Multi-objective Benchmark Problems Research Articles

Co-design of an unmanned cable shovel for structural and control integrated optimization: A highly heterogeneous constrained multi-objective optimization algorithm

Unmanned cable shovel is an intelligence-based large machine used for excavating in open-pit coal mines, with significant implications for the security and development of energy resources. The co-design of unmanned cable shovels, considering elements from both structural and control systems, has demonstrated superior performance in obtaining optimal solutions compared to the traditional approaches that focus on individual systems. However, the inherent complexities of diverse systems often lead to heterogeneous function evaluations, posing challenges for the existing optimization algorithms. To address highly heterogeneous multi-objective optimization problems characterized by surrogate-approximated expensive functions and explicit inexpensive functions, this study proposes a highly heterogeneous constrained multi-objective optimization (HHCMO) algorithm. Leveraging NSGA-III as an evolutionary optimizer and the Kriging model for approximating expensive functions, HHCMO strategically addresses the challenges posed by highly heterogeneous objective evaluations. A Monte Carlo sampling-based expected hypervolume improvement (EHVI) criterion, employing a new nearest point approximation method in a unified sampling space, effectively handles the impact of heterogeneous objective evaluations on the sequential infill of the Kriging model. Thorough evaluations on both unconstrained and constrained multi-objective benchmark problems validate the correctness of HHCMO and its superiority relative to state-of-the-art algorithms. Finally, the effectiveness of the proposed method is successfully demonstrated in an engineering scenario of an unmanned cable shovel, where it adeptly handles expensive functions from the structural system and inexpensive functions from the control system.

Multi-objective Geometric Mean Optimizer (MOGMO): A Novel Metaphor-Free Population-Based Math-Inspired Multi-objective Algorithm

This research introduces a novel multi-objective adaptation of the Geometric Mean Optimizer (GMO), termed the Multi-Objective Geometric Mean Optimizer (MOGMO). MOGMO melds the traditional GMO with an elite non-dominated sorting approach, allowing it to pinpoint Pareto optimal solutions through offspring creation and selection. A Crowding Distance (CD) coupled with an Information Feedback Mechanism (IFM) selection strategy is employed to maintain and amplify the convergence and diversity of potential solutions. MOGMO efficacy and capabilities are assessed using thirty notable case studies. This encompasses nineteen multi-objective benchmark problems without constraints, six with constraints and five multi-objective engineering design challenges. Based on the optimization results, the proposed MOGMO is better 54.83% in terms of GD, 64.51% in terms of IGD, 67.74% in terms of SP, 70.96% in terms of SD, 64.51% in terms of HV and 77.41% in terms of RT. Therefore, MOGMO has a better convergence and diversity for solving un-constraint, constraint and real-world application. Statistical outcomes from MOGMO are compared with those from Multi-Objective Equilibrium Optimizer (MOEO), Decomposition-Based Multi-Objective Symbiotic Organism Search (MOSOS/D), Non-dominated Sorting Genetic Algorithm (NSGA-II), Multi-Objective Multi-Verse Optimization (MOMVO) and Multi-Objective Plasma Generation Optimizer (MOPGO) algorithms, utilizing identical performance measures. This comparison reveals that MOGMO consistently exhibits robustness and excels in addressing an array of multi-objective challenges. The MOGMO source code is available at https://github.com/kanak02/MOGMO.

New solutions to a multi-objective benchmark problem of induction heating: an application of computational biogeography and evolutionary algorithms

New solutions to a multi-objective benchmark problem of induction heating: an application of computational biogeography and evolutionary algorithms

Directed quick search guided evolutionary framework for large-scale multi-objective optimization problems

