True Pareto Front Research Articles

Constrained large-scale multiobjective optimization based on a competitive and cooperative swarm optimizer

Many engineering application problems can be modeled as constrained multiobjective optimization problems (CMOPs), which have attracted much attention. In solving CMOPs, existing algorithms encounter difficulties in balancing conflicting objectives and constraints. Worse still, the performance of the algorithms deteriorates drastically when the size of the decision variables scales up. To address these issues, this study proposes a competitive and cooperative swarm optimizer for large-scale CMOPs. To balance conflict objectives and constraints, a bidirectional search mechanism based on competitive and cooperative swarms is designed. It involves two swarms, approximating the true Pareto front from two directions. To enhance the search efficiency in large-scale space, we propose a fast-converging competitive swarm optimizer. Unlike existing competitive swarm optimizers, the proposed optimizer updates the velocity and position of all particles at each iteration. Additionally, to reduce the search range of the decision space, a fuzzy decision variables operator is used. Comparison experiments have been performed on test instances with 100–1000 decision variables. Experiments demonstrate the superior performance of the proposed algorithm over five peer algorithms.

An Improved MOEA/D with an Auction-Based Matching Mechanism

Multi-objective optimization problems (MOPs) constitute a vital component in the field of mathematical optimization and operations research. The multi-objective evolutionary algorithm based on decomposition (MOEA/D) decomposes a MOP into a set of single-objective subproblems and approximates the true Pareto front (PF) by optimizing these subproblems in a collaborative manner. However, most existing MOEA/Ds maintain population diversity by limiting the replacement region or scale, which come at the cost of decreasing convergence. To better balance convergence and diversity, we introduce auction theory into algorithm design and propose an auction-based matching (ABM) mechanism to coordinate the replacement procedure in MOEA/D. In the ABM mechanism, each subproblem can be associated with its preferred individual in a competitive manner by simulating the auction process in economic activities. The integration of ABM into MOEA/D forms the proposed MOEA/D-ABM. Furthermore, to make the appropriate distribution of weight vectors, a modified adjustment strategy is utilized to adaptively adjust the weight vectors during the evolution process, where the trigger timing is determined by the convergence activity of the population. Finally, MOEA/D-ABM is compared with six state-of-the-art multi-objective evolutionary algorithms (MOEAs) on some benchmark problems with two to ten objectives. The experimental results show the competitiveness of MOEA/D-ABM in the performance of diversity and convergence. They also demonstrate that the use of the ABM mechanism can greatly improve the convergence rate of the algorithm.

Adaptive machine learning surrogate based multiobjective optimization for scavenging residual saltwater behind subsurface dams

High-fidelity physically based groundwater flow and solute transport models have been limited for seawater intrusion remediation design because of computationally intensive evolutionary algorithms. Data-driven machine learning approaches are promising to substitute expensive-cost groundwater numerical models within optimization due to computing efficiency. However, machine learning surrogates may accumulate error of forecasting and thereby result in infeasible optimal solutions. To achieve desired fidelity level, this study proposes a novel adaptive machine learning surrogate based multiobjective optimization method for coastal aquifer desalination. An adaptive modeling algorithm is newly introduced to iteratively retrain poorly-performed machine learning models and enhance predicting accuracy. The method is demonstrated to seek optimal extraction and injection strategies for scavenging residual saltwater trapped in an upstream aquifer behind a subsurface dam. Two conflicting objectives of minimizing the total extraction-injection rate and maximizing saltwater removal effectiveness are considered. Three machine learning models including artificial neural network (ANN), Gaussian process (GP) and response surface regression model (RSR) are developed to replace a high-fidelity seawater intrusion model for predicting chloride concentration and salinity mass. Non-dominated Sorting Genetic Algorithm II (NSGA-II) is employed to derive Pareto fronts. Pareto optimal solutions obtained from machine learning models are compared against those from the seawater intrusion model. Results indicate that the developed machine learning models do not only have strong predicting capability, but also maintain good quality of Pareto optimal solutions, while achieving substantial computational saving up to 95%. Especially, the adaptively retrained RSR model inhibits error accumulation of forecasting and accelerates correct convergence to the true Pareto front. The proposed method is found superior performance in accurate convergence, widely-spread diversity as well as high computing efficiency than conventional methods.

