Many-objective Optimization Algorithm Research Articles

Symmetrical Generalized Pareto Dominance and Adjusted Reference Vector Cooperative Evolutionary Algorithm for Many-Objective Optimization

Symmetrical Generalized Pareto Dominance and Adjusted Reference Vector Cooperative Evolutionary Algorithm for Many-Objective Optimization

Many-objective optimization algorithm based on the similarity principle and multi-mechanism collaborative search

Many-objective optimization algorithm based on the similarity principle and multi-mechanism collaborative search

Adaptive decomposition-based evolutionary algorithm for many-objective optimization with two-stage dual-density judgment

In order to better balance the convergence and diversity of MOEA/D for many objective optimization problems (MaOPs) with various Pareto fronts (PFs), an adaptive decomposition-based evolutionary algorithm for MaOPs with two-stage dual-density judgment is proposed. To solve the problem that weighted Tchebycheff decomposition may produce weakly Pareto optimal solutions when the solution is not unique or the uniqueness is difficult to guarantee, an augmented weighted Tchebycheff decomposition is adopted. To balance the convergence and diversity of non-dominated solutions in the external archive, different sparsity-level evaluations using vector angles or Euclidean distances are used to measure the distribution of solutions at different stages. To improve the diversity of solution sets obtained by MOEA/D for various PFs, an adaptive weight vector adjustment method based on two-stage dual-density judgment is presented. For weight vector addition, the potential search area is found according to the two-stage density judgment, and then a two-stage sparsity level judgment on the solutions of this area is performed for a second density judgment. For weight vector deletion, the degree of crowding is used to delete the weight vectors with a high crowding degree. Compared with nine advanced multi-objective optimization algorithms on DTLZ and WFG problems, the results demonstrate that the performance of the proposed algorithm is significantly better than other algorithms.

A many-objective optimization approach to design office building façade considering energy-daylight balance concept within prevalent climate types of Iran

The concept of energy-daylight balance in architectural design aims to maximize daylighting while reducing energy consumption. Many parameters influence this concept, although they are not necessarily of equal importance in different climates. This study aims to identify the appropriate weight distribution sets (WDSs) for the most influential parameters of the energy-daylight balance concept, namely energy efficiency, daylighting, visual comfort, and thermal comfort, under the prevalent climate types of Iran. To this end, a many-objective optimization algorithm (NSGA-III) is applied to a parametric design procedure involving design variables related to the window system and a proposed shading system, leading to the identification of the most optimal façade designs for office buildings in each climate. Next, a weighted sum method (WSM) evaluates the Pareto-optimal solutions of each climate type based on nine different WDSs. Analyzing the absolute Pareto-optimal solutions, which are the solutions with the highest WSM scores within each WDS in each climate, reveals that, depending on the climate, prioritizing daylighting, thermal comfort, or energy efficiency objectives leads to the highest overall performances within these solutions. Furthermore, the Friedman test results show that in the design of office buildings in Bandar Abbas, Isfahan, Rasht, or Tehran, the most optimal trade-off between the objectives generally occurs by prioritizing energy efficiency, whereas in Dezful, it is generally achieved by prioritizing daylighting. However, pairwise comparisons of the WDSs reveal that in each climate, there are a number of pairs of sets that are not significantly different.

Multi-indicator collaborative evolutionary algorithm for many-objective optimization

Multi-indicator collaborative evolutionary algorithm for many-objective optimization

Advancing SWAT Model Calibration: A U-NSGA-III-Based Framework for Multi-Objective Optimization