For large-scale multi-objective evolutionary algorithms (LSMOEAs), obtaining efficient evolutionary directions in an ultrahigh-dimensional decision space to produce high-quality offspring is a major challenge. This study proposes a novel framework called the directed quick search-guided large-scale evolutionary framework (QSLMOF) to address multi-objective optimization problems on a large scale. The framework contains two innovative strategies: the bidirectional vector-based sampling strategy (BDVS) and the quick search-guided directed reproduction strategy (QS-DRS). BDVS is introduced as an approach to swiftly discern promising solutions that can steer the exploration in the large-scale decision space. This is achieved by formulating two distinct types of sampling directions to rapidly reduce the search space and strike a delicate balance between convergence and diversity. In QS-DRS, we introduced the concept of the potential convergence gradient (PCG), incorporating directional information from historical searches and convergence directions indicated byelite solutions in the current population. With this property, inferior solutions can obtain excellent search directions to explore the decision space, which can improve the convergence rate and prevent the search from falling into a local optimum. The proposed large-scale evolutionary framework incorporates an existing environmental selection mechanism. Comprehensive experiments show that the two novel strategies improve the search efficiency and evolutionary quality of LSMOEAs in ultrahigh-dimensional decision spaces. Moreover, the proposed framework outperformed seven state-of-the-art LSMOEAs for nine large-scale multi-objective benchmark problems LSMOP1-LSMOP9 with up to three objectives and 10000 decision variables.

Evolutionary Multitasking for Optimization Based on Generative Strategies

Evolutionary multitasking (EMT) is one of the emerging topics in evolutionary computation. EMT can solve multiple related optimization tasks simultaneously and enhance the optimization of each task via knowledge sharing among tasks. Many EMT algorithms have been proposed and achieved success in various problems, yet EMT for multi-objective optimization remains a big challenge. The existing multi-objective EMT algorithms tend to suffer from slow convergence and difficulty in generating high-quality knowledge. To alleviate these issues, this paper proposes a new EMT algorithm namely EMT-GS for multi-objective optimization based on two generative strategies. Particularly, generative adversarial networks and inertial differential evolution are introduced to generate transferable knowledge and offspring, respectively. A generative adversarial network is trained periodically for each source-target task pair, based on which helpful knowledge is generated from the source task and transferred to boost the solving of the target task. To accelerate the population convergence, the inertial differential evolution strategy is put forward to generate offspring in a promising direction according to the individuals from the previous generation and the transferred knowledge. The performance of EMT-GS is validated on three multitasking multi-objective benchmark problems. The experimental results highlight the excellent competitiveness of EMT-GS compared to other state-of-the-art multi-objective EMT algorithms.

A framework for inverse surrogate modeling for fitness estimation applied to Multi-Objective Evolutionary Algorithms

Many-Objective Optimization Problems, or MaOPs, are complex optimization problems with more than three objective functions. Traditional Multi-Objective Evolutionary Algorithms (MOEAs) have shown poor scalability in solving this problem. Neuroevolution techniques can be used to overdraw that problem. The use of machine learning techniques to enhance optimization algorithms applied to MaOPs has drawn attention due to their capacity to add domain knowledge during the search process. One method of this kind is inverse surrogate modeling for fitness estimation, which uses machine learning models to enhance MOEAs, mapping the objective function values to the decision variables. Some inverse modeling methods in literature have performed well in various optimization problems. Despite the promising results, new strategies using the generated knowledge during the search can be further improved. The main goal of this work is to propose a framework that uses an inverse modeling approach coupled with an MOEA. The inverse surrogate modeling is used to sample solutions that are combined with the evolutionary search of the MOEA. The framework is evaluated in different scenarios, including multi-objective benchmark problems and real-world problems. The results show that the proposed framework had statistically better or equivalent results in 86% scenarios in the benchmark problems and 100% scenarios in the real-world problem.

Personalized Bayesian optimization for noisy problems

In many real-world applications of interest, several related optimization tasks can be encountered, where each task is associated with a specific context or personalized information. Moreover, the amount of available data for each task may be highly limited due to the expensive cost involved. Although Bayesian optimization (BO) has emerged as a promising paradigm for handling black-box optimization problems, addressing such a sequence of optimization tasks can be intractable due to the cold start issues in BO. The key challenge is to speed up the optimization by leveraging the transferable information, while taking the personalization into consideration. In this paper, optimization problems with personalized variables are formally defined at first. Subsequently, a personalized evolutionary Bayesian algorithm is proposed to consider the personalized information and the measurement noise. Specifically, a contextual Gaussian process is used to jointly learn a surrogate model in different contexts with regard to the varying personalized parameter, and an evolutionary algorithm is tailored for optimizing an acquisition function for handling the presence of personalized information. Finally, we demonstrate the effectiveness of the proposed algorithm by testing it on widely used single- and multi-objective benchmark problems with personalized variables.

