Multimodal Optimization Problems Research Articles

Teaching–Learning-Based Optimization Algorithm with Stochastic Crossover Self-Learning and Blended Learning Model and Its Application

This paper presents a novel variant of the teaching–learning-based optimization algorithm, termed BLTLBO, which draws inspiration from the blended learning model, specifically designed to tackle high-dimensional multimodal complex optimization problems. Firstly, the perturbation conditions in the “teaching” and “learning” stages of the original TLBO algorithm are interpreted geometrically, based on which the search capability of the TLBO is enhanced by adjusting the range of values of random numbers. Second, a strategic restructuring has been ingeniously implemented, dividing the algorithm into three distinct phases: pre-course self-study, classroom blended learning, and post-course consolidation; this structural reorganization and the random crossover strategy in the self-learning phase effectively enhance the global optimization capability of TLBO. To evaluate its performance, the BLTLBO algorithm was tested alongside seven distinguished variants of the TLBO algorithm on thirteen multimodal functions from the CEC2014 suite. Furthermore, two excellent high-dimensional optimization algorithms were added to the comparison algorithm and tested in high-dimensional mode on five scalable multimodal functions from the CEC2008 suite. The empirical results illustrate the BLTLBO algorithm’s superior efficacy in handling high-dimensional multimodal challenges. Finally, a high-dimensional portfolio optimization problem was successfully addressed using the BLTLBO algorithm, thereby validating the practicality and effectiveness of the proposed method.

Differential evolution based on strategy adaptation and deep reinforcement learning for multimodal optimization problems

Multimodal optimization problems (MMOPs) include multiple optima, which are common in practical fields. However, the success of solving MMOPs requires the algorithm to have both exploration and exploitation performance. How to select a reasonable search strategy is a difficult problem facing MMOPs. In this research, a differential evolution based on strategy adaptation and deep reinforcement learning, termed SA-DQN-DE, is proposed to select mutation strategies reasonably and search the optima effectively, which mainly includes three aspects. First, strategy adaptation, which calculates selection probability based on the feedback of different mutation operators in previous evolution, and assigns mutation operations to each individual to provide guidance for the next stage of evolution is developed. Secondly, a new historical individual preservation method is designed to improve search efficiency. Thirdly, deep reinforcement learning is applied to select multiple local search operators to refine the accuracy of potential optimal solutions. The performance of SA-DQN-DE is tested on publicly acknowledged CEC2013 benchmark MMOP set. The experimental results demonstrate that the proposed SA-DQN-DE has competitive performance compared with some of its peer multimodal optimization algorithms.

A multi-role steered artificial bee colony algorithm with variable dimensionality perturbation for multimodal optimization problems

A multi-role steered artificial bee colony algorithm with variable dimensionality perturbation for multimodal optimization problems

Fraction order particle swarm optimization for parameter extraction of triple-diode photovoltaic models

This work proposes an application of Fractional Order Particle Swarm Optimization (FO-PSO), a meta-heuristic method for parameters estimation of photo-voltaic (PV) module as a non-linear, transcendental, multi-modal and implicit optimization problem. The uses single diode model (SDM), double diode model (DDM) and three-diode model (TDM) of PV modules with the constraint that only data-sheet information may be utilized. A fitness function based on the error amongst the computed values of current and voltage and the ones given in characteristic I-V curves of data-sheet, is minimized using the FO-PSO to get the required parameters. The comparative study between the estimated and data-sheet values provided by PV module manufacturers will determine the effectiveness of this research. The effectiveness of the FO-PSO is demonstrated by comparing the fitness values with that of other techniques. The FO-PSO technique makes a novel contribution to the PV power systems industry by making it possible to obtain a nearly realistic model of any commercial PV module. The effectiveness of the FO-PSO is determined by comparing the results for all the three models with the state of the art optimization techniques. The Root Mean Square error values calculated for the TDM is less than 10e–16, producing very consistent FO-PSO results. Therefore, FO-PSO is anticipated to be a competitive method for obtaining PV module specifications.