SEA: Many-objective evolutionary algorithm with selection evolution strategy

Balancing the convergence and diversity of the population is crucial for solving multi-objective problems. As the number of objectives increases, the inherent conflict between maintaining diversity and ensuring convergence becomes more significant. To address this challenge, we propose a novel evolutionary algorithm that selectively emphasizes either convergence or diversity, guided by the convergence and diversity indicators of the current population and a predefined priority criterion. During the iterative process, the algorithm strategically aims to either approach the true Pareto front to improve convergence or foster a more uniform distribution within the current Pareto layer to enhance diversity. Continuous monitoring of these indicators enables the algorithm to effectively manage and fine-tune the convergence and diversity of the population. We meticulously evaluated the performance of the proposed algorithm by comparing it with eight state-of-the-art evolution algorithms on 31 benchmark problems. The experimental results unequivocally demonstrated the outstanding performance of the proposed algorithm in solving multi-objective problems. Furthermore, the algorithm can be seamlessly incorporated into other evolution algorithms to strike a delicate balance between diversity and convergence, thereby empowering them to tackle challenging many-objective optimization tasks with enhanced efficiency and accuracy.

An adaptive reference vector guided many-objective optimization algorithm based on the pareto front density estimation

The performance of evolutionary algorithms using reference vectors to guide the evolution process mainly depends on the adaptive reference vector update strategy. In order to solve the challenging many-objective optimization problems with irregular Pareto fronts, this paper proposes an adaptive reference vector update strategy based on the Pareto front density estimation, which estimates the true Pareto front by finding sparse regions while ensuring the uniform distribution of reference vectors. In addition, an improved environmental selection strategy using the angle-based neighborhood density estimation has been proposed for estimating the neighborhood density to effectively guide the population evolution. On this basis, this paper proposes an adaptive reference vector guided many-objective optimization algorithm based on Pareto front density estimation (MaOEA-PDE). Experimental results on a large number of benchmark problems show MaOEA-PDE achieves better performance compared with some state-of-the-art algorithms in the literature.

Multi-objective trajectory planning for industrial robots using a hybrid optimization approach

Abstract In this paper, a hybrid approach organized in four phases is proposed to solve the multi-objective trajectory planning problem for industrial robots. In the first phase, a transcription of the original problem into a standard multi-objective parametric optimization problem is achieved by adopting an adequate parametrization scheme for the continuous robot configuration variables. Then, in the second phase, a global search is performed using a population-based search metaheuristic in order to build a first approximation of the Pareto front (PF). In the third phase, a local search is applied in the neighborhood of each solution of the PF approximation using a deterministic algorithm in order to generate new solutions. Finally, in the fourth phase, results of the global and local searches are gathered and postprocessed using a multi-objective direct search method to enhance the quality of compromise solutions and to converge toward the true optimal PF. By combining different optimization techniques, we intend not only to improve the overall search mechanism of the optimization strategy but also the resulting hybrid algorithm should keep the robustness of the population-based algorithm while enjoying the theoretical properties of convergence of the deterministic component. Also, the proposed approach is modular and flexible, and it can be implemented in different ways according to the applied techniques in the different phases. In this paper, we illustrate the efficiency of the hybrid framework by considering different techniques available in various numerical optimization libraries which are combined judiciously and tested on various case studies.

Multi-Objective Particle Swarm Optimization Algorithm based on Position Vector Offset

In response to the issue that particle swarm optimization algorithms tend to fall into local optima when dealing with multi-objective optimization tasks, a multi-objective optimization algorithm based on particle swarm is proposed. This algorithm is based on the relationship between the position vectors of particles, changing the selection and movement strategies of particles to find the true Pareto front. Firstly, two additional position vectors are added around the iterating particles to enhance their search capability; then, a non-dominated vector archive is established to record the non-dominated solutions of the iterating particles and the additional position vectors, increasing particle diversity. Finally, additional position vectors with high fitness are selected to produce a shift in the iterating particle's position, accelerating particle convergence. Comparing this algorithm with dMOPSO, SMPSO, NMPSO, and MOPSOCD algorithms, simulation experiments show that the proposed PVSPSO algorithm has stronger optimization ability.