A multiobjective evolutionary algorithm using multi-ecological environment selection strategy

For many-objective optimization problems (MaOPs), the conflict between convergence and diversity becomes more and more serious as the number of objectives increases. This paper proposes the evolutionary algorithm MeEA of multi-ecological environment selection strategy and uses this algorithm to solve MaOPs. Firstly, the objective space is divided into several different types of ecological environments. Secondly, the preference for convergence or diversity in the ecological environment is initially determined during environment selection and then the overall diversity maintenance of the population is ensured. Thirdly, the proposed algorithm is compared with five popular evolutionary algorithms on 44 multi-objective benchmark problems. Finally, it is applied to the optimization design of hydrodynamic lubrication radial sliding bearing of crane gearbox. Experimental results show that the performance of this algorithm is better than other algorithms in solving MaOPs.

Complementary surrogate-assisted differential evolution algorithm for expensive multi-objective problems under a limited computational budget

Different surrogate-assisted strategies can greatly influence the optimization efficiency of surrogate-assisted multi-objective evolutionary algorithms. By hybridizing two complementary surrogate-assisted strategies, this study proposed an efficient surrogate-assisted differential evolution algorithm to optimize expensive multi-objective problems under a limited computational budget. The two proposed surrogate-assisted strategies balance global and local search for multi-objective optimization. Specifically, one strategy is an improved surrogate-based multi-objective local search method that is based on maximin angle-distance sequential sampling. Compared with the previous local search method that is based on Euclidian distance-based sampling, the improved local search method is more efficient because it can mitigate the scale difference of different objectives. The other surrogate-assisted strategy is prescreening based on a diversity-enhanced expected improvement matrix infill criterion. The proposed infill criterion aims to improve the diversity of approximate Pareto optimal solutions by considering distribution of candidate individuals in the objective space in the infill function. Within a limited computational burden, the performance of the proposed algorithm is demonstrated on a large set of multi-objective benchmark problems, as well as a real-world airfoil design problem. Experimental results show that the proposed algorithm performs significantly better than some existing algorithms on most problems investigated in this study.

Local Fitness Landscape Exploration Based Genetic Algorithms

Genetic algorithms (GAs) have been used to evolve optimal/sub-optimal solutions of many problems. When using GAs for evolving solutions, often fitness evaluation is the most computationally expensive, and this discourages researchers from applying GAs for computationally challenging problems. This paper presents an approach for generating offspring based on a local fitness landscape exploration to increase the speed of the search for optimal/sub-optimal solutions and to evolve better fitness solutions. The proposed algorithm, “Fitness Landscape Exploration based Genetic Algorithm” (FLEX-GA) can be applied to single and multi-objective optimization problems. Experiments were conducted on several single and multi-objective benchmark problems with and without constraints. The performance of the FLEX-based algorithm on single-objective problems is compared with a canonical GA and other algorithms. For multi-objective benchmark problems, the comparison is made with NSGA-II, and other multi-objective optimization algorithms. Lastly, Pareto solutions are evolved on eight real-world multi-objective optimization problems, and a comparative performance is presented with NSGA-II. Experimental results show that using FLEX on most of the single and multi-objective problems, the speed of the search improves up to 50% and the quality of solutions also improves. These results provide sufficient evidence of the applicability of fitness landscape approximation-based algorithms for solving real-world optimization problems.

