Many-objective Optimization Problems Research Articles

Bayesian Co-evolutionary Optimization based entropy search for high-dimensional many-objective optimization

Bayesian evolutionary optimization algorithms have been widely employed to solve expensive many-objective optimization problems. However, the existing approaches are generally designed for low-dimensional problems. In high-dimensional problems, the accuracy of the prediction decreases. And the acquisition function becomes ineffective. The combination of these challenges renders existing approaches unsuitable for selecting potential individual solutions for high-dimensional many-objective optimization problems. To address these limitations, we propose a novel Entropy Search-based Bayesian Co-Evolutionary Optimization approach (ESB-CEO). With the co-evolutionary algorithm as the basic optimizer, it executes an adaptive acquisition function combining the Lp-norm and information entropy to efficiently solve computationally expensive many-objective optimization problems. Individual solutions that have a significant effect on different search stages can be effectively identified, which improves the convergence and diversity of the algorithm. Extensive experimental results based on a set of expensive multi/many-objective test problems demonstrate that the proposed approach significantly outperforms five state-of-the-art surrogate-assisted evolutionary algorithms.

Many-objective African vulture optimization algorithm: A novel approach for many-objective problems.

Several optimization problems can be abstracted into many-objective optimization problems (MaOPs). The key to solving MaOPs is designing an effective algorithm to balance the exploration and exploitation issues. This paper proposes a novel many-objective African vulture optimization algorithm (MaAVOA) that simulating the African vultures' foraging and navigation behaviours to solve the MaOPs. MaAVOA is an updated version of the African Vulture Optimization Algorithm (AVOA), which was recently proposed to solve the MaOPs. A new social leader vulture for the selection process is introduced and integrated into the proposed model. In addition, an environmental selection mechanism based on the alternative pool is adapted to improve the selection process to maintain diversity for approximating different parts of the whole Pareto Front (PF). The best-nondominated solutions are saved in an external Archive based on the Fitness Assignment Method (FAM) during the population evolution. FAM is based on a convergence measure that promotes convergence and a density measure that promotes variety. Also, a Reproduction of Archive Solutions (RAS) procedure is developed to improve the quality of archiving solutions. RAS has been designed to help reach out to the missing areas of the PF that the vultures easily miss. Two experiments are conducted to verify and validate the suggested MaAVOA's performance efficacy. First, MaAVOA was applied to the DTLZ functions, and its performance was compared to that of several popular many-objective algorithms and according to the results, MaAVOA outperforms the competitor algorithms in terms of inverted generational distance and hypervolume performance measures and has a beneficial adaptation ability in terms of both convergence and diversity performance measures. Also, statistical tests are implemented to demonstrate the suggested algorithm's statistical relevance. Second, MaAVOA has been applied to solve two real-life constrained engineering MaOPs applications, namely, the series-parallel system and overspeed protection for gas turbine problems. The experiments show that the suggested algorithm can tackle many-objective real-world applications and provide promising choices for decision-makers.

An incremental learning evolutionary algorithm for many-objective optimization with irregular Pareto fronts

Multi objective optimization problems (MOPs) with irregular Pareto Fronts (PFs) appear frequently in a large of practical cases. Proximity and diversity are the key indicators for decomposition-based multi-objective evolutionary algorithms (MOEA/D) to solve MOPs, which are decided by the neighborhood structure and a set of predefined direction vectors. In certain MOPs with irregular PFs, the effectiveness of MOEA/D is restricted by the shape of the PF. This limitation arises because MOEA/D generates direction vectors in a uniform manner, which can result in invalid direction vectors due to the irregularity of the PF. When dealing with many-objective optimization problems (MaOP) that have irregular PFs, such as degenerate or disconnected PFs, this decline in effectiveness becomes particularly problematic. To address this limitation, a copula incremental learning (CIL) scheme has been developed to progressively extract implicit knowledge on the appropriate distribution of direction vectors for generating non-uniform direction vectors. Additionally, a niche hierarchical selection (NHS) methodology is employed to construct the neighborhood structure and prevent the generation of duplicate solutions. These enhancements are designed to improve the overall effectiveness of MOEA/D. Efficiency is further ensured through the use of convergence-guided direction (CGD) to approximate irregular PFs. Statistical analysis indicates that the proposed method outperforms other competitive algorithms across most test benchmarks, particularly in its ability to effectively address MaOPs with irregular PFs.