Optimizing the Three-Dimensional Multi-Objective of Feeder Bus Routes Considering the Timetable

To optimize the evacuation process of rail transit passenger flows, the influence of the feeder bus network on bus demand is pivotal. This study first examines the transportation mode preferences of rail transit station passengers and addresses the feeder bus network’s optimization challenge within a three-dimensional framework, incorporating an elastic mechanism. Consequently, a strategic planning model is developed. Subsequently, a multi-objective optimization model is constructed to simultaneously increase passenger numbers and decrease both travel time costs and bus operational expenses. Due to the NP-hard nature of this optimization problem, we introduce an enhanced non-dominated sorting genetic algorithm, INSGA-II. This algorithm integrates innovative encoding and decoding rules, adaptive parameter adjustment strategies, and a combination of crowding distance and distribution entropy mechanisms alongside an external elite archive strategy to enhance population convergence and local search capabilities. The efficacy of the proposed model and algorithm is corroborated through simulations employing standard test functions and instances. The results demonstrate that the INSGA-II algorithm closely approximates the true Pareto front, attaining Pareto optimal solutions that are uniformly distributed. Additionally, an increase in the fleet size correlates with greater passenger volumes and higher operational costs, yet it substantially lowers the average travel cost per customer. An optimal fleet size of 11 vehicles is identified. Moreover, expanding feeder bus routes enhances passenger counts by 18.03%, raises operational costs by 32.33%, and cuts passenger travel time expenses by 21.23%. These findings necessitate revisions to the bus timetable. Therefore, for a bus network with elastic demand, it is essential to holistically optimize the actual passenger flow demand, fleet size, bus schedules, and departure frequencies.

A Feature Selection Method Based on Feature-Label Correlation Information and Self-Adaptive MOPSO

Feature selection can be seen as a multi-objective task, where the goal is to select a subset of features that exhibit minimal correlation among themselves while maximizing their correlation with the target label. Multi-objective particle swarm optimization algorithm (MOPSO) has been extensively utilized for feature selection and has achieved good performance. However, most MOPSO-based feature selection methods are random and lack knowledge guidance in the initialization process, ignoring certain valuable prior information in the feature data, which may lead to the generated initial population being far from the true Pareto front (PF) and influence the population’s rate of convergence. Additionally, MOPSO has a propensity to become stuck in local optima during the later iterations. In this paper, a novel feature selection method (fMOPSO-FS) is proposed. Firstly, with the aim of improving the initial solution quality and fostering the interpretability of the selected features, a novel initialization strategy that incorporates prior information during the initialization process of the particle swarm is proposed. Furthermore, an adaptive hybrid mutation strategy is proposed to avoid the particle swarm from getting stuck in local optima and to further leverage prior information. The experimental results demonstrate the superior performance of the proposed algorithm compared to the comparison algorithms. It yields a superior feature subset on nine UCI benchmark datasets and six gene expression profile datasets.

An improved differential evolution algorithm for multi-modal multi-objective optimization.

Multi-modal multi-objective problems (MMOPs) have gained much attention during the last decade. These problems have two or more global or local Pareto optimal sets (PSs), some of which map to the same Pareto front (PF). This article presents a new affinity propagation clustering (APC) method based on the Multi-modal multi-objective differential evolution (MMODE) algorithm, called MMODE_AP, for the suit of CEC'2020 benchmark functions. First, two adaptive mutation strategies are adopted to balance exploration and exploitation and improve the diversity in the evolution process. Then, the affinity propagation clustering method is adopted to define the crowding degree in decision space (DS) and objective space (OS). Meanwhile, the non-dominated sorting scheme incorporates a particular crowding distance to truncate the population during the environmental selection process, which can obtain well-distributed solutions in both DS and OS. Moreover, the local PF membership of the solution is defined, and a predefined parameter is introduced to maintain of the local PSs and solutions around the global PS. Finally, the proposed algorithm is implemented on the suit of CEC'2020 benchmark functions for comparison with some MMODE algorithms. According to the experimental study results, the proposed MMODE_AP algorithm has about 20 better performance results on benchmark functions compared to its competitors in terms of reciprocal of Pareto sets proximity (rPSP), inverted generational distances (IGD) in the decision (IGDX) and objective (IGDF). The proposed algorithm can efficiently achieve the two goals, i.e., the convergence to the true local and global Pareto fronts along with better distributed Pareto solutions on the Pareto fronts.