Plotting Impossible? Surveying Visualization Methods for Continuous Multi-Objective Benchmark Problems

Traditionally, visualizing benchmark problems is an integral task in the domain of evolutionary algorithms development. Researchers get inspired for new search heuristics by challenges observed in functional landscapes. Moreover, landscape characteristics, features, and even terminology to describe them are derived from visualizations. And most importantly, benchmark designers need visualizations for identifying diverse problems that potentially challenge different aspects of optimization algorithms. As easy as it is to visualize single-objective problems, until recently there were hardly any approaches for gaining similar insights for multi-objective problems. Also, there have been no seamlessly accessible tools to support such visualizations. This article presents a comprehensive overview of the available visualization techniques from literature, including two interactive techniques to visualize 3-D problems, as well as two novel techniques which are suitable to scale some visualization properties to even higher-dimensional spaces. All presented techniques are integrated into a single tool, the moPLOT-dashboard, which enables users to perform landscape analyses in an interactive manner. Finally, the value of the tool and the visualizations is demonstrated in a series of usage scenarios on well-known benchmark problems.

A robust performance evaluation approach for solution preservation in multiobjective optimization

Many real-world optimization problems have complex features, such as bias, multimodel, etc. Existing evolutionary algorithms mainly utilize solutions’ current performance to decide their survivals, which are not comprehensive enough to describe the evolving trend, and may misguide the evolve decision. In this paper, a novel robust performance evaluation approach for evolutionary multiobjective optimization algorithm is proposed. Here, the robustness refers to the performance fluctuation degree among several generations, which can be expressed by interval values in respect to the decision and objective spaces. Based on the robust performance evaluation, solutions can be selected and preserved considering their historical performance, and thus, the exploration strength in convergence potential areas can be maintained. Meanwhile, to construct an evolutionary algorithm that embeds robustness evaluation, a robust elite managerial method and a learning-based updating strategy are also designed. Experiments on multiobjective benchmark problems and a real-world optimization in a robotic manipulation system have proved the superiority of the proposed approach.

Multi-Objective Search Group Algorithm for engineering design problems

This study proposes a new multi-objective version of the Search Group Algorithm (SGA) called the Multi-Objective Search Group Algorithm (MOSGA). The MOSGA is the combination of the conventional SGA integrated with an elitist non-dominated sorting technique, enabling it to define Pareto optimal solutions via mutation, offspring generation, and selection. The Pareto archive with a selection mechanism is used to preserve and enhance the convergence and diversity of solutions. The MOSGA is validated on twenty-five prominent case studies, including nineteen unconstrained multi-objective benchmark problems, six constrained multi-objective benchmark problems, and five multi-objective engineering design problems to validate its capability and effectiveness. The statistical results are compared to the outcomes of other well-regarded algorithms using the same performance metrics. The comparative results show that MOGSA is robust and superior in handling a wide variety of multi-objective problems.

A novel self-adaptive multi-fidelity surrogate-assisted multi-objective evolutionary algorithm for simulation-based production optimization

Multi-objective production optimization (MOPO) which aims at finding an optimal well control scheme is regarded as a win-win strategy for improving economic and production gains. Generally, the MOPO processes involve computationally expensive numerical simulations and high-dimensional decision variables which limit their application to actual reservoir optimization problems. Surrogate-assisted multi-objective optimization method which uses a simple yet efficient approximation model shows huge potential in solving production optimization problems since it lessens the computational burden by decreasing the use of computationally expensive numerical simulations. In this paper, a novel self-adaptive multi-fidelity surrogate-assisted (SAMFS) multi-objective production optimization algorithm (SAMFS-MOPO) is proposed to reduce the computational burden and enhance the accuracy of the surrogate model. A similar method has been applied to well spacing optimization but the uniqueness of this method is that two fidelity samples are used to establish a multi-fidelity (MF) surrogate model, while i-updating and g-updating strategies are used to renew the MF surrogate model during the optimization process so as to improve its accuracy and reduce the computational burden. To the best of our knowledge, this is the first time a self-adaptive multi-fidelity surrogate model is used for production optimization. Furthermore, three classic multi-objective benchmark problems and two reservoirs with different complexities were applied to illustrate the effectiveness and accuracy of the proposed SAMFS-MOPO method. It was found that the SAMFS-MOPO method had superior performance in convergence, diversity, and efficiency than other conventional methods.