Pareto Dominance Archive and Coordinated Selection Strategy-Based Many-Objective Optimizer for Protein Structure Prediction.

Protein structure prediction (PSP) is predicting the three-dimensional of protein from its amino acid sequence only based on the information hidden in the protein sequence. One of the efficient tools to describe this information is protein energy functions. Despite the advancements in biology and computer science, PSP is still a challenging problem due to its large protein conformation space and inaccurate energy functions. In this study, PSP is treated as a many-objective optimization problem and four conflicting energy functions are used as different objectives to be optimized. A novel Pareto-dominance-archive and Coordinated-selection-strategy-based Many-objective-optimizer (PCM) is proposed to perform the conformation search. In it, convergence and diversity-based selection metrics are used to enable PCM to find near-native proteins with well-distributed energy values, while a Pareto-dominance-based archive is proposed to save more potential conformations that can guide the search to more promising conformation areas. The experimental results on thirty-four benchmark proteins demonstrate the significant superiority of PCM in comparison with other single, multiple, and many-objective evolutionary algorithms. Additionally, the inherent characteristics of iterative search of PCM can also give more insights into the dynamic progress of protein folding besides the final predicted static tertiary structure. All these confirm that PCM is a fast, easy-to-use, and fruitful solution generation method for PSP.

NSGA-II/SDR-OLS: A Novel Large-Scale Many-Objective Optimization Method Using Opposition-Based Learning and Local Search

Recently, many-objective optimization problems (MaOPs) have become a hot issue of interest in academia and industry, and many more many-objective evolutionary algorithms (MaOEAs) have been proposed. NSGA-II/SDR (NSGA-II with a strengthened dominance relation) is an improved NSGA-II, created by replacing the traditional Pareto dominance relation with a new dominance relation, termed SDR, which is better than the original algorithm in solving small-scale MaOPs with few decision variables, but performs poorly in large-scale MaOPs. To address these problems, we added the following improvements to the NSGA-II/SDR to obtain NSGA-II/SDR-OLS, which enables it to better achieve a balance between population convergence and diversity when solving large-scale MaOPs: (1) The opposition-based learning (OBL) strategy is introduced in the initial population initialization stage, and the final initial population is formed by the initial population and the opposition-based population, which optimizes the quality and convergence of the population; (2) the local search (LS) strategy is introduced to expand the diversity of populations by finding neighborhood solutions, in order to avoid solutions falling into local optima too early. NSGA-II/SDR-OLS is compared with the original algorithm on nine benchmark problems to verify the effectiveness of its improvement. Then, we compare our algorithm with six existing algorithms, which are promising region-based multi-objective evolutionary algorithms (PREA), a scalable small subpopulation-based covariance matrix adaptation evolution strategy (S3-CMA-ES), a decomposition-based multi-objective evolutionary algorithm guided by growing neural gas (DEA-GNG), a reference vector-guided evolutionary algorithm (RVEA), NSGA-II with conflict-based partitioning strategy (NSGA-II-conflict), and a genetic algorithm using reference-point-based non-dominated sorting (NSGA-III).The proposed algorithm has achieved the best results in the vast majority of test cases, indicating that our algorithm has strong competitiveness.

