peMOEA/D. The fat partitions generated by the nearest-neighbor partitioning can reduce migration traffic of elitist individuals across subpopulations. Then, hybrid migration of both elitist individuals and utopian points helps separate subpopulations to cooperate in guiding the search quickly towards a complete front. Next, the adaptive neighbor-based topology connects neighbor partitions only and achieves an excellent balance between convergence speed and migration traffic. Experimental results on benchmark multi-objective optimization problems with two or more objectives demonstrate the satisfactory overall performance of the peMOEA/D in terms of both convergence performance and speedup on message-passing clusters.

A Simple and Fast Hypervolume Indicator-Based Multiobjective Evolutionary Algorithm.

To find diversified solutions converging to true Pareto fronts (PFs), hypervolume (HV) indicator-based algorithms have been established as effective approaches in multiobjective evolutionary algorithms (MOEAs). However, the bottleneck of HV indicator-based MOEAs is the high time complexity for measuring the exact HV contributions of different solutions. To cope with this problem, in this paper, a simple and fast hypervolume indicator-based MOEA (FV-MOEA) is proposed to quickly update the exact HV contributions of different solutions. The core idea of FV-MOEA is that the HV contribution of a solution is only associated with partial solutions rather than the whole solution set. Thus, the time cost of FV-MOEA can be greatly reduced by deleting irrelevant solutions. Experimental studies on 44 benchmark multiobjective optimization problems with 2-5 objectives in platform jMetal demonstrate that FV-MOEA not only reports higher hypervolumes than the five classical MOEAs (nondominated sorting genetic algorithm II (NSGAII), strength Pareto evolutionary algorithm 2 (SPEA2), multiobjective evolutionary algorithm based on decomposition (MOEA/D), indicator-based evolutionary algorithm, and S-metric selection based evolutionary multiobjective optimization algorithm (SMS-EMOA)), but also obtains significant speedup compared to other HV indicator-based MOEAs.

Multiobjective nondominated neighbor coevolutionary algorithm with elite population

A nondominated neighbor coevolutionary algorithm (NNCA) with a novel coevolutionary mechanism is proposed for multiobjective optimization, where elite individuals are used to guide the search. All the nondominated individuals are divided into two subpopulations, namely, the elite population and the common population according to their crowding-distance values. The elite individual located in less-crowded region will have more chances to select more team members for its own team and thus this region can be explored more sufficiently. Therefore, the elite population will guide the search to the more promising and less-crowded region. Secondly, to avoid the `search stagnation' situation which means that algorithms fail to find enough nondominated solutions, a size guarantee mechanism (SGM) is proposed for elite population by emigrating some dominated individuals to the elite population when necessary. The SGM can prevent the algorithm from searching around limited nondominated individuals and being trapped into the `search stagnation' situation. In addition, several different kinds of crossover and mutation operator are used to generate offspring, which are benefits for the diversity property. Tests on 20 multiobjective optimization benchmark problems including five ZDT problems, five DTLZ problems and ten unconstrained CEC09 test problems show that NNCA is very competitive compared with seven the state-of-the-art multiobjective optimization algorithms.