Multiobjective Differential Evolution for Higher-Dimensional Multimodal Multiobjective Optimization

In multimodal multiobjective optimization problems(MMOPs),there are several Pareto optimal solutions corre-sponding to the identical objective vector.This paper proposes a new differential evolution algorithm to solve MMOPs with higher-dimensional decision variables.Due to the increase in the dimensions of decision variables in real-world MMOPs,it is diffi-cult for current multimodal multiobjective optimization evolu-tionary algorithms(MMOEAs)to find multiple Pareto optimal solutions.The proposed algorithm adopts a dual-population framework and an improved environmental selection method.It utilizes a convergence archive to help the first population improve the quality of solutions.The improved environmental selection method enables the other population to search the remaining decision space and reserve more Pareto optimal solutions through the information of the first population.The combination of these two strategies helps to effectively balance and enhance conver-gence and diversity performance.In addition,to study the per-formance of the proposed algorithm,a novel set of multimodal multiobjective optimization test functions with extensible decision variables is designed.The proposed MMOEA is certified to be effective through comparison with six state-of-the-art MMOEAs on the test functions.

An Immune-Inspired Approach with Interval Allocation in Solving Multimodal Multi-Objective Optimization Problems with Local Pareto Sets

In practical engineering,multi-objective optimization often encounters situations where multiple Pareto sets(PS)in the decision space correspond to the same Pareto front(PF)in the objective space,known as Multi-Modal Multi-Objective Optimization Problems(MMOP).Locating multiple equivalent global PSs poses a significant challenge in real-world applications,especially considering the existence of local PSs.Effectively identifying and locating both global and local PSs is a major challenge.To tackle this issue,we introduce an immune-inspired reproduction strategy designed to produce more offspring in less crowded,promising regions and regulate the number of offspring in areas that have been thoroughly explored.This approach achieves a balanced trade-off between exploration and exploitation.Furthermore,we present an interval allocation strategy that adaptively assigns fitness levels to each antibody.This strategy ensures a broader survival margin for solutions in their initial stages and progressively amplifies the differences in individual fitness values as the population matures,thus fostering better population convergence.Additionally,we incorporate a multi-population mechanism that precisely manages each subpopulation through the interval allocation strategy,ensuring the preservation of both global and local PSs.Experimental results on 21 test problems,encompassing both global and local PSs,are compared with eight state-of-the-art multimodal multi-objective optimization algorithms.The results demonstrate the effectiveness of our proposed algorithm in simultaneously identifying global Pareto sets and locally high-quality PSs.

Horizontal-to-vertical spectral ratio inversion method based on multimodal forest optimization algorithm

The exploration of urban underground spaces is of great significance to urban planning,geological disaster prevention,resource exploration and environmental monitoring.However,due to the existing of severe interferences,conventional seismic methods cannot adapt to the complex urban environment well.Since adopting the single-node data acquisition method and taking the seismic ambient noise as the signal,the microtremor horizontal-to-vertical spectral ratio(HVSR)method can effectively avoid the strong interference problems caused by the complex urban environment,which could obtain information such as S-wave velocity and thickness of underground formations by fitting the microtremor HVSR curve.Nevertheless,HVSR curve inversion is a multi-parameter curve fitting process.And conventional inversion methods can easily converge to the local minimum,which will directly affect the reliability of the inversion results.Thus,the authors propose a HVSR inversion method based on the multimodal forest optimization algorithm,which uses the efficient clustering technique and locates the global optimum quickly.Tests on synthetic data show that the inversion results of the proposed method are consistent with the forward model.Both the adaption and stability to the abnormal layer velocity model are demonstrated.The results of the real field data are also verified by the drilling information.

Multimodal Fuzzy Downstream Petroleum Supply Chain:A Novel Pentagonal Fuzzy Optimization

The petroleum industry has a complex,inflexible and challenging supply chain(SC)that impacts both the national economy as well as people’s daily lives with a range of services,including transportation,heating,electricity,lubricants,as well as chemicals and petrochemicals.In the petroleum industry,supply chain management presents several challenges,especially in the logistics sector,that are not found in other industries.In addition,logistical challenges contribute significantly to the cost of oil.Uncertainty regarding customer demand and supply significantly affects SC networks.Hence,SC flexibility can be maintained by addressing uncertainty.On the other hand,in the real world,decision-making challenges are often ambiguous or vague.In some cases,measurements are incorrect owing to measurement errors,instrument faults,etc.,which lead to a pentagonal fuzzy number(PFN)which is the extension of a fuzzy number.Therefore,it is necessary to develop quantitative models to optimize logistics operations and supply chain networks.This study proposed a linear programming model under an uncertain environment.The model minimizes the cost along the refineries,depots,multimode transport and demand nodes.Further developed pentagonal fuzzy optimization,an alternative approach is developed to solve the downstream supply chain using themixed-integer linear programming(MILP)model to obtain a feasible solution to the fuzzy transportation cost problem.In this model,the coefficient of the transportation costs and parameters is assumed to be a pentagonal fuzzy number.Furthermore,defuzzification is performed using an accuracy function.To validate the model and technique and feasibility solution,an illustrative example of the oil and gas SC is considered,providing improved results compared with existing techniques and demonstrating its ability to benefit petroleum companies is the objective of this study.