A similarity-detection-based evolutionary algorithm for large-scale multimodal multi-objective optimization

In recent years, there has been a surge in the development of evolutionary algorithms tailored for multimodal multi-objective optimization problems (MMOPs). These algorithms aim to find multiple equivalent Pareto optimal solution sets (PSs). However, little work has been done on MMOPs with large-scale decision variables, especially when the Pareto optimal solutions are sparse. These problems pose significant challenges due to the dimension curse, the unknown sparsity, and the unknown number of equivalent PSs. In this paper, we propose an evolutionary algorithm based on similarity detection called SD-MMEA to solve large-scale MMOPs with sparse Pareto-optimal solutions. Specifically, it employs a multi-population independent evolution to explore multiple PSs and distinguishes different PSs by double detection of the similarity between subpopulations. Simultaneously, develop online scoring mechanisms for decision variables to guide the subpopulations to explore in different directions. In addition, during the latter stage of evolution, the decision variables of individuals are further optimized by a double-layer grouping process. The proposed algorithm is compared with six state-of-the-art algorithms. Experimental results show that SD-MMEA has significant advantages in solving large-scale MMOPs with sparse solutions.

Growing Neural Gas Network-based surrogate-assisted Pareto set learning for multimodal multi-objective optimization

The key issue in handling multimodal multi-objective optimization problems (MMOPs) is to find multiple Pareto sets (PSs) corresponding to one Pareto front (PF). Therefore, learning the PSs is critical to facilitate solving MMOPs while unfortunately, current research only focuses on PF learning which is helpless in finding multiple PSs by the information of one PF. Moreover, since the PSs of an MMOP are usually non-functional, traditional approximative function model-based PF learning is inapplicable. Consequently, developing new PS learning techniques is desired. Inspired by data-driven evolutionary algorithms, data can be used to train surrogate models to assist the algorithm. This article proposes an online data-driven PSs learning technique that aims to learn the topologies of PSs through a surrogate model to facilitate the search for PSs. Specifically, the Growing Neural Gas network is trained using non-dominated solutions to learn the topologies of PSs during the evolutionary process. Then, the nodes of the network are used to generate new solutions and adopted as reference points for environmental selection. A new algorithm is developed based on the PS learning technique for MMOPs. Experimental studies on three benchmark test suites and two different real-world applications demonstrate the superiority of our method over six state-of-the-art algorithms dedicated to MMOPs. The PSs learning technique can obtain the topologies of PSs and facilitate the search for them.

ЕФЕКТИВНЕ РОЗВ’ЯЗУВАННЯ МУЛЬТИМОДАЛЬНИХ ОПТИМІЗАЦІЙНИХ ЗАДАЧ

In this work, we consider multimodal optimization problems. Such problems contain many local extrema. We can say that most practical problems are multimodal. In particular, discrete optimization problems with Boolean and integer variables can easily be transformed into multimodal problems with continuous variables. Multimodal optimization problems can be of small or large dimensions in which the number of variables to be determined is hundreds or thousands of variables. Libraries of test and applied problems have been created to test the effectiveness of new global optimization methods. The author suggests separating problems with unknown optimal solutions in these tests. Then the best method for solving multimodal optimization problems will be the one that will allow obtaining better solutions in most such problems. Currently, this criterion is satisfied only by the exact quadratic regularization method developed by the author. This is confirmed by significant computational experiments on existing tests and applied multimodal optimization problems.

Wireless sensor networks-based adaptive differential evolution for multimodal optimization problems