A Comparative Study of State-of-the-art Metaheuristics for Solving Many-objective Optimization Problems of Fixed Wing Unmanned Aerial Vehicle Conceptual Design

A Comparative Study of State-of-the-art Metaheuristics for Solving Many-objective Optimization Problems of Fixed Wing Unmanned Aerial Vehicle Conceptual Design

A multiobjective evolutionary algorithm using multi-ecological environment selection strategy

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

Objective Space-Based Population Generation to Accelerate Evolutionary Algorithms for Large-Scale Many-Objective Optimization

The generation and updating of solutions, e.g., crossover and mutation, of many existing evolutionary algorithms directly operate on decision variables. The operators are very time consuming for large-scale and many-objective optimization problems. Different from them, this work proposes an objective space-based population generation method to obtain new individuals in the objective space and then map them to decision variable space and synthesize new solutions. It introduces three new objective vector generation methods and uses a linear mapping method to tightly connect objective space and decision one to jointly determine new-generation solutions. A loop can be formed directly between two spaces, which can generate new solutions faster and use more feedback information in the objective space. In order to demonstrate the performance of the proposed algorithm, this work performs a series of empirical experiments involving both large-scale decision variables and many objectives. Compared with the state-of-the-art traditional and large-scale algorithms, the proposed method exceeds or at least reaches its peers’ best level in overall performance while achieving great saving in execution time.

Growing neural gas assisted evolutionary many-objective optimization for handling irregular Pareto fronts

Reference vector adjustment approaches pose a considerable potential to strengthen the capability of decomposition-based evolutionary many-objective algorithms in handling many-objective optimization problems (MaOPs) with irregular Pareto-optimal fronts (PFs). However, most existing approaches often rely on the current population’s distribution, which may not well match MaOPs’ PFs. This can easily mislead the reference vector adjustment and impair algorithms’ performance. To address the abovementioned issue, this paper designs a growing neural gas-assisted evolutionary many-objective optimization algorithm, called GNG-EMO. Specifically, the proposal contains an environmental selection without reference vectors to maintain a well-diversified and converged archive over the PFs. Also, the GNG-EMO leverages a growing neural gas (GNG) network to learn Pareto-optimal fronts’ topologies using the solutions in the archive. In addition, to alleviate the issue that the nodes of the GNG network cannot reach the boundaries of Pareto-optimal fronts, we design an expansion mechanism to further diversify the nodes of the GNG network for constructing a set of better reference vectors. To evaluate the performance of the GNG-EMO, we compare it with five state-of-the-art competitors in the context of 48 test cases with irregular PFs. The numerical results demonstrate the GNG-EMO’s superior competitiveness by significantly outperforming all the five competitors on 27 test cases with respect to the indicator Hypervolume.

Offline and Online Objective Reduction via Gaussian Mixture Model Clustering

The objective reduction has been regarded as a basic issue in many-objective optimization. Existing objective reduction methods identify one set of essential objectives using an approximate nondominated front. However, if the Pareto front (PF) of a many-objective optimization problem (MaOP) is irregular, one single set of essential objectives may not be efficient for objective reduction. This article proposes to produce several different sets of essential objectives in objective reduction. More specifically, we use the Gaussian mixture model clustering to classify the obtained nondominated front into different subsets and perform objective reduction on each subset. Both an offline objective reduction method and an online objective reduction method are developed. The experimental results indicate that our proposed methods work well for MaOPs with degenerate or nondegenerate PFs.

A two-stage adaptive reference direction guided evolutionary algorithm with modified dominance relation for many-objective optimization

Traditional dominance-based multi-objective evolutionary algorithms cease to be effective as the number of objectives increases due to the non-dominated sorting mechanism. Accordingly, a novel clustering indicator founded on the penalty-based adaptive rectangular area (APA) between solution and reference direction is proposed to assist in non-dominated levels sorting to deal with this issue. However, directly predefined reference directions with uniform distribution usually cause deteriorated performance in solving multi-objective problems with irregular Pareto fronts. Thus, an adaptive adjustment method guided by the local population is implemented in this paper. At this rate, a two-stage adaptive reference point guided evolutionary algorithm with APA-based dominance relation for many-objective optimization problems (named AREA-APA) is proposed and tested for solving these multi-objective optimization problems (including constrained and unconstrained problems). The proposed algorithm is proven to achieve comparable performance on scalable benchmark problems compared with state-of-the-art algorithms.