Coevolutionary Framework for Generalized Multimodal Multi-Objective Optimization

Most multimodal multi-objective evolutionary algorithms(MMEAs)aim to find all global Pareto optimal sets(PSs)for a multimodal multi-objective optimization problem(MMOP).However,in real-world problems,decision makers(DMs)may be also interested in local PSs.Also,searching for both global and local PSs is more general in view of dealing with MMOPs,which can be seen as generalized MMOPs.Moreover,most state-of-theart MMEAs exhibit poor convergence on high-dimension MMOPs and are unable to deal with constrained MMOPs.To address the above issues,we present a novel multimodal multiobjective coevolutionary algorithm(Co MMEA)to better produce both global and local PSs,and simultaneously,to improve the convergence performance in dealing with high-dimension MMOPs.Specifically,the Co MMEA introduces two archives to the search process,and coevolves them simultaneously through effective knowledge transfer.The convergence archive assists the Co MMEA to quickly approach the Pareto optimal front.The knowledge of the converged solutions is then transferred to the diversity archive which utilizes the local convergence indicator and the-dominance-based method to obtain global and local PSs effectively.Experimental results show that Co MMEA is competitive compared to seven state-of-the-art MMEAs on fifty-four complex MMOPs.

A quantum inspired genetic algorithm for multimodal optimization of wind disturbance alleviation flight control system

This paper develops a Quantum-inspired Genetic Algorithm(QGA) to find the sets of optimal parameters for the wind disturbance alleviation Flight Control System(FCS). To search the problem domain more evenly and uniformly, the lattice rule based stratification method is used to create new chromosomes. The chromosomes are coded and updated according to quantuminspired strategies. A niching method is used to ensure every chromosome can converge to its corresponding local minimum in the optimization process. A parallel archive system is adopted to monitor the chromosomes on-line and save all potential feasible solutions in the optimization process. An adaptive search strategy is used to gradually adjust the search domain of each niche to finally approach the local minima. The solutions found by the QGA are compared with some other Multimodal Optimization(MO) algorithms and are tested on the FCS of the Boeing 747 to demonstrate the effectiveness of the proposed algorithm.

Zoning Search With Adaptive Resource Allocating Method for Balanced and Imbalanced Multimodal Multi-Objective Optimization

Maintaining population diversity is an important task in the multimodal multi-objective optimization.Although the zoning search(ZS)can improve the diversity in the decision space,assigning the same computational costs to each search subspace may be wasteful when computational resources are limited,especially on imbalanced problems.To alleviate the above-mentioned issue,a zoning search with adaptive resource allocating(ZS-ARA)method is proposed in the current study.In the proposed ZS-ARA,the entire search space is divided into many subspaces to preserve the diversity in the decision space and to reduce the problem complexity.Moreover,the computational resources can be automatically allocated among all the subspaces.The ZS-ARA is compared with seven algorithms on two different types of multimodal multi-objective problems(MMOPs),namely,balanced and imbalanced MMOPs.The results indicate that,similarly to the ZS,the ZS-ARA achieves high performance with the balanced MMOPs.Also,it can greatly assist a“regular”algorithm in improving its performance on the imbalanced MMOPs,and is capable of allocating the limited computational resources dynamically.

Exploring a Promising Region and Enhancing Decision Space Diversity for Multimodal Multi-Objective Optimization

During the past decade,research efforts have been gradually directed to the widely existing yet less noticed multimodal multi-objective optimization problems(MMOPs)in the multi-objective optimization community.Recently,researchers have begun to investigate enhancing the decision space diversity and preserving valuable dominated solutions to overcome the shortage caused by a preference for objective space convergence.However,many existing methods still have limitations,such as giving unduly high priorities to convergence and insufficient ability to enhance decision space diversity.To overcome these shortcomings,this article aims to explore a promising region(PR)and enhance the decision space diversity for handling MMOPs.Unlike traditional methods,we propose the use of non-dominated solutions to determine a limited region in the PR in the decision space,where the Pareto sets(PSs)are included,and explore this region to assist in solving MMOPs.Furthermore,we develop a novel neighbor distance measure that is more suitable for the complex geometry of PSs in the decision space than the crowding distance.Based on the above methods,we propose a novel dual-population-based coevolutionary algorithm.Experimental studies on three benchmark test suites demonstrates that our proposed methods can achieve promising performance and versatility on different MMOPs.The effectiveness of the proposed neighbor distance has also been justified through comparisons with crowding distance methods.