In wireless sensor networks (WSN), we often detect the monitoring areas among different sensors so that the sensors can be switched on and off adaptively to save energy and extend their lifetime. Inspired by the principle of WSN, a WSN-based adaptive differential evolution (WSNADE) algorithm is proposed in this paper, together with a WSN-based adaptive niching technique (WANT) and two novel strategies called protection-based dual-scale mutation (PDM) strategy and multi-level reset (MLR) strategy, for solving multimodal optimization problems (MMOPs). In WANT, each individual is considered as a sensor with its monitoring area. If the monitoring areas of two individuals intersect, which means these two individuals monitor the similar area and should be partitioned into the same niche. In this way, WANT can adaptively form a niche for each individual, avoiding the sensitivity of niching parameters. Based on WANT, the PDM strategy is designed to select the appropriate mutation strategy for each individual. Besides, to save fitness evaluations (FEs) for exploring more promising areas, the MLR strategy is developed to store the promising individuals and reset the stagnant individuals. The experimental results on 20 multimodal benchmark test functions in CEC2015 multimodal competition show that the proposed WSNADE algorithm generally performs better than or at least comparable with other state-of-the-art multimodal algorithms, including the winner of the CEC2015 competition. Finally, WSNADE is applied to a real-world multimodal application in multiple competitive facilities location design (MCFLD) problem to illustrate its practical applicability.

Evolutionary Multimodal Multiobjective Optimization for Traveling Salesman Problems

Multimodal multiobjective optimization problems (MMOPs) are commonly seen in real-world applications. Many evolutionary algorithms have been proposed to solve continuous MMOPs. However, little effort has been made to solve combinatorial (or discrete) MMOPs. Searching for equivalent Pareto optimal solutions in the discrete decision space is challenging. Moreover, the true Pareto optimal solutions of a combinatorial MMOP are usually difficult to know, which has limited the development of its optimizer. In this paper, we first propose a test problem generator for multimodal multiobjective traveling salesman problems (MMTSPs). It can readily generate MMTSPs with known Pareto optimal solutions. Then we propose a novel evolutionary algorithm to solve MMTSPs. In our proposed algorithm, we develop two new edge assembly crossover operators, which are specialized in searching for superior solutions to MMTSPs. Moreover, the proposed algorithm uses a new environmental selection operator to maintain a good balance between the objective space diversity and decision space diversity. We compare our algorithm with five state-of-the-art designs. Experimental results convincingly show that our algorithm is powerful in solving MMTSPs.

Enhancing the performance of hybrid wave-wind energy systems through a fast and adaptive chaotic multi-objective swarm optimisation method

Hybrid offshore renewable energy platforms have been proposed to optimise power production and reduce the levelised cost of energy by integrating or co-locating several renewable technologies. One example is a hybrid wave-wind energy system that combines offshore wind turbines with wave energy converters (WECs) on a single floating foundation. The design of such systems involves multiple parameters and performance measures, making it a complex, multi-modal, and expensive optimisation problem. This paper proposes a novel, robust and effective multi-objective swarm optimisation method (DMOGWA) to provide a design solution that best compromises between maximising WEC power output and minimising the effect on wind turbine nacelle acceleration. The proposed method uses a chaotic adaptive search strategy with a dynamic archive of non-dominated solutions based on diversity to speed up the convergence rate and enhance the Pareto front quality. Furthermore, a modified exploitation technique (Discretisation Strategy) is proposed to handle the large damping and spring coefficient of the Power Take-off (PTO) search space. To evaluate the efficiency of the proposed method, we compare the DMOGWA with four well-known multi-objective swarm intelligence methods (MOPSO, MALO, MODA, and MOGWA) and four popular evolutionary multi-objective algorithms (NSGA-II, MOEA/D, SPEA-II, and PESA-II) based on four potential deployment sites on the South Coast of Australia. The optimisation results demonstrate the dominance of the DMOGWA compared with the other eight methods in terms of convergence speed and quality of solutions proposed. Furthermore, adjusting the hybrid wave-wind model’s parameters (WEC design and PTO parameters) using the proposed method (DMOGWA) leads to a considerably improved power output (average proximate boost of 138.5%) and a notable decline in wind turbine nacelle acceleration (41%) throughout the entire operational spectrum compared with the other methods. This improvement could lead to millions of dollars in additional income per year over the lifespan of hybrid offshore renewable energy platforms.