In recent years, remote sensing data have revealed considerable potential in unraveling crucial information regarding water balance dynamics due to their unique spatiotemporal distribution characteristics, thereby advancing multi-objective optimization algorithms in hydrological model parameter calibration. However, existing optimization frameworks based on the Soil and Water Assessment Tool (SWAT) primarily focus on single-objective or multiple-objective (i.e., two or three objective functions), lacking an open, efficient, and flexible framework to integrate many-objective (i.e., four or more objective functions) optimization algorithms to satisfy the growing demands of complex hydrological systems. This study addresses this gap by designing and implementing a multi-objective optimization framework, Py-SWAT-U-NSGA-III, which integrates the Unified Non-dominated Sorting Genetic Algorithm III (U-NSGA-III). Built on the SWAT model, this framework supports a broad range of optimization problems, from single- to many-objective. Developed within a Python environment, the SWAT model modules are integrated with the Pymoo library to construct a U-NSGA-III algorithm-based optimization framework. This framework accommodates various calibration schemes, including multi-site, multi-variable, and multi-objective functions. Additionally, it incorporates sensitivity analysis and post-processing modules to shed insights into model behavior and evaluate optimization results. The framework supports multi-core parallel processing to enhance efficiency. The framework was tested in the Meijiang River Basin in southern China, using daily streamflow data and Penman–Monteith–Leuning Version 2 (PML-V2(China)) remote sensing evapotranspiration (ET) data for sensitivity analysis and parallel efficiency evaluation. Three case studies demonstrated its effectiveness in optimizing complex hydrological models, with multi-core processing achieving a speedup of up to 8.95 despite I/O bottlenecks. Py-SWAT-U-NSGA-III provides an open, efficient, and flexible tool for the hydrological community that strives to facilitate the application and advancement of multi-objective optimization in hydrological modeling.

A population hierarchical-based evolutionary algorithm for large-scale many-objective optimization

In large-scale many-objective optimization problems (LMaOPs), the performance of algorithms faces significant challenges as the number of objective functions and decision variables increases. The main challenges in addressing this type of problem are as follows: the large number of decision variables creates an enormous decision space that needs to be explored, leading to slow convergence; and the high-dimensional objective space presents difficulties in selecting dominant individuals within the population. To address this issue, this paper introduces an evolutionary algorithm based on population hierarchy to address LMaOPs. The algorithm employs different strategies for offspring generation at various population levels. Initially, the population is categorized into three levels by fitness value: poorly performing solutions with higher fitness (Ph), better solutions with lower fitness (Pl), and excellent individuals stored in the archive set (Pa). Subsequently, a hierarchical knowledge integration strategy (HKI) guides the evolution of individuals at different levels. Individuals in Pl generate offspring by integrating differential knowledge from Pa and Ph, while individuals in Ph generate offspring by learning prior knowledge from Pa. Finally, using a cluster-based environment selection strategy balances population diversity and convergence. Extensive experiments on LMaOPs with up to 10 objectives and 5000 decision variables validate the algorithm’s effectiveness, demonstrating superior performance.

A many-objective evolutionary algorithm based on three states for solving many-objective optimization problem

In recent years, researchers have taken the many-objective optimization algorithm, which can optimize 5, 8, 10, 15, 20 objective functions simultaneously, as a new research topic. However, the current research on many-objective optimization technology also encounters some challenges. For example: Pareto resistance phenomenon, difficult diversity maintenance. Based on the above problems, this paper proposes a many-objective evolutionary algorithm based on three states (MOEA/TS). Firstly, a feature extraction operator is proposed. It can extract the features of the high-quality solution set, and then assist the evolution of the current individual. Secondly, based on Pareto front layer, the concept of “individual importance degree” is proposed. The importance degree of an individual can reflect the importance of the individual in the same Pareto front layer, so as to further distinguish the advantages and disadvantages of different individuals in the same front layer. Then, a repulsion field method is proposed. The diversity of the population in the objective space is maintained by the repulsion field, so that the population can be evenly distributed on the real Pareto front. Finally, a new concurrent algorithm framework is designed. In the algorithm framework, the algorithm is divided into three states, and each state focuses on a specific task. The population can switch freely among these three states according to its own evolution. The MOEA/TS algorithm is compared with 7 advanced many-objective optimization algorithms. The experimental results show that the MOEA/TS algorithm is more competitive in many-objective optimization problems.