Performance Analysis of Multi-Objective Simulated Annealing Based on Decomposition

Simulated annealing is a metaheuristic that balances exploration and exploitation to solve global optimization problems. However, to deal with multi- and many-objective optimization problems, this balance needs to be improved due to diverse factors such as the number of objectives. To deal with this issue, this work proposes MOSA/D, a hybrid framework for multi-objective simulated annealing based on decomposition and evolutionary perturbation functions. According to the literature, the decomposition strategy allows diversity in a population while evolutionary perturbations add convergence toward the Pareto front; however, a question should be asked: What is the effect of such components when included as part of a multi-objective simulated annealing design? Hence, this work studies the performance of the MOSA/D framework considering in its implementation two widely used perturbation operators: classical genetic operators and differential evolution. The proposed algorithms are MOSA/D-CGO, based on classical genetic operators, and MOSA/D-DE, based on differential evolution operators. The main contribution of this work is the performance analysis of MOSA/D using both perturbation operators and identifying the one most suitable for the framework. The approaches were tested using DTLZ on two and three objectives and CEC2009 benchmarks on two, three, five, and ten objectives; the performance analysis considered diversity and convergence measured through the hypervolume (HV) and inverted generational distance (IGD) indicators. The results pointed out that there is a promising improvement in performance in favor of MOSA/D-DE.

A Coevolutionary Algorithm for Many-Objective Optimization Problems with Independent and Harmonious Objectives

Evolutionary algorithm is an effective strategy for solving many-objective optimization problems. At present, most evolutionary many-objective algorithms are designed for solving many-objective optimization problems where the objectives conflict with each other. In some cases, however, the objectives are not always in conflict. It consists of multiple independent objective subsets and the relationship between objectives is unknown in advance. The classical evolutionary many-objective algorithms may not be able to effectively solve such problems. Accordingly, we propose an objective set decomposition strategy based on the partial set covering model. It decomposes the objectives into a collection of objective subsets to preserve the nondominance relationship as much as possible. An optimization subproblem is defined on each objective subset. A coevolutionary algorithm is presented to optimize all subproblems simultaneously, in which a nondominance ranking is presented to interact information among these sub-populations. The proposed algorithm is compared with five popular many-objective evolutionary algorithms and four objective set decomposition based evolutionary algorithms on a series of test problems. Numerical experiments demonstrate that the proposed algorithm can achieve promising results for the many-objective optimization problems with independent and harmonious objectives.

Evolutionary design of marginally robust multivariable PID controller

Two Margin-based design methods for tuning of decentralized PI controllers for multivariable systems are presented. The first method is predefined marginally robust controller (PMR) which is based on solving a constrained optimization problem to satisfy the desired values of the multivariable phase and gain margins that are set based on the designer’s knowledge of the working condition of the plant. The second method is self-setting marginally robust (SMR) that is an automatic controller design procedure. This method, by solving a many-objective optimization problem (MaOP), sets controller coefficients and finds optimum gain and phase margins. It offers a trade-off between tracking, interaction, and robustness. Evolutionary algorithm is applied to solve the problems and disk margin to define the multivariable stability margins. Proposed methods by combining the simplicity of margin-based design and effectiveness of optimization problems offer a practical solution in tuning the industrial multivariable controllers. Simulation results show that these methods are simple, practical, and at the same time, comparable with other robust multivariable controllers.