A Reference Point Selection and Direction Guidance-Based Algorithm for Large-Scale Multi-Objective Optimization

Designing efficient algorithms for the large-scale multi-objective problems (LSMOPs) is a very challenging problem currently. To tackle LSMOPs, we first design a new reference points selection strategy to enhance the diversity of algorithms and avoid the algorithm falling into local minima. The strategy selects not only a part of nondominated solutions with the largest crowding distance, but also a part of relatively uniformly distributed solutions as the reference points. In this way, much better diversity and convergence can be obtained. Second, we propose a direction-guided offspring generation strategy, where a type of potential directions is designed to generate the promising solutions which can balance the convergence and diversity of the obtained solutions and improve the effectiveness of the algorithm significantly. Based on the two proposed strategies, we propose a new effective algorithm for LSMOPs. Numerical experiments are executed on two widely used large-scale multi-objective benchmark problem sets with 200, 500 and 1000 decision variables and a comparison with five state-of-the-art algorithms is made. The experimental results show that our proposed algorithm is effective and can obtain significantly better solutions than the compared algorithms.

A new optimization algorithm to solve multi-objective problems

Simultaneous optimization of several competing objectives requires increasing the capability of optimization algorithms. This paper proposes the multi-objective moth swarm algorithm, for the first time, to solve various multi-objective problems. In the proposed algorithm, a new definition for pathfinder moths and moonlight was proposed to enhance the synchronization capability as well as to maintain a good spread of non-dominated solutions. In addition, the crowding-distance mechanism was employed to select the most efficient solutions within the population. This mechanism indicates the distribution of non-dominated solutions around a particular non-dominated solution. Accordingly, a set of non-dominated solutions obtained by the proposed multi-objective algorithm is kept in an archive to be used later for improving its exploratory capability. The capability of the proposed MOMSA was investigated by a set of multi-objective benchmark problems having 7 to 30 dimensions. The results were compared with three well-known meta-heuristics of multi-objective evolutionary algorithm based on decomposition (MOEA/D), Pareto envelope-based selection algorithm II (PESA-II), and multi-objective ant lion optimizer (MOALO). Four metrics of generational distance (GD), spacing (S), spread (Δ), and maximum spread (MS) were employed for comparison purposes. The qualitative and quantitative results indicated the superior performance and the higher capability of the proposed MOMSA algorithm over the other algorithms. The MOMSA algorithm with the average values of CPU time = 2771 s, GD = 0.138, S = 0.063, Δ = 1.053, and MS = 0.878 proved to be a robust and reliable model for multi-objective optimization.

Automatic preference based multi-objective evolutionary algorithm on vehicle fleet maintenance scheduling optimization

A preference based multi-objective evolutionary algorithm is proposed for generating solutions in an automatically detected knee point region. It is named Automatic Preference based DI-MOEA (AP-DI-MOEA) where DI-MOEA stands for Diversity-Indicator based Multi-Objective Evolutionary Algorithm). AP-DI-MOEA has two main characteristics: firstly, it generates the preference region automatically during the optimization; secondly, it concentrates the solution set in this preference region. Moreover, the real-world vehicle fleet maintenance scheduling optimization (VFMSO) problem is formulated, and a customized multi-objective evolutionary algorithm (MOEA) is proposed to optimize maintenance schedules of vehicle fleets based on the predicted failure distribution of the components of cars. Furthermore, the customized MOEA for VFMSO is combined with AP-DI-MOEA to find maintenance schedules in the automatically generated preference region. Experimental results on multi-objective benchmark problems and our three-objective real-world application problems show that the newly proposed algorithm can generate the preference region accurately and that it can obtain better solutions in the preference region. Especially, in many cases, under the same budget, the Pareto optimal solutions obtained by AP-DI-MOEA dominate solutions obtained by MOEAs that pursue the entire Pareto front.