TRAA: a two-risk archive algorithm for expensive many-objective optimization

Many engineering problems are essentially expensive multi-/many-objective optimization problems, and surrogate-assisted evolutionary algorithms have gained widespread attention in dealing with them. As the objective dimension increases, the error of predicting solutions based on surrogate models accumulates. Existing algorithms do not have strong selection pressure in the candidate solution obtaining and adaptive sampling stages. These make the effectiveness and area of application of the algorithms unsatisfactory. Therefore, this paper proposes a two-risk archive algorithm, which contains a strategy for mining high-risk and low-risk archives and a four-state adaptive sampling criterion. In the candidate solution mining stage, two types of Kriging models are trained, then conservative optimization models and non-conservative optimization models are constructed for model searching, followed by archive selection to obtain more reliable two-risk archives. In the adaptive sampling stage, in order to improve the performance of the algorithms, the proposed criterion considers environmental assessment, demand assessment, and sampling, where the sampling approach involves the improvement of the comprehensive performance in reliable environments, convergence and diversity in controversial environments, and surrogate model uncertainty. Experimental results on numerous benchmark problems show that the proposed algorithm is far superior to seven state-of-the-art algorithms in terms of comprehensive performance.

A many‐objective evolutionary algorithm based on bi‐direction fusion niche dominance

SummaryAlthough some many‐objective optimization algorithms (MaOEAs) have been proposed recently, Pareto dominance‐based MaOEAs still cannot effectively balance convergence and diversity in solving many objective optimization problems (MaOPs) due to insufficient selection pressure. To address this problem, a bi‐directional fusion niche domination is proposed. This method merges the strengths of cone and parallel decomposition directions in comparing dominations for nondominance stratification within the candidate population, augmenting the selection pressure of population. Subsequently, the crowding distance is introduced as an additional selection criterion to further refine the selection of nondominated individuals within the critical layer. Lastly, a MaOEA based on bi‐directional fusion niche dominance (MaOEA/BnD) is proposed, utilizing bi‐directional fusion niche dominance and crowding distance as important components of environmental selection. The performance of MaOEA/BnD was compared with five representative MaOEAs in 20 benchmark problems. Experimental results demonstrate that MaOEA/BnD effectively balances convergence and diversity when handling MaOPs with complex Pareto fronts.

Two-stage many-objective evolutionary algorithm: enhanced dominance relations and control mechanisms for separated balance

Although the multiobjective evolutionary algorithms (MOEAs) have been proved to bring promising prospects for solving multiobjective optimization problems (MOPs), the performance of the algorithm deteriorates sharply in high-dimensional objective space due to the weak selection pressure and the unregulated balance, which is caused by the increase of objective space dimension. Some current MOEAs with two-stage strategy (TS) strive to address above issues by dividing the evolutionary process into two independent stages, in which convergence and diversity are handled separately within successive generations of different stages. However, TS-MOEAs have some weaknesses, such as sensitivity to stage division, and incomplete separation of convergence and diversity. In this paper, TS/KW-MaOEA is proposed for solving many-objective optimization problems (MaOPs), which keeps TS as the central and equips a perfect control mechanism for separated balance. More specifically, TS/KW-MaOEA can automatically adjust the balance trend and provide appropriate selection pressure for MaOPs according to the Kondratiev wave (KW) search model and the objective space dimension. To verify the effectiveness of the proposed algorithm, a series of experiments are carried out against seven state-of-the-art many-objective optimization algorithms on 15 benchmark problems with up to 30 objectives. Experimental results indicate that the proposed algorithm is highly competitive against peer competitors.