Improved multi objective particle swarm optimization based reactive power optimization for ensuring voltage security of power systems

In this study an improved multi objective particle swarm optimization (IMOPSO) algorithm is proposed for power system reactive power optimization with the objective of ensuring voltage security. The multi objective particle swarm optimization (MOPSO) is improved by introducing an adapted binary crossover (ABX) to the new positions obtained by the basic particle swarm optimization (PSO) algorithm. Additionally, diversity maintenance strategy is added to the algorithm by employing crowding distance (CD) calculation. The developed algorithm is tested and compared with standard MOPSO and non dominated sorting genetic algorithm (NASGA II). The comparison is made based on the degree of closeness to the true pareto front, as measured by the inverted generational distance (IGD), and based on diversity, as measured by the CDs . The test is made using ZDT1, ZDT2, and ZDT3 test functions. The IMOPSO showed improved performance over MOPSO and NASGA II algorithms in terms of convergence to the true pareto front (PF) and in terms of the speed of convergence as well as in maintaining diversity. The algorithm is then implemented to reactive power optimization of IEEE 14 bus test system. For the implementation purpose, the voltage stability and voltage deviation components of voltage security are formulated as a multi objective functions. The implementation has resulted diverse options of optimal settings of reactive power controlling parameters. The optimal settings proved to produce an improved voltage security as measured in terms of voltage deviation and voltage stability.

A novel competitive constrained dual-archive dual-stage evolutionary algorithm for constrained multiobjective optimization

In constrained multi-objective optimization problems (CMOPs), the key problem is how to balance convergence and diversity simultaneously and find as many feasible solutions as possible in the reduced search space. To address this issue, a novel competitive constrained dual-archive dual-stage evolutionary algorithm, named c-DADSEA, is proposed with a convergence-driven archive (CA) and a diversity-driven archive (DA). To improve the algorithm’s convergence, instead of simply excluding infeasible solutions, an adaptive penalty function is novelly proposed to maintain competitively infeasible solutions in CA. To overcome the difficulty in finding the true Pareto front for the dual-population framework in some problems, a dual-stage framework is introduced to make DA search in different directions at different stages (i.e., Forward and Backward), enhancing the diversity of the algorithm. DA evolves without considering the constraints to explore all regions as much as possible in the Forward stage. While in the Backward stage, DA degenerates with increasing constraints to search the regions around the true Pareto front by using ϵ-constrained method. Moreover, to balance convergence and diversity, different benchmark rankings are designed for the individuals in different archives. The experiments on three benchmark CMOPs test suites demonstrate that c-DADSEA is better than seven other state-of-the-art constrained evolutionary algorithms, especially for the CMOPs with a constrained Pareto front away from the unconstrained Pareto front.

A matheuristic for tri-objective binary integer linear programming

Many real-world optimisation problems involve multiple objectives. When considered concurrently, they give rise to a set of optimal trade-off solutions, also known as efficient solutions. These solutions have the property that neither objective can be improved without deteriorating another objective. Motivated by the success of matheuristics in the single-objective domain, we propose a linear programming-based matheuristic for tri-objective binary integer linear programming. To achieve a high-quality approximation of the optimal set of trade-off solutions, a lower bound set is first obtained using the vector linear programming solver Bensolve. Then, feasibility pump-based ideas in combination with path relinking are applied in novel ways so as to obtain a high-quality upper bound set. Our matheuristic is compared to a recently suggested algorithm that is, to the best of our knowledge, the only existing matheuristic method for tri-objective binary integer linear programming. In an extensive computational study, we show that our method generates a better approximation of the true Pareto front than the state-of-the-art matheuristic on a large set of tri-objective benchmark instances. Since the developed approach starts from a potentially fractional lower bound set, it may also be used as a primal heuristic in the context of linear relaxation-based multi-objective branch-and-bound algorithms.