An ACO-based Hyper-heuristic for Sequencing Many-objective Evolutionary Algorithms that Consider Different Ways to Incorporate the DM's Preferences

Many-objective optimization is an area of interest common to researchers, professionals, and practitioners because of its real-world implications. Preference incorporation into Multi-Objective Evolutionary Algorithms (MOEAs) is one of the current approaches to treat Many-Objective Optimization Problems (MaOPs). Some recent studies have focused on the advantages of embedding preference models based on interval outranking into MOEAs; several models have been proposed to achieve it. Since there are many factors influencing the choice of the best outranking model, there is no clear notion of which is the best model to incorporate the preferences of the decision maker into a particular problem. This paper proposes a hyper-heuristic algorithm—named HyperACO—that searches for the best combination of several interval outranking models embedded into MOEAs to solve MaOPs. HyperACO is able not only to select the most appropriate model but also to combine the already existing models to solve a specific MaOP correctly. The results obtained on the DTLZ and WFG test suites corroborate that HyperACO can hybridize MOEAs with a combined preference model that is suitable to the problem being solved. Performance comparisons with other state-of-the-art MOEAs and tests for statistical significance validate this conclusion.

A Two-State Dynamic Decomposition-Based Evolutionary Algorithm for Handling Many-Objective Optimization Problems

Decomposition-based many-objective evolutionary algorithms (D-MaOEAs) are brilliant at keeping population diversity for predefined reference vectors or points. However, studies indicate that the performance of an D-MaOEA strongly depends on the similarity between the shape of the reference vectors (points) and that of the PF (a set of Pareto-optimal solutions symbolizing balance among objectives of many-objective optimization problems) of the many-objective problem (MaOP). Generally, MaOPs with expected PFs are not realistic. Consequently, the inevitable weak similarity results in many inactive subspaces, creating huge difficulties for maintaining diversity. To address these issues, we propose a two-state method to judge the decomposition status according to the number of inactive reference vectors. Then, two novel reference vector adjustment strategies, set as parts of the environmental selection approach, are tailored for the two states to delete inactive reference vectors and add new active reference vectors, respectively, in order to ensure that the reference vectors are as close as possible to the PF of the optimization problem. Based on the above strategies and an efficient convergence performance indicator, an active reference vector-based two-state dynamic decomposition-base MaOEA, referred to as ART-DMaOEA, is developed in this paper. Extensive experiments were conducted on ART-DMaOEA and five state-of-the-art MaOEAs on MaF1-MaF9 and WFG1-WFG9, and the comparative results show that ART-DMaOEA has the most competitive overall performance.

A strength pareto evolutionary algorithm based on adaptive reference points for solving irregular fronts

Evolutionary algorithms have proven to be extremely effective at tackling multi-objective optimization problems (MOPs). However, when dealing with many-objective optimization problems (MaOPs), their performance frequently degrades, especially when the Pareto marking irregular shapes. The population pressure to choose the Pareto optimal front and address the generalizability of different Pareto front shapes becomes more challenging as the number of objectives increases. We present a strength Pareto evolutionary algorithm based on adaptive reference points (SPEA/ARP) to address this problem. First, the reference points are updated using current and historical population information. The angles between the current demographic information and the predefined uniform reference points are used to select the active reference points, and the adaptive reference points are selected from the historical population information projected onto the reference plane. Second, the fitness function values are applied to classify the environmental selection criteria into two categories: 1) The angle distance scaling function using adaptive reference points is utilized to increase selection pressure, and the diversity of non-dominated solutions is balanced using the angle-based secondary selection technique. 2) Otherwise, the fitness function values are employed to choose the next generation of non-dominated solutions. Third, an aggregate fitness r-value generated by the angle distance scaling function is employed to construct matching pools that produce valid offsprings. Finally, extensive experiments are carried out to demonstrate SPEA/ARP performance by comparing it with six state-of-the-art many-objective evolutionary algorithms on 5-, 10-, 15-objective of 31 benchmark MaOPs. The experiments show that SPEA/ARP outperforms the compared algorithms.