Many-objective ant lion optimizer (MaOALO): A new many-objective optimizer with its engineering applications

Many-objective optimization (MaO) is an important aspect of engineering scenarios. In many-objective optimization algorithms (MaOAs), a key challenge is to strike a balance between diversity and convergence. MaOAs employs various tactics to either enhance selection pressure for better convergence and/or implements additional measures for sustaining diversity. With increase in number of objectives, the process becomes more complex, mainly due to challenges in achieving convergence during population selection. This paper introduces a novel Many-Objective Ant Lion Optimizer (MaOALO), featuring the widely-popular ant lion optimizer algorithm. This method utilizes reference point, niche preserve and information feedback mechanism (IFM), to enhance the convergence and diversity of the population. Extensive experimental tests on five real-world (RWMaOP1- RWMaOP5) optimization problems and standard problem classes, including MaF1-MaF15 (for 5, 9 and 15 objectives), DTLZ1-DTLZ7 (for 8 objectives) has been carried out. It is shown that MaOALO is superior compared to ARMOEA, NSGA-III, MaOTLBO, RVEA, MaOABC-TA, DSAE, RL-RVEA and MaOEA-IH algorithms in terms of GD, IGD, SP, SD, HV and RT metrics. The MaOALO source code is available at: https://github.com/kanak02/MaOALO.

Many-objective optimization based mutual feed scheduling for energy system of integrated energy station

The battery storage system of the integrated energy station contains electric vehicles as the distributed energy storage node of the regional power system can help the regional power system to achieve peak shaving and valley filling. This paper proposes a many-objective optimization based mutual feed model of the energy system of the integrated energy station. The model includes multiple modules of electric vehicle (EV), wind turbine (WT), photovoltaic (PV) and battery energy storage system (BESS). BESS was modeled according to the two behaviors of EVs charging and swapping. The model ensures the role of peak shaving and valley filling for regional power, reduces the loss of battery life in the battery storage system of the integrated energy station, and reduces the operating cost of the battery storage system. In the optimized results, the regional power system load peak-to-valley ratio by 28.72 %, and the daily battery charge/discharge life loss is minimized to 0.00036 %. In addition, for known many-objective optimization problems, this paper also proposes a new meta-heuristic many-objective optimization algorithm the non-dominated sorting hunter prey optimization (NSHPO for short) and compared with other four many-objective optimization algorithms by three indexes (Distribution range, Hypervolume index, and Running time). The simulation results show that compared with other algorithms, the HV metrics of NSHPO are lower by about 1.56 %-7.26 %, Spread metrics are better by about 0.39 %-2.73 %, and the proposed NSHPO algorithm is better in terms of convergence and distributivity.

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.

Identifying Pareto Fronts Reliably Using a Multistage Reference-Vector-Based Framework

Evolutionary multi-objective and many-objective optimization (EMO and EMaO) algorithms are increasingly being used to identify the true shape and location of the Paretooptimal front using a few representative well-converged and welldistributed solutions. The reason for their popularity is due to their ability to provide a better understanding of objective relationships for optimal solutions, and also to facilitate the choice of a preferred solution using an interactive or postoptimal multi-criterion decision analysis. However, since EMO and EMaO algorithms are stochastic, a single application may not provide a true representative set with a desired number of Pareto solutions reliably in repetitive runs and importantly with a well-distributed set of solutions. In this paper, we propose a multistage framework involving reference-vector based evolutionary multi-and many-objective algorithms (MuSt-EMO and MuSt-EMaO) that attempts to recursively rectify shortcomings of previous stages by careful executions of subsequent stages so that a prescribed number of well-distributed and well-converged solutions are achieved at the end. The proposed multi-stage approach is implemented to a number of popular reference vector based EMO/EMaO algorithms and is applied on various multiand many-objective test and real-world problems.

Activation Function-Assisted Objective Space Mapping to Enhance Evolutionary Algorithms for Large-Scale Many-Objective Optimization

Activation Function-Assisted Objective Space Mapping to Enhance Evolutionary Algorithms for Large-Scale Many-Objective Optimization

A Reference Vector-Assisted Many-Objective Optimization Algorithm with Adaptive Niche Dominance Relation

A Reference Vector-Assisted Many-Objective Optimization Algorithm with Adaptive Niche Dominance Relation

Balancing competing objectives in bigel formulations using many-objective optimization algorithms and different decision-making methods