Rotation forest based on multimodal genetic algorithm

In machine learning,randomness is a crucial factor in the success of ensemble learning,and it can be injected into tree-based ensembles by rotating the feature space.However,it is a common practice to rotate the feature space randomly.Thus,a large number of trees are required to ensure the performance of the ensemble model.This random rotation method is theoretically feasible,but it requires massive computing resources,potentially restricting its applications.A multimodal genetic algorithm based rotation forest(MGARF)algorithm is proposed in this paper to solve this problem.It is a tree-based ensemble learning algorithm for classification,taking advantage of the characteristic of trees to inject randomness by feature rotation.However,this algorithm attempts to select a subset of more diverse and accurate base learners using the multimodal optimization method.The classification accuracy of the proposed MGARF algorithm was evaluated by comparing it with the original random forest and random rotation ensemble methods on 23 UCI classification datasets.Experimental results show that the MGARF method outperforms the other methods,and the number of base learners in MGARF models is much fewer.

Hybrid Operator and Strengthened Diversity Improving for Multimodal Multi-Objective Optimization

Multimodal multi-objective optimization problems(MMOPs)contain multiple equivalent Pareto subsets(PSs)corresponding to a single Pareto front(PF),resulting in difficulty in maintaining promising diversities in both objective and decision spaces to find these PSs.Widely used to solve MMOPs,evolutionary algorithms mainly consist of evolutionary operators that generate new solutions and fitness evaluations of the solutions.To enhance performance in solving MMOPs,this paper proposes a multimodal multi-objective optimization evolutionary algorithm based on a hybrid operator and strengthened diversity improving.Specifically,a hybrid operator mechanism is devised to ensure the exploration of the decision space in the early stage and approximation to the optima in the latter stage.Moreover,an elitist-assisted differential evolution mechanism is designed for the early exploration stage.In addition,a new fitness function is proposed and used in environmental and mating selections to simultaneously evaluate diversities for PF and PSs.Experimental studies on 11 widely used benchmark instances from a test suite verify the superiority or at least competitiveness of the proposed methods compared to five state-of-the-art algorithms tailored for MMOPs.

A Novel Multiobjective Fireworks Algorithm and Its Applications to Imbalanced Distance Minimization Problems

Recently,multimodal multiobjective optimization problems(MMOPs)have received increasing attention.Their goal is to find a Pareto front and as many equivalent Pareto optimal solutions as possible.Although some evolutionary algorithms for them have been proposed,they mainly focus on the convergence rate in the decision space while ignoring solutions diversity.In this paper,we propose a new multiobjective fireworks algorithm for them,which is able to balance exploitation and exploration in the decision space.We first extend a latest single-objective fireworks algorithm to handle MMOPs.Then we make improvements by incorporating an adaptive strategy and special archive guidance into it,where special archives are established for each firework,and two strategies(i.e.,explosion and random strategies)are adaptively selected to update the positions of sparks generated by fireworks with the guidance of special archives.Finally,we compare the proposed algorithm with eight state-of-the-art multimodal multiobjective algorithms on all 22 MMOPs from CEC2019 and several imbalanced distance minimization problems.Experimental results show that the proposed algorithm is superior to compared algorithms in solving them.Also,its runtime is less than its peers'.

A Dynamic Resource Allocation Strategy with Reinforcement Learning for Multimodal Multi-objective Optimization

Many isolation approaches, such as zoning search, have been proposed to preserve the diversity in the decision space of multimodal multi-objective optimization(MMO). However, these approaches allocate the same computing resources for subspaces with different difficulties and evolution states. In order to solve this issue, this paper proposes a dynamic resource allocation strategy(DRAS)with reinforcement learning for multimodal multi-objective optimization problems(MMOPs). In DRAS, relative contribution and improvement are utilized to define the aptitude of subspaces, which can capture the potentials of subspaces accurately. Moreover, the reinforcement learning method is used to dynamically allocate computing resources for each subspace. In addition, the proposed DRAS is applied to zoning searches. Experimental results demonstrate that DRAS can effectively assist zoning search in finding more and better distributed equivalent Pareto optimal solutions in the decision space.