Cooperative-competitive two-stage game mechanism assisted many-objective evolutionary algorithm

It is critical to maintain significant convergence and diversity in many-objective optimization problems (MaOPs) for the performance of many-objective evolutionary algorithms (MaOEAs). However, some issues pose serious challenges to MaOEAs, such as the intensification of the conflict between convergence and diversity, and the lack of Pareto selection pressure. To address the above issues, we propose a cooperative-competitive two-stage game mechanism assisted many-objective evolutionary algorithms (MaOEA-GM). The algorithm is divided into two stages, such as competition and cooperative. In competitive game stage, a strategy pool is constructed, including angle penalty distance strategy and favorable convergence strategy. In addition, a new game utility function is designed to balance convergence and diversity. This method promotes the selection of genetically superior parents for inheritance, so that the population can quickly approach the true Pareto front. In cooperative game stage, individuals choose their preferred environmental selection mechanism by voting. The final scheme is determined using the criterion of minority obedience to the majority, thereby increasing the algorithm Pareto selection pressure. Experimental results demonstrate that compared with five advanced MaOEAs, The MaOEA-GM algorithm has not only preponderance in convergence and diversity indicators, but also higher competitiveness in solving MaOPs.

A Many-Objective Evolutionary Algorithm with Local Shifted Density Estimation Based on Dynamic Decomposition

Pareto dominance-based many-objective evolutionary algorithms (PDMaOEAs) are challenging in dealing with many-objective problems (MaOPs) encountering many incomparable nondominated solutions. Recently, convergence-related metrics have been incorporated into PDMaOEAs to enhance the selection pressure approaching the true Pareto front and furtherly improve the balance of diversity and convergence, however these approaches still have limitations. To address the drawbacks of the previous approaches, a many objective evolutionary algorithm with local shifted density estimation based on dynamic decomposition (MaOEA/LSD-DD) is proposed in this paper. The proposed MaOEA/LSD-DD maintains the convergence and diversity of populations through the dynamic synergy of dynamic decomposition and shifted density estimation. First, identifying potential regions through dynamic decomposition can reduce redundant evaluation to save computational resources and maintain good distribution. Then, shifted density estimation is executed in potential regions to selected the individuals with good diversity and convergence into next generation population. Finally, this process is repeated until the population size is satisfied. The performance of the MaOEA/LSD-DD is investigated extensively on 28 test problem from three popular benchmark problem suites and a practical problem of water resource planning problem by comparing them with seven excellent MaOEAs. The experimental results show the effectiveness of the proposed MaOEA/LSD-DD in keeping a good tradeoff between diversity and convergence over other MaOEAs when solving most of these MaOPs.

An enhanced bacterial colony optimization with dynamic multi‐leader co‐evolution for multiobjective optimization problems

AbstractThe information transfer mechanism within the population is an essential factor for population‐based multiobjective optimization algorithms. An efficient leader selection strategy can effectively help the population to approach the true Pareto front. However, traditional population‐based multiobjective optimization algorithms are restricted to a single global leader and cannot transfer information efficiently. To overcome those limitations, in this paper, a multiobjective bacterial colony optimization with dynamic multi‐leader co‐evolution (MBCO/DML) is proposed, and a novel information transfer mechanism is developed within the group for adaptive evolution. Specifically, to enhance convergence and diversity, a multi‐leaders learning mechanism is designed based on a dynamically evolving elite archive via direction‐based hierarchical clustering. Finally, adaptive bacterial elimination is proposed to enable bacteria to escape from the local Pareto front according to convergence status. The results of numerical experiments show the superiority of the proposed algorithm in comparison with related population‐based multiobjective optimization algorithms on 24 frequently used benchmarks. This paper demonstrates the effectiveness of our dynamic leader selection in information transfer for improving both convergence and diversity to solve multiobjective optimization problems, which plays a significant role in information transfer of population evolution. Furthermore, we confirm the validity of the co‐evolution framework to the bacterial‐based optimization algorithm, greatly enhancing the searching capability for bacterial colony.