Many-objective optimization, which deals with balancing multiple competing objectives to find compromised solutions, is essential for solving complex problems. This study explores evolutionary algorithms for optimizing the microstructural, rheological, stability, and drug release properties of bigel systems formulated using structured almond oil, mixed organogelators, and carbopol. The oleogel was identified as the dispersed phase, with droplet sizes ranging from 1.43 µm to 7.37 µm, indicating improved characteristics compared to other bigels. Each formulation exhibited non-Newtonian shear-thinning and thixotropic behaviors, which were positively influenced by the proportions of the excipients. After undergoing multiple stress cycles, highly concentrated bigels exhibited phase separation. Unexpectedly, bigels with lower viscosity exhibited reduced rates of drug release. FT-IR and HPLC analyses confirmed the compatibility and stability of drug-excipient interactions, with impurities remaining below 4%. This study emphasizes the complex interactions within mixed lipid-based bigels, requiring many-objective optimization techniques to address conflicting objectives. The objectives of optimization involve simultaneously minimizing microstructural properties while maximizing structural recovery and drug release properties. This led to conflicting objectives, where achieving higher structural recovery did not align with the desired drug release rate. Additionally, more stable formulations did not meet the optimal microstructural objectives. To resolve these conflicts, an RSM-MaOEAs approach was applied, employing various decision-making methods. Among EAs, RSM-RVEA notably achieved exceptional convergence. Furthermore, three MaOEAs-integrated decision-making methods—WSM, WPM, NED—and the RSM-desirability, offered potential solutions. Overall, this research proposes a robust framework for compromising the bigels’ performance and stability, with broader applications in drug delivery and related fields.

A solution potential-based adaptation reference vector evolutionary algorithm for many-objective optimization

Decomposition-based multiobjective evolutionary algorithms are widely utilized for solving multiobjective optimization problems (MOPs). These algorithms have a higher degree of diversity due to a uniform distribution of the reference vectors in the objective space. However, these algorithms may struggle to select a suitable candidate set for solving many-objective optimization problems (MaOPs) with irregular Pareto fronts (PFs). To solve this problem, a solution potential-based adaptation reference vector evolutionary algorithm (SPARVEA) is proposed in this paper. In the algorithm, a concept called solution potential is presented to assess whether the direction of convergence of the solution to the ideal solution has potential. The solution potential is obtained with an evaluation function to calculate the potentials of the corresponding solutions. A solution potential-based adaptation strategy is then designed for adapting the reference vector to better guide the direction of convergence of the solution when solving MaOPs with irregular PFs. A modified angle penalty distance (mAPD) is adjusted to improve the convergence rate of the solution set in many-objective optimization problems. The proposed algorithm SPARVEA is compared with state-of-the-art many-objective evolutionary algorithms on four test sets with irregular PFs and four practical problems. Experimental results demonstrate the superior performance of SPARVEA on many-objective optimization problems. This contribution helps to advance the development of efficient algorithms for solving many-objective optimization problems.

An Expensive Many-Objective Optimization Algorithm Based on Efficient Expected Hypervolume Improvement

The expected hypervolume improvement (EHVI) is one of the most popular infill criteria for multiobjective optimization problems. Although it has a significant advantage in exploring potential Pareto-optimal solutions, it has rarely been applied in many-objective problems due to its high computational cost. To address this issue, this paper proposes an expensive many-objective optimization algorithm based on the framework of NSGA-III and assisted by the kriging surrogate models. In the proposed algorithm, the Monte Carlo sampling method for EHVI estimation is improved by importance sampling, in which only one sampling process is required during the entire optimization process using a uniform distribution in normalized objective space. Considering the predicted uncertainty from the kriging model, an uncertainty-assisted nondominated sorting approach is proposed to substitute for the conventional approach in NSGA-III. In the proposed method, the predicted uncertainty is incorporated into the objective space as one independent dimension for nondominated sorting, which can enable the exploration of potential points with desirable EHVI values. In addition, the proposed algorithm considers the diversity of the solutions by de-emphasizing the pursuit of the best EHVI. The experimental results on benchmark problems demonstrate that the proposed EHVI calculation method can save computational costs compared with Monte Carlo sampling and indicate the superiority of the proposed algorithm over the others.