IMDE-UGAN: Improved Memetic Direction Exploitation Optimized U-Net Generative Adversarial Network for Classification of Diabetic Retinopathy

The most important human organ is the eye which is essential for the processing and absorbing of external visual information. Diabetic retinopathy (DR) is the primary factor contributing to the global increase in blindness. To maintain eye health, people have been paying lots of attention to the problem. However, there are still lots of people who have eye problems all around the world. Thus, we proposed the Improved Memetic Direction Exploitation Optimized U-Net Generative Adversarial Network (IMDE-optimized U-Net GAN) model for DR grading. Also, for Multimodal Optimization Problems (MMOPs), a Niching Competition-based Memetic Direction Exploitation (NC-MDE) method is proposed with adaptive local search operations. Additionally, SPatially-ADaptivE (SPADE)-based conditional GANs are used to segment DR. The U-Net GAN model is used to categorize DR images. In the discriminator design, the standard classification network is modified to create the decoder-encoder network known as U-Net. The complete network error of the U-Net GAN architecture is modeled using the IMDE archives. The various performance metrics are used in the method for testing the proposed including accuracy, specificity, sensitivity, and ROC curve. As a result, the proposed method achieves 98.89% of accuracy, 98.54% for sensitivity, and specificity, it achieves 98.67%.

A Bi-Objective Evolutionary Algorithm for Multimodal Multiobjective Optimization

Multimodal multi-objective optimization problems (MMOPs) possess multiple Pareto optimal sets corresponding to the identical Pareto optimal front (PF). To handle MMOPs, we propose a bi-objective evolutionary algorithm (BOEA), which transforms an MMOP into a bi-objective optimization problem. This problem is constructed by the penalty boundary intersection technique and a diversity indicator to obtain multiple Pareto optimal sets. The first objective reflects the population convergence and factors in the population diversity in the objective space, while the other objective concentrates more on the population diversity in the decision space. Furthermore, an environmental selection strategy is designed to choose the offspring solutions, which consists of non-dominated sorting based on the transformed optimization problem and hierarchical clustering for selecting promising solutions. Experiments on 34 MMOPs demonstrate that BOEA performs better than selected state-of-the-art representatives, including 22 MMOPs from CEC2019 and 12 imbalanced MMOPs. In addition, the effectiveness of BOEA is further validated by six feature selection problemsin real-world applications.

A Surrogate-Assisted Evolutionary Algorithm for Seeking Multiple Solutions of Expensive Multimodal Optimization Problems

A Surrogate-Assisted Evolutionary Algorithm for Seeking Multiple Solutions of Expensive Multimodal Optimization Problems

Multimodal network path optimization based on a two‐stage algorithm in the perspective of sustainable transport development

AbstractThe environmental issues brought on by carbon emissions from transport have risen to prominence in recent years. More and more academics are using the multi‐objective path optimization method to solve the multimodal optimization problem from the standpoint of sustainable development in order to address the environmental issues brought on by the transport process. The research proposes a two‐stage method to handle multi‐objective optimization convergence and simplify multimodal transport path optimization. In the first stage, a fuzzy C clustering model is established, and based on the clustering results, the multimodal transport network nodes are identified. In the second stage, a multimodal transport multi‐objective path optimization model is established, and the optimal path is solved using a genetic algorithm. The research method was applied in the Bohai Rim region. Results indicated that the fuzzy C‐clustering method and the genetic method were able to select the optimal node city, thus solving the actual site selection problem of multimodal transportation networks. Using the FCM model, the 86 city nodes were categorized into four types, leading to the establishment of the most proficient multimodal transportation network in the Bohai Rim region. Using a genetic algorithm for optimization, a stable state is reached after 25 iterations. In the validation experiment on path optimization, the cost was reduced by 47.12% compared to the minimum single objective time, and transportation carbon emissions saw a reduction of 28.23%. Similarly, compared to the lowest target for transportation carbon emissions, the cost was reduced by 39.48% and the time was reduced by 38.12%. Compared to the lowest target for transportation carbon emissions, the time was reduced by 32.02% and the carbon emissions were reduced by 19.23%. Notably, the transportation multi‐objective path optimization model showed significant improvement compared to the single‐target model. The research method has been proven to be superior, and can offer the most optimal transportation route guidance for participants in multimodal transportation. Furthermore, it can effectively tackle the issue of node selection convergence and multi‐objective optimization, while also serving as a valuable source of data to support the theoretical advancement of multimodal transportation network path optimization.