Many-objective optimal power flow problems based on distributed power flow calculations for hierarchical partition-managed power systems

It is more and more challenging to obtain the global optimal power flow (OPF) information of a whole system with the developing trend of power systems toward hierarchical partition-managed ones. Traditional centralized OPF problems are not adaptive to modern partition-managed power systems because of the emergence of distributed sources and multi-stakeholders and the protection of their private data. This paper extends the traditional OPF to a many-objective optimization problem which is helpful to the independent power flow calculation (PFC) of the various subregions in partition-managed networks. To address the coordination of various regions, the individual distributed PFC problem is transformed into a whole optimization problem through node tearing and line disconnection methods, and then the pattern search algorithm is employed to address its solution. Then, the distributed PFC is integrated into the improved NSGA-III (I-NSGA-III) approach to solve the entire many-objective OPF (Ma-OPF) for partition-managed power systems. The effectiveness and feasibility of the proposed distributed approach are demonstrated in multiple benchmarks of IEEE 30/39/118-bus systems and a practical partitioned power system in Guizhou Province of China. The results show that the average voltage difference between distributed PFC and centralized PFC is within 1.12% while the phase angle difference is within 0.44%. In the application of OPF problems, the proposed distributed approach shows the high ability to yield approximate solutions of the original I-NSGA-III with centralized PFC. It is also found from the simulation that the distributed approach shows competitive performance compared to other considered centralized PFC-based evolutionary algorithms and weighted interior point method while it does not require complete information about various regions and is beneficial to the privacy protection of multi-stakeholders.

A Reference Point-Based Evolutionary Algorithm Solves Multi and Many-Objective Optimization Problems: Method and Validation.

The integration of a decision maker's preferences in evolutionary multi-objective optimization (EMO) has been a common research scope over the last decade. In the published literature, several preference-based evolutionary approaches have been proposed. The reference point-based non-dominated sorting genetic (R-NSGA-II) algorithm represents one of the well-known preference-based evolutionary approaches. This method mainly aims to find a set of the Pareto-optimal solutions in the region of interest (ROI) rather than obtaining the entire Pareto-optimal set. This approach uses Euclidean distance as a metric to calculate the distance between each candidate solution and the reference point. However, this metric may not produce desired solutions because the final minimal Euclidean distance value is unknown. Thus, determining whether the true Pareto-optimal solution is achieved at the end of optimization run becomes difficult. In this study, R-NSGA-II method is modified using the recently proposed simplified Karush-Kuhn-Tucker proximity measure (S-KKTPM) metric instead of the Euclidean distance metric, where S-KKTPM-based distance measure can predict the convergence behavior of a point from the Pareto-optimal front without prior knowledge of the optimum solution. Experimental results show that the algorithm proposed herein is highly competitive compared with several state-of-the-art preference-based EMO methods. Extensive experiments were conducted with 2 to 10 objectives on various standard problems. Results show the effectiveness of our algorithm in obtaining the preferred solutions in the ROI and its ability to control the size of each preferred region separately at the same time.

MaOMFO: Many-objective moth flame optimizer using reference-point based non-dominated sorting mechanism for global optimization problems

Many-objective optimization (MaO) deals with a large number of conflicting objectives in optimization problems to acquire a reliable set of appropriate non-dominated solutions near the true Pareto front, and for the same, a unique mechanism is essential. Numerous papers have reported multi-objective evolutionary algorithms to explain the absence of convergence and diversity variety in many-objective optimization problems. One of the most encouraging methodologies utilizes many reference points to segregate the solutions and guide the search procedure. The above-said methodology is integrated into the basic version of the Moth Flame Optimization (MFO) algorithm for the first time in this paper. The proposed Many-Objective Moth Flame Optimization (MaOMFO) utilizes a set of reference points progressively decided by the hunt procedure of the moth flame. It permits the calculation to combine with the Pareto front yet synchronize the decent variety of the Pareto front. MaOMFO is employed to solve a wide range of unconstrained and constrained benchmark functions and compared with other competitive algorithms, such as non-dominated sorting genetic algorithm, multi-objective evolutionary algorithm based on dominance and decomposition, and novel multi-objective particle swarm optimization using different performance metrics. The results demonstrate the superiority of the algorithm as a new many-objective algorithm for complex many-objective optimization problems.