Vertical distance-based clonal selection mechanism for the multiobjective immune algorithm

Traditional multiobjective immune algorithms (MOIAs) widely use the domination relationship and crowding distance metric to run the cloning operator, which places more attention on nondominated solutions with larger crowding distance values. This “convergence-first” cloning principle may hamper population diversity. To better maintain population diversity in most traditional MOIAs, we propose a novel vertical distance-based clonal selection mechanism for MOIAs, called VD-MOIA in this paper, which arranges the cloning numbers for each solution based on the vertical distance value between the solution and its corresponding weight vector. First, the decomposition approach is used to decompose the target MOP into a set of subproblems, which are then optimized cooperatively and simultaneously. Second, the vertical distance-based clonal selection mechanism is performed, which puts more emphasis on the solutions with small vertical distance values. A large number of clones is allocated to the solutions that show good performance in terms of diversity, as these solutions are closer to their corresponding weight vectors than solutions that result in poor diversity. Moreover, the aggregated values quantified by the Chebyshev function are used to ensure convergence, while differential evolution (DE) is employed to strengthen the exploration ability of the algorithm, which further improves the population diversity as well. To validate the effectiveness of our proposed algorithm, twenty-eight multiobjective benchmark problems are adopted with complicated Pareto-optimal sets and fronts. The experimental comparison results validate the superiority of our proposed VD-MOIA when compared with four competitive MOIAs and four state-of-the-art multiobjective evolutionary algorithms.

Multi-objective search group algorithm for thermo-economic optimization of flat-plate solar collector

This study aims to develop a multi-objective version of the search group algorithm (SGA) called the multi-objective search group algorithm (MOSGA) to help determine thermo-economic optimization of flat-plate solar collector (FPSC) systems. Search mechanisms of the SGA were modified to determine non-dominated solutions through mutation, generation, and selection stages. Authors also mined the Pareto archive with a selection mechanism to maintain and intensify convergence and distribution of solutions. The study tested the proposed MOSGA with well-known multi-objective benchmark problems. Results were compared with outcomes from conventional algorithms using the same performance metrics to validate the capability and performance of the MOSGA. Afterward, MOSGA was applied to find the best design parameters to simultaneously optimize thermal efficiency and the total annual cost of FPSC systems. Four case studies were conducted with four different working fluids (pure water, SiO2, Al2O3, and CuO nanofluids). Optimization results obtained by the MOSGA were analyzed and compared with solutions provided by other algorithms. The findings revealed relative improvement in thermal efficiency and reduced annual cost for all nanofluids compared to pure water. Thermal efficiency was improved by 2.2748%, 2.4298%, and 2.7948% for SiO2, Al2O3, and CuO case studies, respectively, compared to pure water. Meanwhile, TAC rates were increased by 2.4111%, 2.3403%, and 2.9133% for these case studies, respectively. Comparative results also demonstrated that MOGSA was robustly effective and superior in the selection of appropriate design parameters of FPSC systems.

Cognizant Multitasking in Multiobjective Multifactorial Evolution: MO-MFEA-II.

Humans have the ability to identify recurring patterns in diverse situations encountered over a lifetime, constantly understanding relationships between tasks and efficiently solving them through knowledge reuse. The capacity of artificial intelligence systems to mimic such cognitive behaviors for effective problem solving is deemed invaluable, particularly when tackling real-world problems where speed and accuracy are critical. Recently, the notion of evolutionary multitasking has been explored as a means of solving multiple optimization tasks simultaneously using a single population of evolving individuals. In the presence of similarities (or even partial overlaps) between high-quality solutions of related optimization problems, the resulting scope for intertask genetic transfer often leads to significant performance speedup-as the cost of re-exploring overlapping regions of the search space is reduced. While multitasking solvers have led to recent success stories, a known shortcoming of existing methods is their inability to adapt the extent of transfer in a principled manner. Thus, in the absence of any prior knowledge about the relationships between optimization functions, a threat of predominantly negative (harmful) transfer prevails. With this in mind, this article presents a realization of a cognizant evolutionary multitasking engine within the domain of multiobjective optimization. Our proposed algorithm learns intertask relationships based on overlaps in the probabilistic search distributions derived from data generated during the course of multitasking-and accordingly adapts the extent of genetic transfers online. The efficacy of the method is substantiated on multiobjective benchmark problems as well as a practical case study of knowledge transfers from low-fidelity optimization tasks to substantially reduce the cost of high-fidelity optimization.