Two-stage evolutionary algorithm with fuzzy preference indicator for multimodal multi-objective optimization

Multimodal multi-objective optimization problems (MMOPs) are multi-objective optimization problems (MOPs) involving multiple equivalent global or local Pareto optimal solution sets (PSs). For decision-makers, not only the global optimal solution sets need to be found, but also the value of local optimal solution sets cannot be ignored. However, most multimodal multi-objective evolutionary algorithms (MMOEAs) tend to select solutions with better convergence, and it is difficult to obtain the global PSs and local PSs at the same time. Therefore, we propose a fuzzy preference indicator-based two-stage evolutionary algorithm (FPITSEA) in this paper. To evaluate more comprehensively the potential of each solution in the population for locating the global and local PS during the evolutionary process, a fuzzy preference indicator is designed in FPITSEA. The fuzzy preference indicator is used to guide the evolution of the population in the first stage to find the global and local Pareto optimal regions. Subsequently, an independent evolution strategy is implemented in the second stage to distinguish different PSs as accurately as possible while also ensuring the convergence quality of the solution set. In addition, an improved distance-based subset selection method is proposed, aiming to simultaneously improve the distribution of the solution set in the decision space and objective space. Experimental results on several test sets of MMOPs show the advantages of FPITSEA over several state-of-the-art MMOEAs.

Multimodal transportation routing optimization based on multi-objective Q-learning under time uncertainty

Multimodal transportation is a modern way of cargo transportation. With the increasing demand for cargo transportation, higher requirements are being placed on multimodal transportation multi-objective routing optimization. In multimodal transportation multi-objective routing optimization, in response to the limitations of classical algorithms in solving large-scale problems with multiple nodes and modes of transport, the limitations of directed transportation networks in the application, and the uncertainty of transport time, this paper proposes an optimization framework based on multi-objective weighted sum Q-learning, combined with the proposed undirected multiple-node network, and characterizes the uncertainty of time with a positively skewed distribution. The undirected multiple-node transportation network can better simulate cargo transportation and characterize transfer information, facilitate the modification of origin and destination, and avoid suboptimal solutions due to the manual setting of wrong route directions. The network is combined with weighted sum Q-learning to solve multimodal transportation multi-objective routing optimization problems faster and better. When modeling the uncertainty of transport time, a positively skewed distribution is used. The three objectives of transport cost, carbon emission cost, and transport time were studied and compared with PSO, GA, AFO, NSGA-II, and MOPSO. The experimental results show that compared with PSO, GA, and AFO using a directed transportation network, the proposed method has a significant improvement in optimization results and running time, and the running time is shortened by 26 times. The proposed method can better solve the boundary of the Pareto front and dominate the partial solutions of NSGA-II and MOPSO. The effect of time uncertainty on the performance of the algorithm is more significant in transport orders with high time weight. With the increase in uncertainty, the reliability of the route decreases. The effectiveness of the proposed method is verified.

A particle swarm optimization algorithm based on modified crowding distance for multimodal multi-objective problems

Multimodal multi-objective optimization problems (MMOPs) are commonly encountered in practice. The difficulty in solving MMOPs is obtaining all of the Pareto optimal sets without degrading the performance of the Pareto optimal front. To address this challenge, this study proposes a particle swarm optimization algorithm based on modified crowding distance (MOPSO_MCD). In MOPSO_MCD, a modified method for calculating the crowding distance (MCD) is devised, which allows for a more comprehensive assessment of the crowding relationship between individuals in the decision space and the objective space. Moreover, a cosine similarity-based elite selection mechanism is designed to identify the neighborhood optimal individuals of the individuals in the population and improve the decision space diversity. Additionally, an offspring competition mechanism is proposed to keep the population from trapping in the local optimum and enhance the global search ability of the MOPSO_MCD algorithm. Experimental results and statistical analysis show that MOPSO_MCD performs better than the other comparison algorithms on sixteen test functions and a map-based practical problem.