MOIMPA: multi-objective improved marine predators algorithm for solving multi-objective optimization problems

This paper introduces a multi-objective variant of the marine predators algorithm (MPA) called the multi-objective improved marine predators algorithm (MOIMPA), which incorporates concepts from Quantum theory. By leveraging Quantum theory, the MOIMPA aims to enhance the MPA’s ability to balance between exploration and exploitation and find optimal solutions. The algorithm utilizes a concept inspired by the Schrödinger wave function to determine the position of particles in the search space. This modification improves both exploration and exploitation, resulting in enhanced performance. Additionally, the proposed MOIMPA incorporates the Pareto dominance mechanism. It stores non-dominated Pareto optimal solutions in a repository and employs a roulette wheel strategy to select solutions from the repository, considering their coverage. To evaluate the effectiveness and efficiency of MOIMPA, tests are conducted on various benchmark functions, including ZDT and DTLZ, as well as using the evolutionary computation 2009 (CEC’09) test suite. The algorithm is also evaluated on engineering design problems. A comparison is made between the proposed multi-objective approach and other well-known evolutionary optimization methods, such as MOMPA, multi-objective ant lion optimizer, and multi-objective multi-verse optimization. The statistical results demonstrate the robustness of the MOIMPA approach, as measured by metrics like inverted generational distance, generalized distance, spacing, and delta. Furthermore, qualitative experimental results confirm that MOIMPA provides highly accurate approximations of the true Pareto fronts.

A two-stage maintenance and multi-strategy selection for multi-objective particle swarm optimization

In multi-objective particle swarm optimization, it is very important to select the personal best and the global best. These leaders are expected to effectively guide the population toward the true Pareto front. In this paper, we propose a two-stage maintenance and multi-strategy selection for multi-objective particle swarm optimization (TMMOPSO), which adaptively selects the global best and updates the personal best by means of hyper-cone domain and aggregation, respectively. This strategy enhances the global exploration and local exploitation abilities of the population. In addition, the excellent particles are perturbed and a two-stage maintenance strategy is used for the external archive. This strategy not only improves the quality of the solutions in the population but also accelerates the convergence speed of the population. In this paper, the proposed algorithm is compared with several multi-objective optimization algorithms on 29 benchmark problems. The experimental results show that TMMOPSO is effective and outperforms the comparison algorithms on most of the 29 benchmark problems.

An effective multi-objective bald eagle search algorithm for solving engineering design problems

In this paper, a multi-objective bald eagle search algorithm (MOBES) is proposed. The MOBES introduces an archive mechanism to store the non-dominated solutions obtained by the algorithm. When the archive overflows, remove the most crowded solutions by using the roulette method. The MOBES also adds elite selection strategy to guide other individuals to optimize by selecting elite individuals in the population. The efficiency of MOBES is validated on CEC 2020 benchmark functions, and the results demonstrate that the proposed algorithm is more efficient than its competitors in terms of convergence, diversity and distribution of solutions. The MOBES is also applied to two-objective, tri-objective and four-objective engineering design problems in real world. The results show its superiority in handling challenging multi-objective optimization problems with unknown true Pareto optimal solutions and fronts, and it is more competitive than other algorithms.

Solution Set Augmentation for Knee Identification in Multiobjective Decision Analysis.

In multiobjective decision making, most knee identification algorithms implicitly assume that the given solutions are well distributed and can provide sufficient information for identifying knee solutions. However, this assumption may fail to hold when the number of objectives is large or when the shape of the Pareto front is complex. To address the above issues, we propose a knee-oriented solution augmentation (KSA) framework that converts the Pareto front into a multimodal auxiliary function whose basins correspond to the knee regions of the Pareto front. The auxiliary function is then approximated using a surrogate and its basins are identified by a peak detection method. Additional solutions are then generated in the detected basins in the objective space and mapped to the decision space with the help of an inverse model. These solutions are evaluated by the original objective functions and added to the given solution set. To assess the quality of the augmented solution set, a measurement is proposed for the verification of knee solutions when the true Pareto front is unknown. The effectiveness of KSA is verified on widely used benchmark problems and successfully applied to a hybrid electric vehicle controller design problem.