A reinforcement learning-based neighborhood search operator for multi-modal optimization and its applications

In this paper, a reinforcement learning-based neighborhood search operator (RLNS) is proposed for multi-modal optimization problems where the main novelties lie in the reinforcement learning-based neighborhood range selection strategy, the neighborhood subpopulation generation strategy and the local vector encirclement model. The reinforcement learning-based neighborhood range selection strategy is proposed to dynamically adjust the subpopulation size to address the issue of too many parameters to be adjusted in the multi-modal optimization algorithm based on the niching methods, while the neighborhood subpopulation generation strategy and the local vector encirclement model are designed with the hope of enhancing the individual’s ability to local exploitation to obtain more accurate solutions. To verify the effectiveness of the proposed RLNS, SSA-RLNS, PSO-RLNS and EO-RLNS are proposed by integrating the proposed RLNS with the existing sparrow search algorithm, particle swarm optimization and equilibrium optimizer. The performances of the proposed SSA-RLNS, PSO-RLNS, EO-RLNS and existing multi-modal optimization algorithms are tested in CEC2015 multi-niche benchmark functions. The experimental results show that the SSA-RLNS, PSO-RLNS and EO-RLNS could locate multiple global optimal solutions with satisfactory accuracy, which illustrate that the proposed RLNS could be successfully used to deal with multi-modal optimization problems by integrating with common population-based optimization algorithms. Finally, the SSA-RLNS is successfully applied in the inverse kinematics of robot manipulator.

Hybrid beluga whale optimization algorithm with multi-strategy for functions and engineering optimization problems

Beluga Whale Optimization (BWO) is a new metaheuristic algorithm that simulates the social behaviors of beluga whales swimming, foraging, and whale falling. Compared with other optimization algorithms, BWO shows certain advantages in solving unimodal and multimodal optimization problems. However, the convergence speed and optimization performance of BWO still have some performance deficiencies when solving complex multidimensional problems. Therefore, this paper proposes a hybrid BWO method called HBWO combining Quasi-oppositional based learning (QOBL), adaptive and spiral predation strategy, and Nelder-Mead simplex search method (NM). Firstly, in the initialization phase, the QOBL strategy is introduced. This strategy reconstructs the initial spatial position of the population by pairwise comparisons to obtain a more prosperous and higher quality initial population. Subsequently, an adaptive and spiral predation strategy is designed in the exploration and exploitation phases. The strategy first learns the optimal individual positions in some dimensions through adaptive learning to avoid the loss of local optimality. At the same time, a spiral movement method motivated by a cosine factor is introduced to maintain some balance between exploration and exploitation. Finally, the NM simplex search method is added. It corrects individual positions through multiple scaling methods to improve the optimal search speed more accurately and efficiently. The performance of HBWO is verified utilizing the CEC2017 and CEC2019 test functions. Meanwhile, the superiority of HBWO is verified by utilizing six engineering design examples. The experimental results show that HBWO has higher feasibility and effectiveness in solving practical problems than BWO and other optimization methods.

A multi-modal multi-objective evolutionary algorithm based on scaled niche distance

Multi-modal multi-objective optimization problems (MMOPs) refer to several solutions in the decision space that share the same or similar objective value. Balancing the diversity of the objective space and decision space while maintaining the convergence of the population is a challenging and important problem. To address this issue, a novel multi-modal multi-objective evolutionary algorithm (MMEA) named MMEA-SND is proposed in this study. In the MMEA-SND, to locate Pareto-optimal solutions, and improve the diversity of solutions in the decision space, a diversity fitness is designed by the niche method to calculate the fitness of solutions in the diversity archive. In order to balance the diversity of solutions in the objective space and decision space, a scaled niche distance (SND) method is proposed in environmental selection. In this context, SND are utilized to measure the distances between each solution in the objective space and decision space. Furthermore, a parameter is implemented to avoid disregarding locally optimal solutions. To verify the performance of MMEA-SND, six state-of-the-art MMEAs are adopted to make a comparison on 42 benchmark problems. The experimental results show that the proposed MMEA-SND achieves a competitive performance in solving MMOPs.